Contraherent cosheaves of contramodules on Noetherian formal schemes

CONTRAHERENT COSHEA VES OF CONTRAMODULES ON NOETHERIAN F ORMAL SCHEMES LEONID POSITSELSKI Abstract. W e deﬁne the exact category of con traherent cosheav es of contramod- ules on a locally No etherian formal sc heme, as w ell as the exact categories of locally con traherent coshea ves of contramodules (with resp ect to a giv en op en co vering). The exp osition in the section of preliminaries in adic comm utativ e algebra is w ork ed out in the greater generality of arbitrary commutativ e rings with adic top ologies (of ﬁnitely generated ideals). Contents 1. In tro duction 2 Ac kno wledgement 2 2. Preliminaries in Adic Commutativ e Algebra 2 2.1. Adic top ological rings and their maps 2 2.2. T orsion mo dules 6 2.3. Adic completions 7 2.4. Con tramo dules 9 2.5. Injectiv e torsion mo dules, pro jective and ﬂat contramodules 11 2.6. Change of scalars 18 2.7. T ensor pro ducts of adic top ological rings 25 2.8. F ormal op en immersions and formal op en co v erings 29 2.9. V ery ﬂat and contraadjusted contramodules 35 2.10. Cotorsion con tramo dules 40 2.11. The co extension of scalars is quotseparated 42 2.12. Colo calization of con traadjustedness and colo cality of cotorsion 45 2.13. Colo calit y of exactness 50 3. Con traheren t Cosheav es of Contramodules 57 3.1. The basics 57 3.2. Coshea v es of mo dules o v er a ringed space 58 3.3. Coshea v es of con tramo dules o ver a formal sc heme 60 3.4. Quasi-coheren t torsion shea v es 61 3.5. Con traheren t cosheav es of contramodules 64 3.6. Lo cally con traheren t coshea ves of con tramo dules 65 3.7. Lo cally cotorsion locally contraheren t cosheav es 69 References 70 1 1. Intr oduction One of the main original goals of the theory of contraheren t coshea v es, developed in [17, 29] and discussed in [28], was to globalize contramodules ov er formal schemes and ind-sc hemes. In the present pap er, w e accomplish this goal in the mo dest gen- eralit y of (lo cally) No etherian formal sc hemes with adic top ologies on the rings of functions on aﬃne op en formal subschemes. These are the formal schemes in the sense of Hartshorne’s textb o ok [12, Section I I.9], as distinguished from the more gen- eral approaches of Grothendieck’s EGAI [11, Sections 0.7 and I.10] and The Stacks pr oje ct [13, Chapter T ag 0AHW]. Wh y c ontr aher ent c oshe aves of c ontr amo dules ? Wh y not quasi-c oher ent she aves of c ontr amo dules ? This question w as raised in [28, Section 2] and answered in [32, Sec- tion 1.10], with a deﬁnitive counterexample in [32, Remark 10.7 and Example 10.8]. Simply put, the localization functors in contramodule categories do not ha v e the requisite exactness prop erties (whic h they hav e in the usual mo dule categories). The exactness prop erties of the colo calization functors in the mo dule categories are b et- ter. Another explanation is that one wan ts to w ork with inﬁnite direct pro ducts of con tramo dules, and the lo calization functors do not commute with direct pro ducts. The main results of the pap er [32] establish the existence, an explicit construction, and the exactness properties of the colocalization functors in con tramodule categories in a generalit y far surpassing the needs of the presen t pap er. This pap er starts with a section “Preliminaries in adic comm utativ e algebra”, where w e discuss the aﬃne geometry of con tramo dules and quotseparated contramod- ules in the generality of commutativ e rings with the adic top ology of a ﬁnitely gen- erated ideal. Then follows a section where we discuss coshea v es of con tramo dules and deﬁne the exact category of lo cally contraheren t cosheav es of con tramo dules on a lo cally Noetherian formal scheme X with resp ect to its op en co v ering W . W e also consider the dual-analogous ab elian category of quasi-coherent torsion shea v es on X , whic h is w ell-kno wn in the literature [1, 2]. Ac kno wledgemen t. I am grateful to Jan ˇ St ’o v ´ ı ˇ cek and Michal Hrb ek for helpful dis- cussions. The author is supp orted b y the GA ˇ CR pro ject 26-22734S and the Institute of Mathematics, Czec h Academ y of Sciences (research plan R VO: 67985840). 2. Preliminaries in Adic Commut a tive Algebra 2.1. Adic top ological rings and their maps. Let R b e a commutativ e ring. By an adic top olo gy on R w e mean a ring top ology for which there exists a ﬁnitely generated ideal I ⊂ R suc h that the ideal I n ⊂ R is op en for every n ≥ 1 and the descending sequence I n , n ≥ 1, of p ow ers of the ideal I is a base of neighborho o ds of zero in R . If this is the case, then the top ology on R is said to b e I -adic , the ring R is said to b e an adic top olo gic al ring , and the ideal I is said to b e the ide al of deﬁnition of the adic topological ring R . 2 Let R b e an adic topological ring and I ⊂ R b e an ideal of deﬁnition of R . Then, for ev ery n ≥ 1, the ideal I n ⊂ R is an ideal of deﬁnition, to o. Another giv en ideal J ⊂ R is an ideal of deﬁnition if and only if there exist integers n , m ≥ 1 s uc h that I n ⊂ J and J m ⊂ I . F or any t w o ideals of deﬁnition I , J ⊂ R , the ideals I + J , I ∩ J , and I J ⊂ R are also ideals of deﬁnition in R . Examples 2.1.1. The discrete top ology on an y commutativ e ring R is adic. The zero ideal (0) ⊂ R is an ideal of deﬁnition of R in the discrete top ology . The indiscrete top ology on R is adic as w ell. The unit ideal (1) ⊂ R is an ideal of deﬁnition of R in the indiscrete topology . Lemma 2.1.2. L et I and J b e two ide als of deﬁnition of an adic top olo gic al ring R . Then the quotient ring R/I is No etherian if and only if the quotient ring R /J is No etherian. Pr o of. The assertion holds due to the assumption that the ideals I and J are ﬁnitely generated. If so, then the ring R/I is No etherian if and only if the ring R/I n is (for all, or equiv alently , an y giv en n ≥ 1). □ W e will say that an adic top ological ring R is adic al ly No etherian (cf. [31, Sec- tion 9]) if it satisﬁes the equiv alen t conditions of Lemma 2.1.2. If the adic top ological ring R is adically No etherian, then the nilradical of R/I is a nilp otent ideal. In other words, the ideal I con tains a suitable p o wer of its radical: there exists n ≥ 1 s uc h that ( √ I ) n ⊂ I . Consequently , there exists a unique maximal ide al of deﬁnition I mx ⊂ R , c haracterized by the prop erties that I mx is an ideal of deﬁnition of R and the quotien t ring R /I mx is reduced (i. e., con tains no nonzero nilp oten t elements). The maximal ideal of deﬁnition I mx can b e constructed as the radical of an y other ideal of deﬁnition, I mx = √ I ⊂ R . Lemma 2.1.3. L et R and S b e two c ommutative rings with adic top olo gies, and let f : R − → S b e a ring homomorphism. Then the fol lowing four c onditions ar e e quivalent: (1) the ring homomorphism f is c ontinuous; (2) ther e exist an ide al of deﬁnition I ⊂ R and an ide al of deﬁnition J ⊂ S such that f ( I ) ⊂ J ; (3) for every ide al of deﬁnition J ⊂ S ther e exists an ide al of deﬁnition I ⊂ R such that f ( I ) ⊂ J ; (4) for every ide al of deﬁnition I ⊂ R ther e exists an ide al of deﬁnition J ⊂ S such that f ( I ) ⊂ J . If the top olo gic al rings R and S ar e adic al ly No etherian, then c onditions (1–4) ar e also e quivalent to the fol lowing c ondition: (5) denoting by I mx ⊂ R and J mx ⊂ S the maximal ide als of deﬁnition of R and S , one has f ( I mx ) ⊂ J mx . Pr o of. (1) = ⇒ (3) The ideal of deﬁnition J ⊂ S is op en, so there exists an op en ideal I ′ ⊂ R such that f ( I ′ ) ⊂ J . No w an y op en ideal I ′ ⊂ R contains some ideal of deﬁnition, i. e., there exists an ideal of deﬁnition I suc h that I ⊂ I ′ . 3 (3) = ⇒ (2) Obvious. (2) = ⇒ (1) Let I ′ ⊂ R and J ′ ⊂ S b e ideals of deﬁnition for which f ( I ′ ) ⊂ J ′ , and let J ⊂ S b e an op en ideal. Then there exists n ≥ 1 suc h that ( J ′ ) n ⊂ J . Hence f (( I ′ ) n ) ⊂ ( J ′ ) n ⊂ J , so ( I ′ ) n ⊂ f − 1 ( J ). Thus f − 1 ( J ) is an op en ideal in R . (2) = ⇒ (4) Let I ′ ⊂ R and J ′ ⊂ S b e ideals of deﬁnition for which f ( I ′ ) ⊂ J ′ , and let I ⊂ R be another given ideal of deﬁnition. Then there exists n ≥ 1 such that I n ⊂ I ′ , hence f ( I n ) ⊂ J ′ . Put J = S f ( I ) + J ′ ⊂ S , where S f ( I ) is the extension of the ideal I ⊂ R in the ring S . Then we hav e J ′ ⊂ J and J n ⊂ J ′ , hence J is an ideal of deﬁnition of S . By construction, f ( I ) ⊂ J . (4) = ⇒ (2) Obvious. (2) = ⇒ (5) W e ha ve I mx = √ I and J mx = √ J . Thus f ( I ) ⊂ J implies f ( I mx ) ⊂ J mx . (5) = ⇒ (2) T ak e I = I mx and J = J mx . □ Lemma 2.1.4. L et R and S b e two c ommutative rings with adic top olo gies, and let f : R − → S b e a ring homomorphism. Then the fol lowing thr e e c onditions ar e e quivalent: (1) ther e exist an ide al of deﬁnition I ⊂ R and an ide al of deﬁnition J ⊂ S such that J ⊂ S f ( I ) ; (2) for every ide al of deﬁnition J ⊂ S ther e exist an ide al of deﬁnition I ⊂ R and and inte ger n ≥ 1 such that J n ⊂ S f ( I ) ; (3) for any two ide als of deﬁnition I ⊂ R and J ⊂ S ther e exists an inte ger n ≥ 1 such that J n ⊂ S f ( I ) ; (4) for every ide al of deﬁnition I ⊂ R ther e exists an ide al of deﬁnition J ⊂ S such that J ⊂ S f ( I ) . Pr o of. (1) = ⇒ (2) Let I ′ ⊂ R and J ′ ⊂ S b e ideals of deﬁnition for whic h J ′ ⊂ S f ( I ′ ), and let J ⊂ S b e another given ideal of deﬁnition. Then there exists n ≥ 1 such that J n ⊂ J ′ , hence J n ⊂ S f ( I ′ ). It remains to put I = I ′ . (2) = ⇒ (3) Let I ′ ⊂ R b e an ideal of deﬁnition and n ′ ≥ 1 b e an in teger such that J n ′ ⊂ S f ( I ′ ). Pick an in teger m ≥ 1 suc h that ( I ′ ) m ⊂ I . Then J n ′ m ⊂ S f (( I ′ ) m ) ⊂ S f ( I ). It remains to put n = n ′ m . (3) = ⇒ (4) = ⇒ (1) Ob vious. □ Remark 2.1.5. Notice that the equiv alen t conditions of Lemma 2.1.4 do not imply that for ev ery ideal of deﬁnition J ⊂ S there exists an ideal of deﬁnition I ⊂ R such that J ⊂ S f ( I ). F or a counterexample, it suﬃces to take R = k to b e a ﬁeld and S = k [ ϵ ] / ( ϵ 2 ) to b e the k -algebra of dual num bers. Endo w b oth R and S with the discrete top ologies, and let f : R − → S b e the iden tit y inclusion. Then I = 0 is the only ideal of deﬁnition in R , and J = ( ϵ ) is an ideal of deﬁnition in S , but J ⊂ S f ( I ). On the other hand, J ′ = (0) is also an ideal of deﬁnition in S , and J ′ = S f ( I ). W e will say that a ring homomorphism f : R − → S acting b et w een tw o rings with adic top ologies is a tight ring map if the equiv alen t conditions of Lemma 2.1.4 hold. Notice that a tight ring map ne e d not b e contin uous in our deﬁnition. Using the 4 criterion of Lemma 2.1.4(4), one can easily chec k that the comp osition of an y tw o tigh t ring maps of adic top ological rings is tight. Giv en an I -adic top ological ring R and an R -mo dule M , the I -adic top olo gy on M is deﬁned b y the prop ert y that the submo dules I n M ⊂ M , n ≥ 1, form a base of op en neighborho o ds of zero. Clearly , if I ′ and I ′′ are tw o ideals of deﬁnition of R , then the I ′ -adic and the I ′′ -adic top ologies on M coincide. Lemma 2.1.6. L et R and S b e two c ommutative rings with adic top olo gies, and let f : R − → S b e a ring homomorphism. Then the fol lowing ﬁve c onditions ar e e quivalent: (1) the ring homomorphism f is c ontinuous and tight; (2) ther e exists an ide al of deﬁnition I ⊂ R such that the top olo gy on S is the I -adic top olo gy; (3) for every ide al of deﬁnition I ⊂ R , the top olo gy on S is the I -adic top olo gy; (4) ther e exists an ide al of deﬁnition I ⊂ R such that J = S f ( I ) is an ide al of deﬁnition in S ; (5) for every ide al of deﬁnition I ⊂ R , the ide al J = S f ( I ) ⊂ S is an ide al of deﬁnition in S . Pr o of. (2) ⇐ ⇒ (3) F ollo ws from the discussion in the paragraph preceding the lemma. (1) = ⇒ (3) Since f is con tinuous, Lemma 2.1.3(3) sa ys that for every ideal of deﬁnition J ⊂ S there exists an ideal of deﬁnition I ⊂ R such that S f ( I ) ⊂ J . Since f is tight, Lemma 2.1.4(4) says that for every ideal of deﬁnition I ⊂ R there exists an ideal of deﬁnition J ⊂ S such that J ⊂ S f ( I ). Thus the J -adic and the I -adic top ologies on S coincide. (3) = ⇒ (1) Under (3), b oth the top ologies on R and S are the I -adic top ologies. As ev ery R -mo dule map M − → N is contin uous with respect to the I -adic top ologies on M and N , it follo ws that f is contin uous. T o chec k that f is tigh t, let I ⊂ R and J ⊂ S b e an y t w o ideals of deﬁnition. In the I -adic top ology , S f ( I ) is an op en subgroup in S . Thus (2) implies that there exists n ≥ 1 for which J n ⊂ S f ( I ). So the condition of Lemma 2.1.4(3) is satisﬁed. (1) = ⇒ (5) By Lemma 2.1.4(4), there exists an ideal of deﬁnition J ′ ⊂ S such that J ′ ⊂ S f ( I ). By Lemma 2.1.3(3), there exists an ideal of deﬁnition I ′ ⊂ R suc h that S f ( I ′ ) ⊂ J ′ . Pic k n ≥ 1 suc h that I n ⊂ I ′ ; then w e ha v e ( S f ( I )) n = S f ( I n ) ⊂ J ′ ⊂ S f ( I ). Since J ′ is an ideal of deﬁn tion in S , it follo ws that S f ( I ) is an ideal of deﬁnition in S , to o. (5) = ⇒ (4) = ⇒ (2) Ob vious. □ It is clear from Lemma 2.1.6 that, for any homomorphism of commutativ e rings f : R − → S and any adic top ology on the ring R , there exists a unique adic top ology on the ring S making the ring map f contin uous and tight. It also follows that any tigh t con tin uous ring map that is an isomorphism of abstract comm utative rings is an isomorphism of top ological rings. Examples 2.1.7. Let k b e a ﬁeld and R = S = k [ x ] b e the ring of p olynomials in one v ariable x ov er k . Let f : R − → S b e the identit y map. Consider the ideals 5 I = ( x ) ⊂ R , J = ( x ) ⊂ S , I 0 = (0) ⊂ R , and J 0 = (0) ⊂ S . Then the map f is con tin uous and tight with resp ect to the I -adic top ology on R and the J -adic top ology on S . The map f is also contin uous and tight with resp ect to the I 0 -adic top ology on R and the J 0 -adic top ology on S . With resp ect to the I 0 -adic top ology on R and the J -adic top ology on S , the map f is contin uous but not tigh t. With resp ect to the I -adic top ology on R and the J 0 -adic top ology on S , the map f is tigh t but not con tin uous (in our terminology). T o give another example, let R = k [ x ] b e the ring of p olynomials in one v ariable and S = k [ x, y ] b e the ring of polynomials in t wo v ariables. Consider the ideals I = ( x ) ⊂ R and J = ( x, y ) ⊂ S . Let g : R − → S b e the identit y inclusion map and h : S − → R b e the k -algebra homomorphism taking x to x and y to 0. Then the map g is con tinuous but not tight, while the map h is con tinuous and tight. F or a p erhaps more complicated example of a tigh t contin uous map, see Remark 2.1.5 (in fact, an y morphism of discrete commutativ e rings is contin uous and tight). Lemma 2.1.8. L et R and S b e two c ommutative rings with adic top olo gies, and let f : R − → S b e a tight c ontinuous ring homomorphism. Then the fol lowing two c onditions ar e e quivalent: (1) ther e exists an ide al of deﬁnition I ⊂ R such that the R/I n -mo dule S/S f ( I n ) is ﬂat for every n ≥ 1 ; (2) for every op en ide al I ⊂ R , the R /I -mo dule S/S f ( I ) is ﬂat. Pr o of. The p oin t is that if I ⊂ I ′ ⊂ R are t w o ideals in R suc h that the R/I -mo dule S/S f ( I ) is ﬂat, then the R/I ′ -mo dule S/S f ( I ′ ) = R/I ′ ⊗ R/I S/S f ( I ) is ﬂat, to o. □ W e will sa y that a tight con tin uous homomorphism of adic top ological rings f : R − → S is ﬂat if it satisﬁes the equiv alen t conditions of Lemma 2.1.8. Clearly , if a tight contin uous map of adic top ological rings f is ﬂat as a map of abstract rings (i. e., the R -mo dule S is ﬂat), then f is also ﬂat as as a tigh t contin uous map of adic top ological rings. But the con verse is not true in general. Using the criterion of Lemma 2.1.8(2), one can easily chec k that the comp osition of an y t w o ﬂat tigh t con tinuous ring maps of adic topological rings is ﬂat. 2.2. T orsion mo dules. Let R b e a comm utative ring and s ∈ R b e an element. F ollo wing the notation in [17, Section 1.1], [20, Section 2.1], etc., we denote b y R [ s − 1 ] the ring R with the elemen t s formally in verted. In other w ords, R [ s − 1 ] = S − 1 R is the lo calization of R at the m ultiplicativ e subset S = { 1 , s, s 2 , s 3 , . . . } ⊂ R . Let M b e an R -mo dule. An element x ∈ M is said to b e s -torsion if there exists n ≥ 1 suc h that s n x = 0 in M . The R -mo dule M is said to b e s -torsion if all the elemen ts of M are s -torsion. Equiv alently , an R -mo dule M is s -torsion if and only if R [ s − 1 ] ⊗ R M = 0. Let I ⊂ R b e an ideal. W e sa y that an R -mo dule M is I -torsion if M is s -torsion for all s ∈ R . One can easily see that it suﬃces to c heck this condition for the elemen t s ranging o v er an y c hosen set of generators of the ideal I [30, Lemma 1.2]. Other deﬁnitions of the term “ I -torsion R -mo dule” exist in the literature (see, e. g., [14, Section 3]), but the diﬀerence b etw een them do es not manifest itself for ﬁnitely 6 gener ate d ideals I , which w e are really interested in. So, for a ﬁnitely generated ideal I ⊂ R , an R -mo dule M is I -torsion if and only if, for every element x ∈ M , there exists an in teger n ≥ 1 such that I n x = 0 in M [30, Lemma 1.3]. F or any R -mo dule M and any ideal J ⊂ R , we denote by J M ⊂ M the submo dule of all elements annihilated by J in M . So J M ≃ Hom R ( R/J, M ). According to the previous paragraph, an R -mo dule M is I -torsion if and only if M = S n ≥ 1 I n M . Lemma 2.2.1. L et R b e a c ommutative ring, I ⊂ R b e an ide al, M b e an I -torsion R -mo dule, and N b e an R -mo dule. Then the R -mo dule M ⊗ R N is I -torsion. Pr o of. The assertions is obvious; see [20, Lemma 6.1(a)] for some details. □ No w let R b e a commutativ e ring endo w ed with an adic top olo gy in the sense of Section 2.1. Let I , J ⊂ R b e tw o ideals of deﬁnition of the ring R . Then an R -mo dule M is I -torsion if and only if it is J -torsion. If this is the case, we say that M is a torsion R -mo dule (with resp ect to the given adic top ology of R ). An R -mo dule M is torsion if and only if, for every element x ∈ M , the annihilator of x in R is an open ideal. In other words, an R -mo dule M is torsion if and only if it is discr ete (as a mo dule o ver the topological ring R ) in the sense of [39, Section VI.4], [3, Section 1.4], [15, Section D.5.1], [17, Section E.2], [18, Sections 1.4 and 2.1], [22, Section 2.3], [23, Section 2.4], [35, Section 7.2], [36, Section 1], etc. F or an y ring R , w e denote b y R – Mo d the ab elian category of (left) R -mo dules. Giv en an ideal I in a commutativ e ring R , the notation R – Mo d I - to rs ⊂ R – Mo d stands for the full sub category of I -torsion R -mo dules. F or an adic top ological ring R , w e will write simply R – Mo d tors instead of R – Mo d I - to rs , presuming an y ideal of deﬁnition I of the top ological ring R . W e will also write R – T ors = R – Mo d tors . F or an y ideal I in a commutativ e ring R , the full sub category R – Mo d I - to rs is closed under submo dules, quotien ts, extensions, and inﬁnite direct sums in R – Mo d . In other w ords, R – Mo d I - to rs is a lo c alizing sub c ate gory , or in a diﬀerent terminology , a her e ditary torsion class [39, Sections VI.2–3] in R – Mo d . It follows that the category R – Mo d I - to rs is a Grothendieck ab elian category . The iden tit y inclusion functor R – Mo d I - to rs − → R – Mo d is exact and preserv es inﬁnite direct sums (hence also all colimits), but not inﬁnite pro ducts. The inﬁnite direct pro duct functors in R – Mo d I - to rs are usually not exact. 2.3. Adic completions. Let R b e a commutativ e ring and I ⊂ R b e a ﬁnitely generated ideal. Then the construction of the I -adic c ompletion Λ I ( M ) = lim ← − n ≥ 1 M /I n M of an arbitrary R -module M is reasonably well-behav ed [40]. The I -adic completion Λ I ( M ) is a naturally a mo dule o ver the I -adic completion Λ I ( R ) = lim ← − n ≥ 1 R/I n of the ring R . There is a natural c ompletion map λ I ,M : M − → Λ I ( M ). W e say that an R -module M is I -adic al ly sep ar ate d if the map λ I ,M is injectiv e, and that M is I -adic al ly c omplete if λ I ,M is surjective. F or an y R -mo dule M , the R -mo dule Λ I ( M ) is I -adically separated and complete [40, Corollary 3.6]. The latter assertion dep ends on the 7 assumption that the ideal I ⊂ R is ﬁnitely generated [40, Example 1.8]. The functor Λ I is left adjoin t to the inclusion of the full sub category of I -adically separated and complete R -mo dules into R – Mo d , and the map λ I ,M is the adjunction unit [20, Theorem 5.8]. Put R = Λ I ( R ) = lim ← − n ≥ 1 R/I n . The extension R I of the ideal I ⊂ R to the ring R coincides with the I -adic completion I = lim ← − n ≥ 2 I /I n ⊂ R of the ideal I . The commutativ e ring R is naturally endow ed with the pro jective limit top ology (of discrete rings R /I n ), which coincides with the I -adic top ology and with the I -adic top ology on R [20, pro of of Theorem 5.8]. So R is an adic top ological ring, to o; and I is an ideal of deﬁnition of R . The top ological ring R is determined by the adic top ological ring R and do es not dep end on the choice of a particular ideal of deﬁnition I ⊂ R (while the ideal I ⊂ R dep ends on the ideal I ⊂ R , of course). Giv en an adic topological ring R , the completion functor Λ I : R – Mo d − → R – Mo d and the natural map λ I ,M dep end on the adic top ology on the ring R only , and not on the c hoice of a particular ideal of deﬁnition I ⊂ R . Consequen tly , the prop erties of an R -module to b e I -adically separated or I -adically complete are also determined b y the I -adic top ology on R and do not dep end on the choice of an ideal of deﬁnition I for the given adic top ology . So we will write Λ = Λ I and λ M = λ I ,M , presuming an y ideal of deﬁnition I of the topological ring R . F ollo wing the terminology ab ov e, an adic top ological ring R is said to b e sep ar ate d if the map λ R : R − → R is injective and c omplete if the map λ R is surjectiv e. F or an y adic top ological ring R , the adic top ological ring R = Λ( R ) is separated and complete, and the completion homomorphism of top ological rings λ R : R − → R is con tin uous, tight, and ﬂat (in the sense of Section 2.1). Let R b e an adic top ological ring and R = Λ( R ) b e its completion. Then the rules I = R λ R ( I ) and I = λ − 1 R ( I ) establish a natural bijection b etw een the ideals of deﬁnition I ⊂ R and the ideals of deﬁnition I ⊂ R . More generally , the same rules provide a bijection b etw een op en ideals I ⊂ R and op en ideals I ⊂ R . The map λ R induces ring isomorphisms R /I ≃ R / I for op en ideals I ⊂ R and I ⊂ R corresp onding to eac h other under this bijection. An y contin uous homomorphism of adic top ological rings f : R − → S induces a con tin uous homomorphism of the completions Λ( f ) : Λ( R ) − → Λ( S ). Lemma 2.3.1. L et f : R − → S b e a tight c ontinuous homomorphism of adic top o- lo gic al rings. Then (a) the induc e d c ontinuous ring map of the c ompletions Λ( f ) : Λ( R ) − → Λ( S ) is also tight; (b) the tight c ontinuous ring map f is ﬂat if and only if the tight c ontinuous ring map Λ( f ) is ﬂat. Pr o of. Put R = Λ( R ), S = Λ( S ), and f = Λ( f ). P art (a): let I ⊂ R b e an ideal of deﬁnition in R . Then, b y Lemma 2.1.6(5), the ideal J = S f ( I ) ⊂ S is an ideal of deﬁnition in S . F ollo wing the discussion ab ov e, the ideal I = R λ R ( I ) ⊂ R is an ideal of deﬁnition in R and the ideal J = S λ S ( J ) 8 is an ideal of deﬁnition in S . Clearly , we hav e J = Sf ( I ). By Lemma 2.1.6(4), it follo ws that the ring map f is tight. P art (b): in the notation of the previous paragraph, w e ha v e R /I ≃ R / I and S/J ≃ S / J . Hence S /J is a ﬂat mo dule ov er R /I if and only if S / J is a ﬂat mo dule o v er R / I . □ Remark 2.3.2. The con v erse assertion to Lemma 2.3.1(a) is not true: There exist con tin uous maps of adic top ological rings f : R − → S such that the contin uous ring map Λ( f ) is tight but the map f is not. F or a coun terexample, it suﬃces to take R = S = k [ x ] and f = id k [ x ] as in Examples 2.1.7, with the discrete ((0)-adic) top ology on R and the indiscrete ((1)-adic) top ology on S . Then one has Λ( R ) = R and Λ( S ) = 0; so the ring map f = Λ( f ) is tight; but the ring map f is not. Let R b e an adic top ological ring and R = Λ( R ) b e its completion. The adic completion functor Λ : R – Mo d − → R – Mo d or Λ : R – Mo d − → R – Mo d is neither left, nor right exact [40, Example 3.20]. This fact is closely related to the fact that the category of separated and complete R -mo dules is not abelian [20, Example 2.7(1)]. T o o vercome these diﬃculties, one considers contramodule R -mo dules instead of the separated and complete R -mo dules; see the next section. 2.4. Con tramo dules. Let R be a commutativ e ring and s ∈ R b e an element. An R -mo dule P is said to b e an s -c ontr amo dule if Hom R ( R [ s − 1 ] , P ) = 0 = Ext 1 R ( R [ s − 1 ] , P ). The ring R [ s − 1 ] is considered as an R -mo dule here. The deﬁnition ab o v e do es not mention any Ext i with i ≥ 2 because the pro jectiv e dimension of the R -mo dule R [ s − 1 ] never exceeds 1 [17, Section 1.1], [16, pro of of Lemma B.7.1], [20, pro of of Lemma 2.1]. A detailed discussion of this deﬁnition, whic h go es back to [15, Remark A.1.1], [16, Theorem B.1.1], [17, Section D.1], and [19, Section 2], can b e found in [20, Sections 1–3]. Let I ⊂ R b e an ideal. An R -mo dule P is said to b e an I -c ontr amo dule (or an I -c ontr amo dule R -mo dule ) if P is an s -contramodule for ev ery s ∈ I . It suﬃces to c hec k this condition for the elemen t s ranging o v er an y c hosen set of generators of the ideal I [16, Theorem B.1.1], [19, Section 2]. A detailed discussion with t w o pro ofs can b e found in [20, Theorem 5.1]. Lemma 2.4.1. L et R b e a c ommutative ring and I ⊂ R b e an ide al. Then (a) for any I -torsion R -mo dule M and any R -mo dule P , the R -mo dule Hom R ( M , P ) is an I -c ontr amo dule; (b) for any R -mo dule M and any I -c ontr amo dule R -mo dule P , the R -mo dule Hom R ( M , P ) is an I -c ontr amo dule. Pr o of. This is [20, Lemma 6.1(b)]. □ Assume that the ideal I ⊂ R is ﬁnitely generated. Then an y I -contramodule R -mo dule is I -adically complete [20, Theorem 5.6], but it ne e d not b e I -adically sep- arated [37, Example 2.5], [40, Example 3.20], [14, Example 4.33], [15, Section A.1.1], [18, Section 1.5], [20, Example 2.7(1)]. An y I -adically separated and complete R -mo dule is an I -con tramo dule [20, Lemma 5.7]. 9 An I -con tramo dule R -mo dule is said to b e quotsep ar ate d [34, Section 5.5], [24, Section 1] if it is a quotient R -mo dule of an I -adically separated and complete R -mo dule. The references in the previous paragraph pro vide examples of quotsepa- rated I -contramodule R -mo dules that are not I -adically separated. F or an example of an I -contramodule R -mo dule that is not quotseparated, see [19, Example 2.6], [24, Examples 1.8]. Ov er a No etherian comm utativ e ring R , all I -contramodule R -mo dules are quotseparated [16, Theorem B.1.1], [24, Corollary 3.7 and Remark 3.8]. A p erhaps more detailed discussion can b e found in [31, Section 1]. No w let R b e a commutativ e ring endo w ed with an adic top olo gy in the sense of Section 2.1. Let I , J ⊂ R b e tw o ideals of deﬁnition of the ring R . Then an R -mo dule P is an I -contramodule if and only if it is a J -contramodule [20, Remark 5.5]. If this is the case, w e will sa y that P is a c ontr amo dule R -mo dule (with resp ect to the giv en adic top ology of R ). Moreo v er, a contramodule R -mo dule is quotseparated as an I -contramodule R -mo dule if and only if it is quotseparated as a J -contramodule R -mo dule; so one can sp eak of quotsep ar ate d c ontr amo dule R -mo dules . Giv en an ideal I in a commutativ e ring R , w e denote b y R – Mo d I - ctra ⊂ R – Mo d the full subcategory of I -con tramo dule R -mo dules. When the ideal I ⊂ R is ﬁnitely generated, the notation R – Mo d qs I - ctra ⊂ R – Mo d I - ctra stands for the full sub category of quotseparated I -contramodule R -mo dules. F or an adic top ological ring R , we will write simply R – Mo d qs ctra ⊂ R – Mo d ctra instead of R – Mo d qs I - ctra ⊂ R – Mo d I - ctra , presum- ing an y ideal of deﬁnition I of the top ological ring R . F or an y ideal I in a comm utativ e ring R , the full sub category R – Mo d I - ctra is closed under kernels, cokernels, extensions, and inﬁnite direct products in R – Mo d [7, Propo- sition 1.1], [19, Section 2], [20, Theorem 1.2(a)]. It follows that R – Mo d I - ctra is an ab elian category . In fact, R – Mo d I - ctra is a lo cally ℵ 1 -presen table ab elian category with enough pro jectiv e ob jects [33, Example 4.1(3)] (see also [21, Example 1.3(4)]). The identit y inclusion functor R – Mo d I - ctra − → R – Mo d is exact and preserves inﬁnite pro ducts (hence also all limits) and ℵ 1 -directed colimits, but not inﬁnite direct sums. The inﬁnite direct sum functors in R – Mo d I - ctra are usually not exact. In the case of a ﬁnitely generated ideal I ⊂ R , the full sub category R – Mo d qs I - ctra is closed under k ernels, cok ernels, and inﬁnite direct pro ducts in R – Mo d I - ctra and in R – Mo d , and also under subob jects and quotien t ob jects in R – Mo d I - ctra [24, Lemma 1.2]. (Ho w ever, the full sub category of quotseparated contramodules is not closed under extensions; in fact, an y I -contramodule R -mo dule is an extension of t w o quotseparated I -con tramo dule R -mo dules [24, Prop osition 1.6].) It follo ws that R – Mo d qs I - ctra is an abelian category . In fact, R – Mo d I - ctra is also a lo cally ℵ 1 -presen table ab elian category with enough pro jectiv e ob jects [24, end of Section 1]. The iden tit y inclusion functors R – Mo d qs I - ctra − → R – Mo d I - ctra and R – Mo d qs I - ctra − → R – Mo d are exact and preserve inﬁnite pro ducts (hence also all limits) and ℵ 1 -directed colimits. The inclusion functor R – Mo d qs I - ctra − → R – Mo d do es not preserv e inﬁnite direct sums. The inﬁnite direct sum functors in R – Mo d qs I - ctra are usually not exact (the global dimension 1 case is one ma jor exception [16, Remark 1.2.1]). 10 T o an y complete, separated top ological ring R where op en righ t ideals form a base of neighborho o ds of zero, the ab elian category R – Contra of left R -c ontr amo dules is naturally assigned [15, Remark A.3], [16, Section 1.2], [18, Section 2.1], [33, Sec- tions 1.2 and 5], [22, Section 2.7], [23, Sections 2.6–2.7], [35, Section 6], [36, Section 1]. The category R – Contra comes equipp ed with a natural exact, faithful forgetful func- tor R – Contra − → R – Mo d , preserving inﬁnite pro ducts (hence all limits). In the case of an adic top ological ring R , the comp osition of forgetful functors R – Contra − → R – Mo d − → R – Mo d is fully faithful, and its essential image coincides with the full sub category of quotseparated con tramo dule R -mo dules [24, Prop osi- tion 1.5]. So there is a natural equiv alence (in fact, isomorphism) of categories R – Contra ≃ R – Mo d qs ctra . Lemma 2.4.2. L et R b e an adic top olo gic al ring. Then (a) for any torsion R -mo dule M and any R -mo dule P , the R -mo dule Hom R ( M , P ) is a quotsep ar ate d c ontr amo dule R -mo dule (in fact, a sep ar ate d and c omplete R -mo dule); (b) for any R -mo dule M and any c omplete, sep ar ate d R -mo dule P , the R -mo dule Hom R ( M , P ) is c omplete and sep ar ate d; (c) for any R -mo dule M and any quotsep ar ate d c ontr amo dule R -mo dule P , the R -mo dule Hom R ( M , P ) is a quotsep ar ate d c ontr amo dule R -mo dule. Pr o of. Part (a) is the assertion in parentheses in [31, Lemma 1.1(b)]. T o prov e parts (b) and (c), represent M as a quotient R -module of a free R -mo dule F . Then Hom R ( M , P ) is an R -submodule of the R -mo dule Hom R ( F , P ), whic h an inﬁnite pro duct of copies of P . F urthermore, Hom R ( M , P ) is a contramodule R -mo dule b y Lemma 2.4.1(b). No w assertion (b) follows from the fact that the full sub category of separated R -mo dules is closed under sub ob jects and inﬁnite pro ducts in R – Mo d . Since the full sub category R – Mo d qs ctra is closed under sub ob jects, quotient ob jects, and inﬁnite pro ducts in R – Mo d ctra , part (c) follows as well. F or an argumen t based on the notion of a contramodule ov er a top ological ring, see [16, Section B.2]. □ 2.5. Injectiv e torsion mo dules, pro jectiv e and ﬂat con tramo dules. F or any ab elian category A , we denote by A inj ⊂ A the full sub category of injectiv e ob jects in A . More generally , the same notation applies to any exact category A (in the sense of Quillen). Dually , the notation B prj ⊂ B stands for the full subcategory of pro jective ob jects in an ab elian or exact category B . Let I b e an ideal in a comm utativ e ring R . Giv en an R -mo dule M , w e denote b y Γ I ( M ) ⊂ M the submo dule of all I -torsion elements in M . The functor M 7− → Γ I ( M ) : R – Mo d − → R – Mo d I - to rs is right adjoin t to the identit y inclusion functor R – Mo d I - to rs − → R – Mo d . As an y Grothendieck category , the ab elian category R – Mo d I - to rs has enough in- jectiv e ob jects. F or every injectiv e R -mo dule K , the torsion R -mo dule Γ I ( K ) is injectiv e as an ob ject of R – Mo d I - to rs . Indeed, the maximal torsion submo dule func- tor Γ I is right adjoint to an exact functor, so it takes injective ob jects to injectiv e ob jects. There are enough injectiv e ob jects of the form Γ I ( K ) in R – Mo d I - to rs , where 11 K ∈ R – Mo d inj . Hence an I -torsion R -mo dule K is injectiv e in R – Mo d I - to rs if and only if it is a direct summand of the I -torsion R -mo dule Γ I ( K ) for some injective R -mo dule K . Let R b e an adic top ological ring with an ideal of deﬁnition I ⊂ R . Clearly , the submo dule Γ( M ) = Γ I ( M ) ⊂ M only dep ends on the adic top ology on R , and not on the c hoice of a particular ideal of deﬁnition I . Using Baer’s criterion of injectivity , one can easily chec k that a torsion R -mo dule K is an injectiv e ob ject of R – Mo d tors if and only if the R/I n -mo dule I n K is injectiv e for every n ≥ 1. Equiv alen tly , a torsion R -mo dule K is injective (as a torsion R -mo dule) if and only if the R /J -mo dule J K is injectiv e for ev ery ideal of deﬁnition J ⊂ R . If this is the case, then the R /J -mo dule J K is also injective for every op en ideal J ⊂ R . When the ring R is No etherian (as an abstract ring), one can easily prov e using the Artin–Rees lemma that an y injective ob ject of R – Mo d tors is also injective in R – Mo d . Th us one has R – Mo d inj tors = R – Mo d tors ∩ R – Mo d inj in this case. The following lemma summarizes the observ ations ab o ve. Lemma 2.5.1. L et R b e a No etherian c ommutative ring endowe d with an adic top ol- o gy, let I ⊂ R b e an ide al of deﬁnition of the top olo gic al ring R , and let K b e a torsion R -mo dule. Then the fol lowing six c onditions ar e e quivalent: (1) K is an inje ctive obje ct of R – Mo d tors ; (2) ther e exists an inje ctive R -mo dule K such that K is a dir e ct summand of the torsion R -mo dule Γ I ( K ) ; (3) for every n ≥ 1 , the R/I n -mo dule I n K is inje ctive; (4) for every ide al of deﬁnition J ⊂ R , the R/J -mo dule J K is inje ctive; (5) for every op en ide al J ⊂ R , the R/J -mo dule J K is inje ctive; (6) K is inje ctive as an obje ct of R – Mo d . Corollary 2.5.2. L et R b e a No etherian c ommutative ring endowe d with an adic top olo gy. Then the ful ly faithful inclusion of ab elian c ate gories R – Mo d tors − → R – Mo d induc es isomorphisms on al l the Ext gr oups/mo dules. Pr o of. The assertion follows from Lemma 2.5.1 (1) ⇔ (6). F or a stronger result con- cerning the full-and-faithfulness of the induced functor b etw een the unbounded de- riv ed categories, see [19, Theorem 1.3]. The No etherianit y assumption can b e weak- ened, but it c annot b e simply dropp ed; see [24, Theorem 4.1]. □ F or an y ideal I in a comm utative ring R , the identit y inclusion functor R – Mo d I - ctra − → R – Mo d has a left adjoint functor ∆ I : R – Mo d − → R – Mo d I - ctra [33, Exam- ples 4.1(2–3)], [21, Example 1.3(4)]. An explicit construction of the functor ∆ I in the case of a ﬁnitely generated ideal I ⊂ R can b e found in [19, Proposition 2.1]; see [20, Sections 6–7] for a more detailed discussion. F or every pro jectiv e R -mo dule P , the I -con tramo dule R -mo dule ∆ I ( P ) is pro- jectiv e as an ob ject of R – Mo d I - ctra . Indeed, the functor ∆ I is left adjoin t to an exact functor, so it takes pro jectiv e ob jects to pro jectiv e ob jects. The R -mo dules ∆ I ( F ), where F ranges ov er the free R -mo dules, are called the fr e e I -c ontr amo dule R -mo dules . There are enough free I -con tramo dule R -mo dules in R – Mo d I - ctra . Hence 12 an I -con tramo dule R -module P is pro jective in R – Mo d I - ctra if and only if it is a direct summand of some free I -con tramo dule R -mo dule. Let R b e an adic topological ring. Then the full sub category R – Mod I - ctra ⊂ R – Mo d do es not depend on the c hoice of an ideal of deﬁnition I ⊂ R of the top ological ring R . So we will write ∆ = ∆ I , and sp eak of fr e e c ontr amo dule R -mo dules presuming an y ideal of deﬁnition I of the adic topology on R . The 0-th left deriv ed functor L 0 Λ of the (neither left nor right exact) adic com- pletion functor Λ is the left adjoint functor L 0 Λ : R – Mo d − → R – Mo d qs ctra to the iden- tit y inclusion functor R – Mo d qs ctra − → R – Mo d [24, Prop osition 1.3]. Similarly to the discussion ab o v e, for every pro jectiv e R -mo dule P , the quotseparated contramod- ule R -mo dule L 0 Λ( P ) is pro jectiv e as an ob ject of R – Mo d qs ctra . Notice that, by the deﬁnition of the 0-th deriv ed functor, one has L 0 Λ( P ) = Λ( P ) for all pro jectiv e R -mo dules P . The R -mo dules Λ( F ) = L 0 Λ( F ), where F ranges o v er the free R -mo dules, are called the fr e e quotsep ar ate d c ontr amo dule R -mo dules . There are enough free quot- separated contramodule R -mo dules in R – Mo d qs ctra . Hence quotseparated con tramo d- ule R -mo dule P is pro jectiv e in R – Mo d qs ctra if and only if it is a direct summand of some free quotseparated contramodule R -mo dule. Recall some notation from the theory of contramodules ov er top ological rings [16, Section 1.2], [33, Sections 1.2 and 5], [18, Section 2.1], [22, Section 2.7], [23, Sec- tions 2.6–2.7], [35, Section 6], [36, Section 1]. F or an y abelian group A and an y set X , w e denote by A [ X ] = A ( X ) the direct sum of copies of A indexed by X . The elemen ts of A [ X ] are in terpreted as ﬁnite formal linear com binations of elemen ts of X with the co eﬃcien ts in A . F or any complete, separated top ological ab elian group A where op en subgroups form a base of neighborho o ds of zero, w e put A [[ X ]] = lim ← − U ⊂ A ( A / U )[ X ], where U ranges ov er the open subgroups of A . The elements of A [[ X ]] are interpreted as inﬁnite formal linear combinations of elemen ts of X with the families of co eﬃcien ts con v erging to zero in A . F or an y complete, separated top ological ring R where open righ t ideals form a base of neighborho o ds of zero, and for an y set X , the set/ab elian group R [[ X ]] has a natural structure of left R -contramodule, called the fr e e R -c ontr amo dule sp anne d by X . The free R -contramodules are pro jectiv e ob jects in R – Contra , there are enough of them, and all pro jectiv e R -con tramo dules are direct summands of free ones. Giv en an adic top ological ring R with an ideal of deﬁnition I ⊂ R , consider the adic completion R = Λ( R ) of the ring R . Then, for any set X , one has an ob vious isomorphism of R -mo dules R [[ X ]] = lim ← − n ≥ 1 (( R/I n )[ X ]) = Λ( R [ X ]), which is also an isomorphism in R – Contra ≃ R – Mo d qs ctra . Th us the free quotseparated contramodule R -mo dules are the same things as free R -contramodules in the sense of the general theory of con tramo dules o v er top ological rings, and our terminology is consistent. When the ring R is No etherian (as an abstract ring), one has R – Mo d ctra = R – Mo d qs ctra (see Section 2.4). Hence a natural isomorphism ∆ ≃ L 0 Λ of functors R – Mo d − → R – Mo d ctra . 13 Lemma 2.5.3. L et R b e an adic top olo gic al ring, let I ⊂ R b e an op en ide al, and let P b e an R -mo dule. Then the natur al adjunction maps P − → ∆( P ) − → L 0 Λ( P ) − → Λ( P ) induc e isomorphisms of R /I -mo dules R/I ⊗ R P ≃ R/I ⊗ R ∆( P ) ≃ R/I ⊗ R L 0 Λ( P ) ≃ R /I ⊗ R Λ( P ) . Pr o of. F ollows from the facts that the functor P 7− → R /I ⊗ R P is left adjoint to the inclusion of full sub category R/I – Mo d − → R – Mo d , and that all R /I -mo dules are separated and complete R -mo dules. □ Corollary 2.5.4. L et R b e an adic top olo gic al ring, let M b e a torsion R -mo dule, and let P b e an R -mo dule. Then the natur al adjunction maps P − → ∆( P ) − → L 0 Λ( P ) − → Λ( P ) induc e isomorphisms of R -mo dules M ⊗ R P ≃ M ⊗ R ∆( P ) ≃ M ⊗ R L 0 Λ( P ) ≃ M ⊗ R Λ( P ) . Pr o of. Any torsion R -mo dule M is a direct union of R /I -mo dules, where I ranges o v er the ideals of deﬁnition of R . Since the tensor pro duct functor ⊗ R preserv es direct limits, it suﬃces to consider the case of an R /I -mo dule M . In that case, the assertion follows from Lemma 2.5.3 in view of the natural isomorphism M ⊗ R Q ≃ M ⊗ R/I ( R/I ⊗ R Q ) for an y R -mo dule Q . □ Lemma 2.5.5. L et R b e a No etherian c ommutative ring endowe d with an adic top ol- o gy, let I ⊂ R b e an ide al of deﬁnition of the top olo gic al ring R , and let P b e a c ontr amo dule R -mo dule. Then the fol lowing seven c onditions ar e e quivalent: (1) P is a pr oje ctive obje ct of R – Mo d ctra = R – Mo d qs ctra ; (2) ther e exists a fr e e R -mo dule F such that P is a dir e ct summand of the c on- tr amo dule R -mo dule ∆( F ) = L 0 Λ( F ) = Λ( F ) ; (3) for every n ≥ 1 , the R/I n -mo dule P /I n P is pr oje ctive; (4) for every ide al of deﬁnition J ⊂ R , the R/J -mo dule P /J P is pr oje ctive; (5) for every op en ide al J ⊂ R , the R/J -mo dule P /J P is pr oje ctive; (6) for every n ≥ 1 , the R /I n -mo dule P /I n P is ﬂat, and the R/I -mo dule P /I P is pr oje ctive; (7) P is a ﬂat R -mo dule and the R/I -mo dule P /I P is pr oje ctive. Pr o of. (1) ⇐ ⇒ (2) Explained ab ov e. No No etherianity assumption on R is needed here. (In the context of R – Mo d I - ctra and ∆ I , the assertion ev en holds for inﬁnitely generated ideals I ⊂ R .) (3) ⇐ ⇒ (4) ⇐ ⇒ (5) This is ob vious and holds for an y adic top ological ring R (do es not need the No etherianity assumption). (3) ⇐ ⇒ (6) This is a particular case of a quite general lemma ab out an asso ciative ring with a nilp otent ideal [16, Lemma B.10.2]. (2) = ⇒ (3) Both in the contexts of R – Mo d ctra and R – Mo d qs ctra , this holds for any adic top ological ring R and do es not need the No etherianity assumption. The fact that the R/I -mo dule R/I ⊗ R F is free whenev er F is either a free con tramodule R -mo dule or a free quotseparated con tramo dule R -mo dule follows from Lemma 2.5.3 (applied to a free R -mo dule M ). 14 (1) ⇐ ⇒ (3) In the context of R – Mo d qs ctra , this also holds for any adic top ological ring R and do es not need the No etherianity assumption. See [17, Corollary E.1.10(a)] for an ev en more general result. (7) = ⇒ (6) Obvious. (1), (3), or (6) = ⇒ (7) The assumption that R is No etherian is essen tial here. See [16, Corollary B.8.2] or [20, Theorem 10.5] for a complete pro of of (1) ⇐ ⇒ (3) ⇐ ⇒ (6) ⇐ ⇒ (7) for No etherian rings R . □ A discussion of the functor of c ontr atensor pr o duct ⊙ R of discrete right R -mo dules and left R -contramodules, for a complete, separated top ological ring R where op en righ t ideals form a base of neigh b orho o ds of zero, can b e found in [18, Section 3.3], [33, Section 5], [22, Section 2.8], [23, Sections 2.8], [35, Section 7.2], [36, Section 1]. In the con text of an adic top ological ring R and its adic completion R , the natural maps M ⊗ R P − − → M ⊗ R P − − → M ⊙ R P are isomorphisms for any torsion R -mo dule M and any quotseparated contramodule R -mo dule P . This assertion, whic h is easy to c hec k directly , follo ws also from the fact that the forgetful functor R – Contra − → R – Mo d is fully faithful [35, Lemma 7.11]. F or this reason, we will not dwell on the con tratensor pro duct functor ⊙ R , w orking with the tensor pro duct ⊗ R or ⊗ R instead. So, for an adic top ological ring R , we are interested in the tensor pro duct functor ⊗ R : R – Mo d tors × R – Mod ctra − − → R – Mo d tors , whic h is w ell-deﬁned b y Lemma 2.2.1. A contramodule R -mo dule F is said to b e ( c ontr a ) ﬂat (as a contramodule R -mo dule) if the tensor pro duct functor − ⊗ R F : R – Mo d tors − → R – Mo d tors is exact. Let I ⊂ R b e an ideal of deﬁnition. Since an y short exact sequence of torsion R -mo dules is a direct limit of short exact se- quences of R/I n -mo dules, n ≥ 1, a con tramo dule R -mo dule F is ﬂat if and only if the R/I n -mo dule F /I n F is ﬂat for all n ≥ 1. F or any set X and any torsion R -mo dule M , we hav e a natural isomorphism of torsion R -mo dules M ⊗ R R [[ X ]] ≃ M [ X ] = M ( X ) . Therefore, the free quotseparated contramodule R -mo dules are ﬂat (as contramod- ule R -mo dules), and it follows that the pro jectiv e quotseparated contramodule R -mo dules are ﬂat, to o. The pro jectiv e con tramo dule R -mo dules, i. e., the pro jectiv e ob jects of the ab elian category R – Mo d ctra are ﬂat contramodule R -mo dules as w ell [31, Section 3] (see also the pro of of Lemma 2.5.5 (2) ⇒ (3) ab ov e). So ﬂat contramodule R -mo dules ne e d not b e separated if they are not quotseparated. Ho w ev er, the follo wing assertion holds. Lemma 2.5.6. F or any adic top olo gic al ring R , any ﬂat quotsep ar ate d c ontr amo dule R -mo dule is (c omplete and) sep ar ate d. In p articular, if the ring R is No etherian (as an abstr act ring), then al l ﬂat c ontr amo dule R -mo dules ar e sep ar ate d. 15 Pr o of. This is a sp ecial case of the muc h more general result of [17, Corollary E.1.7], or of the ev en more general [33, Corollary 6.15]. The No etherian adic case can b e found in [16, pro of of Lemma B.9.2] or [20, Corollary 10.3(b)]. □ The follo wing corollary is a partial generalization of [14, Lemma 3.5 and Prop osi- tion 3.6]. Corollary 2.5.7. L et R b e an adic top olo gic al ring with an ide al of deﬁnition I ⊂ R , and let F b e an R -mo dule. Then the fol lowing four c onditions ar e e quivalent: (1) the R /I n -mo dule F /I n F is ﬂat for every n ≥ 1 ; (2) the c ontr amo dule R -mo dule ∆( F ) is ﬂat; (3) the c ontr amo dule R -mo dule L 0 Λ( F ) is ﬂat; (4) the c ontr amo dule R -mo dule Λ( F ) is ﬂat. If any one of the c onditions (1–4) holds, then the R -mo dule L 0 Λ( F ) is c omplete and sep ar ate d, and the natur al R -mo dule map L 0 Λ( F ) − → Λ( F ) is an isomorphism. Conse quently, the c ompletion map F − → Λ( F ) induc es an isomorphism of R -mo dules Hom R (Λ( F ) , P ) ≃ Hom R ( F , P ) for any quotsep ar ate d c ontr amo dule R -mo dule P in this c ase. Pr o of. The equiv alence of the four conditions (1–4) follows from Lemma 2.5.3. If condition (3) holds, then the R -mo dule L 0 Λ( F ) is complete and separated b y Lemma 2.5.6. P assing to the pro jectiv e limit of the isomorphisms from Lemma 2.5.3 o ver the integers n ≥ 1, w e conclude that L 0 Λ( F ) ≃ Λ( F ). Since the functor L 0 Λ : R – Mod − → R – Mo d qs ctra is left adjoin t to the inclusion functor R – Mo d qs ctra − → R – Mo d , the last assertion of the corollary follo ws. □ Corollary 2.5.8. L et R b e an adic top olo gic al ring with an ide al of deﬁnition I ⊂ R , and let F b e a quotsep ar ate d c ontr amo dule R -mo dule. Assume that the sep ar ate d c ontr amo dule R -mo dule Λ( F ) is ﬂat. Then the R -mo dule F is sep ar ate d, so F ≃ Λ( F ) . Pr o of. This can b e deduced from the next-to-last assertion of Corollary 2.5.7 b y noticing that F ≃ L 0 Λ( F ) for any quotseparated contramodule R -mo dule F . Alter- nativ ely , one can say that the quotseparated contramodule R -mo dule F is ﬂat by Corollary 2.5.7 (4) ⇒ (1), and refer to Lemma 2.5.6. □ Lemma 2.5.9. L et R b e an adic top olo gic al ring and M b e a torsion R -mo dule. Then (a) the ful l sub c ate gory of ﬂat quotsep ar ate d c ontr amo dule R -mo dules is close d un- der extensions and kernels of epimorphisms in R – Mo d qs ctra ; (b) for any short exact se quenc e of quotsep ar ate d c ontr amo dule R -mo dules 0 − → H − → G − → F − → 0 with a ﬂat quotsep ar ate d c ontr amo dule R -mo dule F , the short se quenc e of torsion R -mo dules 0 − → M ⊗ R H − → M ⊗ R G − → M ⊗ R F − → 0 is exact. Pr o of. The argument is based on the category equiv alence/isomorphism R – Mo d qs ctra ≃ R – Contra . P art (a) is a sp ecial case of [17, Lemmas E.1.4 and E.1.5]. A further generalization can b e found in [33, Corollaries 6.8 and 6.13]. P art (b) is a sp ecial case of [33, Lemma 6.10]. □ 16 When the ring R is No etherian (as an abstract ring), one can sho w that a con- tramo dule R -mo dule is ﬂat as a contramodule R -mo dule if and only if it is ﬂat as R -mo dule. This is the main assertion of the follo wing lemma. Lemma 2.5.10. L et R b e a No etherian c ommutative ring endowe d with an adic top olo gy, let I ⊂ R b e an ide al of deﬁnition of the top olo gic al ring R , let R = Λ( R ) b e the adic c ompletion of R , and let F b e a c ontr amo dule R -mo dule. Then the fol lowing seven c onditions ar e e quivalent: (1) F is a ﬂat c ontr amo dule R -mo dule; (2) F is a ﬂat c ontr amo dule R -mo dule; (3) for every n ≥ 1 , the R/I n -mo dule F /I n F is ﬂat; (4) for every ide al of deﬁnition J ⊂ R , the R/J -mo dule F /J F is ﬂat; (5) for every op en ide al J ⊂ R , the R/J -mo dule F /J F is ﬂat; (6) F is a ﬂat R -mo dule; (7) F is a ﬂat R -mo dule. Pr o of. (1) ⇐ ⇒ (3) Explained ab ov e. No No etherianity assumption on R is needed here. (3) ⇐ ⇒ (4) ⇐ ⇒ (5) This is ob vious and holds for an y adic top ological ring R (do es not need the No etherianity assumption). (1) ⇐ ⇒ (2) Without the Noetherianity assumption, one has to be careful. There is a natural isomorphism of the categories of quotsep ar ate d con tramo dules R – Mo d qs ctra ≃ R – Mo d qs ctra , as b oth the categories are isomorphic to R – Contra . Simply put, an y quotseparated con tramo dule R -mo dule admits a unique extension of its R -mo dule structure to an R -mo dule structure, making it a quotseparated con- tramo dule R -mo dule; and an y R -mo dule morphism of quotseparated contramodule R -mo dules is an R -mo dule morphism. How ev er, a nonquotseparated con tramo dule R -mo dule do es not admit an extension of its R -mo dule structure to an R -mo dule structure, in general. Instead, the nonquotseparated con tramo dule R -mo dules hav e natural mo dule structures ov er the ring ∆( R ) [21, Example 5.2(3)]. Nev ertheless, in the context of quotseparated contramodules, the equiv alence (1) ⇐ ⇒ (2) holds for an y adic top ological ring R and follows immediately from the equiv alence (1) ⇐ ⇒ (3). (6) = ⇒ (5) Obvious. (3) = ⇒ (6) The assumption that R is No etherian is essential here. A pro of can b e found in [16, Lemma B.9.2] or in [20, Corollary 10.3(a)]. (2) ⇐ ⇒ (7) This is a particular case of (1) ⇐ ⇒ (6). □ Corollary 2.5.11. L et R and S b e No etherian c ommutative rings endowe d with adic top olo gies, and let f : R − → S b e a tight c ontinuous ring map. Then the fol lowing c onditions ar e e quivalent: (1) f is ﬂat as a tight c ontinuous map of adic top olo gic al rings; (2) Λ( S ) is a ﬂat R -mo dule; (3) Λ( S ) is a ﬂat Λ( R ) -mo dule. 17 Pr o of. Notice that S = Λ( S ) is a separated and complete module o v er the adic top ological ring R (as the adic/completion top ology on S coincides with the topology induced on S b y the adic top ology on R , due to the assumptions that f is tight and con tin uous). Therefore, S is a contramodule R -mo dule. F urthermore, for every open ideal I ⊂ R , the ideal S f ( I ) ⊂ S is op en in S , hence S/S f ( I ) ≃ S / S λ S ( f ( I )). The assertion of the Corollary can b e obtained b y applying Lemma 2.5.10 (5) ⇔ (6) ⇔ (7) to the con tramo dule R -mo dule F = S . □ Prop osition 2.5.12. L et R b e a No etherian c ommutative ring endowe d with an adic top olo gy. Then the ful ly faithful inclusion of ab elian c ate gories R – Mo d qs ctra = R – Mo d ctra − → R – Mo d induc es isomorphisms on al l the Ext gr oups/mo dules. Pr o of. This is [16, Theorem B.8.1]. F or a stronger result concerning the full-and- faithfulness of the induced functor betw een the un bounded deriv ed categories, see [19, Theorem 2.9]. The No etherianity assumption can b e weak ened, but it c annot b e simply dropp ed; see [24, Theorems 4.2 and 4.3]. □ 2.6. Change of scalars. Giv en a homomorphism of asso ciative rings f : R − → S , w e denote by f ∗ : S – Mo d − → R – Mo d the functor of r estriction of sc alars , taking ev ery left S -mo dule N to its underlying left R -mo dule N . The left adjoint functor to f ∗ , taking every left R -mo dule M to the left S -mo dule S ⊗ R M , will b e denoted by f ∗ : R – Mo d − → S – Mo d ; it is called the functor of extension of sc alars with resp ect to f . The righ t adjoin t functor to f ∗ , taking every left R -mo dule P to the left S -module Hom R ( S, P ), will b e denoted by f ! : R – Mo d − → S – Mo d and called the functor of c o extension of sc alars . Let R and S b e adic top ological rings, and let f : R − → S b e a ring homomorphism. P arts (b) and (d) of the following lemma are generalizations of [17, Lemma D.4.1(a)]. Lemma 2.6.1. Assume that the ring homomorphism f is c ontinuous. Consider the functor of r estriction of sc alars f ∗ : S – Mo d − → R – Mod . Then (a) the for getful functor f ∗ takes torsion S -mo dules to torsion R -mo dules; (b) the for getful functor f ∗ takes c ontr amo dule S -mo dules to c ontr amo dule R -mo d- ules; (c) the for getful functor f ∗ takes sep ar ate d S -mo dules to sep ar ate d R -mo dules; (d) the for getful functor f ∗ takes quotsep ar ate d c ontr amo dule S -mo dules to quot- sep ar ate d c ontr amo dule R -mo dules. Pr o of. Recall that, by Lemma 2.1.3(2), there exist an ideal of deﬁnition I ⊂ R and an ideal of deﬁnition J ⊂ S suc h that f ( I ) ⊂ J . P art (a): more generally , for any homomorphism of comm utativ e rings f : R − → S and any tw o ideals I ⊂ R and J ⊂ S suc h that f ( I ) ⊂ J , the functor S – Mo d − → R – Mo d takes J -torsion S -modules to I -torsion R -mo dules. Indeed, for any S -mo dule N and any ﬁxed element r ∈ R , an element y ∈ N is f ( r )-torsion (as an element of the S -module N ) if and only if y is r -torsion (as an elemen t of the underlying R -mo dule of N ). P art (b): once again, for any homomorphism of commutativ e rings f : R − → S and an y tw o ideals I ⊂ R and J ⊂ S such that f ( I ) ⊂ J , the functor S – Mo d − → R – Mo d 18 tak es J -contramodule S -mo dules to I -con tramo dule R -mo dules. The p oin t is that, for any S -module Q and any ﬁxed element r ∈ R , there is a natural isomorphism of the Ext groups Ext ∗ S ( S [ f ( r ) − 1 ] , Q ) ≃ Ext ∗ R ( R [ r − 1 ] , Q ), b ecause R [ r − 1 ] is a ﬂat R -mo dule and S ⊗ R R [ r − 1 ] ≃ S [ f ( r ) − 1 ]. See, e. g., [34, Lemma 4.1(a)]; cf. [20, Lemma 2.1]. So the S -module Q is an f ( r )-contramodule if and only if the underlying R -mo dule of Q is an r -contramodule. P art (c): for every S -mo dule C and every n ≥ 1, one has I n C ⊂ J n C ⊂ C . Hence T n ≥ 1 J n C = 0 implies T n ≥ 1 I n C = 0. P art (d) follows from parts (b) and (c). Recall that all contramodule S -mo dules are complete, and all separate complete S -mo dules are con tramo dules, as men tioned in Section 2.4. So the class of separated and complete S -mo dules coincides with the class of separated contramodule S -mo dules. Let Q b e a quotseparated con tramo dule S -module. Then Q is a contramodule S -module and a quotien t S -module of a separated contramodule S -mo dule C . Then, b y (b) and (c), the underlying R -mo dule of Q is a con tramo dule R -mo dule and a quotien t R -mo dule of a separated con tramo dule R -mo dule C . □ Let f : R − → S b e a con tinuous homomorphism of adic top ological rings. Then Lemma 2.6.1(a) pro vides an exact, faithful functor of restriction of scalars f ⋄ : S – Mo d tors − − → R – Mo d tors . The functor f ⋄ has a righ t adjoin t functor f ⋄ : R – Mo d tors − − → S – Mo d tors , whic h can be computed by the formula f ⋄ ( M ) = Γ J (Hom R ( S, M )) for an y M ∈ R – Mo d tors , where J is an ideal of deﬁnition in S . Moreov er, for any R -mo dule M one has a natural isomorphism of torsion S -mo dules f ⋄ (Γ I ( M )) ≃ Γ J ( f ! ( M )) , where I is an ideal of deﬁnition in R . As a righ t adjoint functor to an exact functor, the functor f ⋄ tak es injective torsion R -mo dules to injectiv e torsion S -mo dules. The same assertions hold in the greater generality of a comm utative ring homomorphism f : R − → S and an y t w o ideals I ⊂ R , J ⊂ S suc h that f ( I ) ⊂ J [31, pro of of Lemma 15.1(a)]. See [23, Section 2.9] and [32, Section 6] for a generalization to the categories of discrete mo dules ov er top ological asso ciative rings. W e call f ⋄ the functor of c o extension of sc alars (for torsion mo dules, with resp ect to a con tin uous homomorphism of adic top ological rings). F urthermore, Lemma 2.6.1(b) provides an exact, faithful functor of restriction of scalars f # : S – Mo d ctra − − → R – Mo d ctra . The functor f # has a left adjoint functor f # : R – Mo d ctra − − → S – Mo d ctra , 19 whic h can be computed b y the form ula f # ( P ) = ∆ J ( S ⊗ R P ) for an y P ∈ R – Mo d ctra , where J is an ideal of deﬁnition in S . Moreov er, for any R -mo dule P one has a natural isomorphism of con tramo dule S -modules f # (∆ I ( P )) ≃ ∆ J ( f ∗ ( P )) , where I is an ideal of deﬁnition in R . As a left adjoin t functor to an exact functor, the functor f # tak es pro jectiv e con tramo dule R -mo dules to pro jectiv e contramodule S -modules. Speciﬁcally , one has f # (∆ I ( R [ X ])) = ∆ J ( S [ X ]) for any set X (so the functor f # also tak es free con tramodule R -mo dules to free con tramo dule S -mo dules). The same assertions hold in the greater generality of a comm utative ring homomor- phism f : R − → S and any t wo ideals I ⊂ R , J ⊂ S suc h that f ( I ) ⊂ J [31, pro of of Lemma 15.1(b)]. W e call f # the functor of c ontr aextension of sc alars (for con tramodules, with resp ect with respect to a contin uous homomorphism of adic top ological rings). Finally , Lemma 2.6.1(d) provides an exact, faithful functor of restriction of scalars f ♯ : S – Mo d qs ctra − − → R – Mo d qs ctra . The functor f ♯ has a left adjoint functor f ♯ : R – Mo d qs ctra − − → S – Mo d qs ctra , whic h can b e computed b y the formula f ♯ ( P ) = L 0 Λ J ( S ⊗ R P ) for an y P ∈ R – Mo d qs ctra , where J is an ideal of deﬁnition in S [31, pro of of Lemma 15.1(c)]. Moreov er, for any R -mo dule P one has a natural isomorphism of quotseparated contramodule S -modules f ♯ ( L 0 Λ I ( P )) ≃ L 0 Λ J ( f ∗ ( P )) , where I is an ideal of deﬁnition in R . As a left adjoin t functor to an exact functor, the functor f ♯ tak es pro jectiv e quotseparated con tramo dule R -mo dules to pro jec- tiv e quotseparated contramodule S -mo dules. Sp eciﬁcally , one has f ♯ (Λ I ( R [ X ])) = Λ J ( S [ X ]) for an y set X (so the functor f ♯ also tak es free quotseparated contramod- ule R -mo dules to free quotseparated contramodule S -mo dules). See [23, Section 2.9] and [32, Section 6] for a generalization to the categories of con tramo dules ov er top o- logical asso ciativ e rings. The functor f ♯ is called the functor of c ontr aextension of sc alars (for quotseparated con tramo dules, with resp ect with resp ect to a con tinuous homomorphism of adic top ological rings). Lemma 2.6.2. L et R b e an adic top olo gic al ring, S b e a discr ete (i. e., (0) -adic) c ommutative ring, and f : R − → S b e a c ontinuous ring homomorphism. Then (a) for every torsion R -mo dule M , ther e is a natur al isomorphism of S -mo dules f ⋄ ( M ) ≃ f ! ( M ) = Hom R ( S, M ); (b) for every c ontr amo dule R -mo dule P , ther e is a natur al isomorphism of S -mo dules f # ( P ) ≃ f ∗ ( P ) = S ⊗ R P ; 20 (c) for every quotsep ar ate d c ontr amo dule R -mo dule P , ther e is a natur al isomor- phism of S -mo dules f ♯ ( P ) ≃ f ∗ ( P ) = S ⊗ R P . Pr o of. Part (a): more generally , giv en a contin uous homomorphism of adic top ological rings f : R − → S and a torsion R -mo dule M , one has f ⋄ ( M ) = f ! ( M ) whenever f ! ( M ) is a torsion S -mo dule. When the ring S is discrete, all S -mo dules are torsion. P art (b): more generally , giv en a contin uous homomorphism of adic top ological rings f : R − → S and a contramodule R -mo dule P , one has f # ( P ) = f ∗ ( P ) whenever f ∗ ( P ) is a con tramo dule S -mo dule. When the ring S is discrete, all S -mo dules are con tramo dules. P art (c): more generally , given a contin uous homomorphism of adic top ological rings f : R − → S and a quotseparated con tramo dule R -mo dule P , one has f ♯ ( P ) = f ∗ ( P ) whenever f ∗ ( P ) is a quotseparated contramodule S -mo dule. When the ring S is discrete, all S -modules are quotseparated con tramo dules. □ Lemma 2.6.3. L et f : R − → S b e a c ontinuous homomorphism of adic top olo gic al rings, and let I ⊂ R and J ⊂ S b e two ide als of deﬁnition such that f ( I ) ⊂ J . Then (a) for every torsion R -mo dule M , ther e is a natur al isomorphism of S/J -mo dules J f ⋄ ( M ) ≃ Hom R/I ( S/J, I M ); (b) for every c ontr amo dule R -mo dule P , ther e is a natur al isomorphism of S/J -mo dules f # ( P ) /J f # ( P ) ≃ S/J ⊗ R/I ( P /I P ); (c) for every quotsep ar ate d c ontr amo dule R -mo dule P , ther e is a natur al isomor- phism of S/J -mo dules f ♯ ( P ) /J f ♯ ( P ) ≃ S/J ⊗ R/I ( P /I P ) . Pr o of. Let us view R/I and S /J as adic top ological rings with the discrete top ologies. Denote the related contin uous homomorphisms of adic top ological rings by p : R − → R/I , p ′ : S − → S /J , and f ′ : R/I − → S/J . Then w e ha v e a commutativ e square diagram of con tinuous homomorphisms of adic top ological rings p ′ f = f ′ p . Hence the comm utativ e diagrams of the related functors of coextension and con traextension of scalars p ′ ⋄ f ⋄ ≃ f ′ ⋄ p ⋄ , p ′ # f # ≃ f ′ # p # , and p ′ ♯ f ♯ ≃ f ′ ♯ p ♯ . It remains to use Lemma 2.6.2 for the computation of the functors of co/contraextension of scalars with resp ect to p , p ′ , and f ′ in order to obtain the desired natural isomorphisms. □ The follo wing corollary is a generalization of [17, Lemma D.4.3]. Corollary 2.6.4. L et f : R − → S b e a c ontinuous homomorphism of adic top olo gic al rings, and let I ⊂ R and J ⊂ S b e two ide als of deﬁnition such that f ( I ) ⊂ J . Then, for any ﬂat quotsep ar ate d c ontr amo dule R -mo dule F , ther e is a natur al isomorphism of quotsep ar ate d c ontr amo dule S -mo dules f ♯ ( F ) ≃ lim ← − n ≥ 1  ( S/J n ) ⊗ R/I n ( F /I n F )  . 21 In p articular, the quosep ar ate d c ontr amo dule S -mo dule f ♯ ( F ) is ﬂat. The functor f ♯ takes short exact se quenc es of ﬂat quotsep ar ate d c ontr amo dule R -mo dules to short exact se quenc es of ﬂat quotsep ar ate d c ontr amo dule S -mo dules. F urthermor e, the c ontr amo dule S -mo dule f # ( F ) is ﬂat for any ﬂat c ontr amo dule S -mo dule F . Pr o of. All the assertions are based on Lemma 2.6.3(b–c). The ﬂatness of f ♯ ( F ) and f # ( F ) can b e established, under the resp ectiv e assumptions, using the facts that a con tramo dule R -mo dule F is ﬂat if and only if the R/I n -mo dule F /I n F is ﬂat for every n ≥ 1, and similarly for contramodule S -mo dules (see Section 2.5). The formula for f ♯ ( F ) follo ws by virtue of Lemma 2.5.6. The exactness assertion holds in view of Lemma 2.5.9(b), as the functor of pro jectiv e limit of sequences of surjective maps of ab elian groups is exact. Alternatively , one can also apply the argument from [17, pro of of Lemma D.4.3] using Lemma 2.5.9. The result of [41, Theorem 1.2 or 2.8] or [17, Lemma E.1.3] is relev an t here. □ Corollary 2.6.5. L et f : R − → S b e a c ontinuous homomorphism of adic top olo gic al rings, and let 0 − → Q − → P − → F − → 0 b e a short exact se quenc e of quotsep- ar ate d c ontr amo dule R -mo dules with a ﬂat quotsep ar ate d c ontr amo dule R -mo dule F . Then the short se quenc e of quotsep ar ate d c ontr amo dule S -mo dules 0 − → f ♯ ( Q ) − → f ♯ ( P ) − → f ♯ ( F ) − → 0 is exact. Pr o of. This is a sp ecial case of the second assertion of [32, Lemma 6.2]. The claim in question follows purely formally from the exactness assertion of Corollary 2.6.4. One needs to use the facts that the functor f ♯ is right exact (as a left adjoint functor), there are enough pro jectiv e quotseparated contramodule R -mo dules in R – Mo d qs ctra , the pro jectiv e quotseparated con tramo dule R -mo dules are ﬂat (see Section 2.5), and the k ernels of surjectiv e morphisms of ﬂat quotseparated con tramo dule R -mo dules are ﬂat (see Lemma 2.5.9(a)). □ No w we pass from the contin uous maps of adic top ological rings to the tigh t ones. P art (b) of the follo wing lemma is a generalization of [17, Lemma D.4.1(b)]. Lemma 2.6.6. L et R and S b e adic top olo gic al rings, and let f : R − → S b e a tight ring map. In this c ontext: (a) if N is an S -mo dule such that the R -mo dule f ∗ ( N ) is torsion, then the S -mo dule N is torsion; (b) if Q is an S -mo dule such that the R -mo dule f ∗ ( Q ) is a c ontr amo dule, then the S -mo dule Q is a c ontr amo dule. Pr o of. Recall that, by Lemma 2.1.4(1), there exist an ideal of deﬁnition I ⊂ R and an ideal of deﬁnition J ⊂ S such that J ⊂ S f ( I ). Put J ′ = S f ( I ); so we hav e J ⊂ J ′ . P art (a): as men tioned in Section 2.2, an S -mo dule N is J ′ -torsion wheneve r it is s -torsion for all s ranging ov er some chosen set of generators of the ideal J ′ ⊂ S . The ideal J ′ is generated by its subset f ( I ) ⊂ J ′ . F or any elemen t r ∈ I , the underlying R -mo dule of N is r -torsion b y assumption. According to the proof of Lemma 2.6.1(a), 22 it follows that the S -mo dule N is f ( r )-torsion. Thus N is J ′ -torsion, and therefore also J -torsion. P art (b): as mentioned in Section 2.4, an S -mo dule Q is a J ′ -con tramo dule when- ev er it is an s -con tramo dule for all s ranging o v er some chosen set of generators of the ideal J ′ ⊂ S . The ideal J ′ is generated by its subset f ( I ) ⊂ J ′ . F or an y elemen t r ∈ I , the underlying R -mo dule of Q is an r -contramodule b y assumption. According to the pro of of Lemma 2.6.1(b), it follo ws that the S -mo dule Q is an f ( r )-contramodule. Th us Q is a J ′ -con tramo dule, and therefore also a J -contramodule. □ A quotseparated v ersion of Lemma 2.6.6(b) holds under an additional ﬂatness assumption on the morphism f . See Theorem 2.11.2 and Remark 2.11.3 b elow. P art (b) of the follo wing lemma is a generalization of [17, Lemma D.4.2(a)]. Lemma 2.6.7. L et R and S b e adic top olo gic al rings, and let f : R − → S b e a tight ring map. Then (a) for any torsion R -mo dule M , the S -mo dule f ∗ ( M ) = S ⊗ R M is also torsion; (b) for any c ontr amo dule R -mo dule P , the S -mo dule f ! ( P ) = Hom R ( S, P ) is also a c ontr amo dule. Pr o of. Part (a): by Lemma 2.2.1, the R -mo dule S ⊗ R M is torsion. By Lemma 2.6.6(a), it follo ws that the S -module S ⊗ R M is torsion. P art (b): by Lemma 2.4.1(b), the R -mo dule Hom R ( S, P ) is a contramodule. By Lemma 2.6.6(b), it follows that the S -mo dule Hom R ( S, P ) is a con tramo dule. □ Let f : R − → S b e a tight map of adic top ological rings. Then Lemma 2.6.7(a) claims that the functor of extension of scalars f ∗ : R – Mo d − → S – Mo d restricts to a functor f ∗ : R – Mo d tors − − → S – Mo d tors , whic h we will also call the extension of sc alars . When the map f is b oth contin u- ous and tigh t, the functor f ∗ is left adjoin t to the functor of restriction of scalars f ⋄ : S – Mo d tors − → R – Mo d tors . Dual-analogously , for an y tigh t map of adic top ological rings f : R − → S , Lemma 2.6.7(b) claims that the functor of co extension of scalars f ! : R – Mo d − → S – Mo d restricts to a functor f ! : R – Mo d ctra − − → S – Mo d ctra , whic h we will also call the c o extension of sc alars . When the map f is b oth contin- uous and tigh t, the functor f ! is right adjoin t to the functor of restriction of scalars f # : S – Mo d ctra − → R – Mo d ctra . F or a quotseparated version of Lemma 2.6.7(b), which w e can only pro ve under a ﬂatness assumption on the map f , see Prop osition 2.11.5 b elow. Remark 2.6.8. F or a con tin uous map of adic top ological rings f : R − → S that is not tigh t, the functor of restriction of scalars f ⋄ : S – Mo d tors − → R – Mo d tors do es not ha v e a left adjoint functor, generally sp eaking. It suﬃces to consider the case of the ring of p olynomials in one v ariable R = S = k [ x ] ov er a ﬁeld k , with the 23 iden tit y map f = id k [ x ] : R − → S , the (0)-adic (i. e., discrete) top ology on R , and the ( x )-adic top ology on S . Then the forgetful functor f ⋄ : S – Mo d tors − → R – Mo d tors is fully faithful, but it do es not preserv e inﬁnite direct pro ducts (b ecause the full sub category of x -torsion mo dules is not closed under inﬁnite pro ducts in k [ x ]– Mo d ). Accordingly , the functor f ⋄ is not a right adjoint. Similarly , for a contin uous map of adic top ological rings f : R − → S that is not tigh t, the functors of restriction of scalars f # : S – Mo d ctra − → R – Mo d ctra and f ♯ : S – Mo d qs ctra − → R – Mod qs ctra do not hav e righ t adjoint functors, generally sp eaking. In the same example of adic top ological rings R , S and the map f as ab ov e, the forgetful functor f # = f ♯ : S – Mo d ctra − → R – Mo d ctra is fully faithful, but it do es not preserv e inﬁnite direct sums (b ecause the full sub category of x -con tramo dule mo d- ules is not closed under inﬁnite direct sums in k [ x ]– Mo d ). Accordingly , the functor f # = f ♯ is not a left adjoint. (See the discussion in [23, Section 2.9] and the details in [32, Example 10.1].) The deﬁnition of a ﬂat map of adic topological rings w as giv en at the end of Section 2.1. Lemma 2.6.9. L et f : R − → S b e a ﬂat tight c ontinuous map of adic top olo gic al rings. Then (a) the functor f ∗ : R – Mo d tors − → S – Mo d tors is exact; (b) the functor f ⋄ : S – Mo d tors − → R – Mo d tors takes inje ctive torsion S -mo dules to inje ctive torsion R -mo dules; (c) the functor f # : S – Mo d ctra − → R – Mo d ctra takes ﬂat c ontr amo dule S -mo dules to ﬂat c ontr amo dule R -mo dules; (d) the functor f ♯ : S – Mo d qs ctra − → R – Mo d qs ctra takes ﬂat quotsep ar ate d c ontr amo dule S -mo dules to ﬂat quotsep ar ate d c ontr amo dule R -mo dules. Pr o of. The assertions of parts (a) and (b) are equiv alent restatements of eac h other, b ecause there are enough injectiv e ob jecs in S – Mo d tors . Nevertheless, we prov e them separately b elow. Let I ⊂ R b e an ideal of deﬁnition in R ; then, by Lemma 2.1.6(5), S f ( I ) ⊂ S is an ideal of deﬁnition in S . P art (a): ev ery short exact sequence of R -mo dules is a direct limit of short exact sequences of R/I n -mo dules, n ≥ 1. F or an y R/I n -mo dule E , one has f ∗ ( E ) = S ⊗ R E ≃ S/S f ( I n ) ⊗ R/I n E . Since S/S f ( I n ) is a ﬂat mo dule o v er R/I n , the functor f ∗ tak es short exact sequences of R/I n -mo dules to short exact sequences of S/S f ( I n )-mo dules. P art (b): for an y S -mo dule K , we hav e f ∗ ( S f ( I ) K ) = I f ∗ ( K ). F or an injective torsion S -mo dule K , the S/S f ( I )-mo dule S f ( I ) K is injectiv e. Since S /S f ( I ) is a ﬂat R/I -mo dule, it follows that I f ⋄ ( K ) is injectiv e as an R /I -mo dule, as desired. P art (c): for an y S -mo dule P , w e hav e f ∗ ( P /S f ( I ) P ) = f ∗ ( P ) /I f ∗ ( P ). F or a ﬂat con tramo dule S -module F , the S/S f ( I )-mo dule F /S f ( I ) F is ﬂat. Since S/S f ( I ) is a ﬂat R /I -mo dule, it follo ws that f # ( F ) /I f # ( F ) is ﬂat as an R /I -mo dule, as desired. P art (d) is the particular case of part (c) for quotseparated con tramo dules. □ 24 Let R b e an adic top ological ring. Then the completion map λ R : R − → R is a ﬂat tigh t con tinuou s map of adic top ological rings (as mentioned in Section 2.3). Clearly , an y torsion R -mo dule admits a unique extension of its R -mo dule struc- ture to an R -mo dule structure. The functor λ R ⋄ : R – Mo d tors − → R – Mo d tors is an equiv alence (in fact, isomorphism) of ab elian categories. Consequently , the functors λ ∗ R : R – Mo d tors − → R – Mo d tors and λ ⋄ R : R – Mo d tors − → R – Mo d tors adjoin t to λ R ⋄ on the left and on the righ t are also equiv alences of categories, isomorphic to each other and in v erse to λ R ⋄ . F urthermore, it is clear from the discussion of the category of contramodules ov er the top ological ring R in Section 2.4 that the functor λ R ♯ : R – Mo d qs ctra − → R – Mo d qs ctra is an equiv alence (in fact, isomorphism) of ab elian categories. Consequen tly , the functors λ ♯ R : R – Mo d qs ctra − → R – Mo d qs ctra and λ ! R : R – Mo d qs ctra − → R – Mo d qs ctra adjoin t to λ R ♯ on the left and on the righ t are also equiv alences of categories, isomorphic to eac h other and in v erse to λ R ♯ . Notice that the functor λ R # : R – Mo d ctra − → R – Mo d ctra is alw ays fully faithful, but it ne e d not b e an equiv alence of categories in general. See the discussion in the pro of of Lemma 2.5.10. 2.7. T ensor pro ducts of adic top ological rings. Let R b e a comm utative ring, let S and T b e tw o adic top ological rings, and let f : R − → S and g : R − → T b e tw o ring homomorphisms. Consider the ring W = T ⊗ R S , and denote by f ′ : T − → T ⊗ R S and g ′ : S − → T ⊗ R S the tw o induced ring homomorphisms. Lemma 2.7.1. (a) Ther e exists a unique ﬁnest ring top olo gy on the ring W with a b ase of neighb orho o ds of zer o forme d by op en ide als such that the ring maps f ′ and g ′ ar e c ontinuous. (b) Ther e exists a unique c o arsest ring top olo gy on the ring W with a b ase of neigh- b orho o ds of zer o forme d by op en ide als such that, for every top olo gic al c ommutative ring V with a b ase of neighb orho o ds of zer o forme d by op en ide als, and for every ring homomorphism h : W − → V for which the c omp ositions hf ′ : T − → V and hg ′ : S − → V ar e c ontinuous, the ring homomorphism h is c ontinuous as wel l. (c) The ring top olo gies on W deﬁne d in p arts (a) and (b) c oincide, and it is an adic top olo gy. (d) L et J ⊂ S b e an ide al of deﬁnition in S and K ⊂ T b e an ide al of deﬁnition in T . Then L = W ( f ′ ( K ) + g ′ ( J )) ⊂ W is an ide al of deﬁnition in the adic top olo gy on W deﬁne d in p arts (a–c) . Pr o of. Consider the adic top ology on W with the ideal of deﬁnition constructed in part (d). Let us c hec k that this top ology on W satisﬁes the conditions stated in parts (a) and (b). P art (a): the ring maps f ′ and g ′ are contin uous b y Lemma 2.1.3(2). Conv ersely , let τ b e a ring topology on W , with a base of neigh b orho o ds of zero formed by op en righ t ideals, such that the maps f ′ and g ′ are contin uous with resp ect to τ . T o sho w that τ is equal to or coarser than the adic top ology on W deﬁned ab o v e, w e need to c hec k that all the ideals in W that are op en in τ are also op en in our adic top ology . 25 In other words, w e need to chec k that for every ideal E ⊂ W that is op en in τ there exists an in teger n ≥ 1 such L n ⊂ E . Indeed, since the map f ′ : T − → W is contin uous with resp ect to τ , there exists an integer n 1 ≥ 1 such that f ′ ( K n 1 ) ⊂ E . Similarly , since the map g ′ : S − → W is con tin uous with resp ect to τ , there exists an in teger n 2 ≥ 1 suc h that g ′ ( J n 2 ) ⊂ E . Put n = n 1 + n 2 − 1. Then w e ha v e ( f ′ ( K ) + g ′ ( J )) n ⊂ E , hence L n ⊂ E . P art (b): ﬁrstly we assume that the comp ositions hf ′ and hg ′ are contin uous, and pro v e that the map h is contin uous with resp ect to our adic top ology on W . Let E ⊂ V be an op en ideal. W e need to sho w that there exists an integer n ≥ 1 such that h ( L n ) ⊂ E . Indeed, since the map hf ′ is contin uous, there exists an integer n 1 ≥ 1 suc h that hf ′ ( K n 1 ) ⊂ E . Similarly , since the map hg ′ is contin uous, there exists an in teger n 2 ≥ 1 suc h that hg ′ ( J n 2 ) ⊂ E . Put n = n 1 + n 2 − 1. Then we hav e ( hf ′ ( K ) + hg ′ ( J )) n ⊂ E , hence h ( L n ) ⊂ E . No w let τ b e a ring top ology on W such that, for every top ological comm utative ring V with a base of neighborho o ds of zero formed b y op en ideals, and for every ring homomorphism h : W − → V for whic h the comp ositions hf ′ : T − → V and hg ′ : S − → V are contin uous, the map h is contin uous with resp ect to τ . T o show that τ is equal to or ﬁner than the adic top ology on W deﬁned ab o ve, w e need to c hec k that all the ideals in W that are open in our adic top ology are also op en in τ . Put V = W , and endo w the ring V with our adic top ology on W . Let h : W − → V b e the identit y map. Then the maps hf ′ and hg ′ are contin uous, as we hav e shown in the very b eginning of the pro of of part (a) ab ov e. So the map h is contin uous with resp ect to the top ology τ on W and our adic top ology on V . This means precisely that τ is equal to or ﬁner than our adic top ology on W . □ The adic top ology on the ring W = T ⊗ R S constructed in Lemma 2.7.1 will b e called the tensor pr o duct top olo gy . In the sequel, we will sometimes abuse the notation and write L = K ⊗ R S + T ⊗ R J ⊂ T ⊗ R S = W instead of L = W ( f ′ ( K ) + g ′ ( J )) ⊂ W in the context of Lemma 2.7.1(d). Notice that ideals of the form L = K ⊗ R S + T ⊗ R J , where J ranges ov er the ideals of deﬁnition in S and K ranges ov er the ideals of deﬁnition in T , form a base of neigh b orho o ds of zero in the tensor pro duct top ology on T ⊗ R S . Lemma 2.7.2. (a) L et f : R − → S b e a c ontinuous homomorphism of adic top olo gic al rings. Endow the tensor pr o duct R ⊗ R S with the tensor pr o duct top olo gy. Then the natur al ring isomorphisms S − → R ⊗ R S − → S ar e isomorphisms of top olo gic al rings. (b) L et R and S b e c ommutative rings, T , U , and V b e adic top olo gic al rings, and R − → T , R − → U , S − → U , and S − → V b e ring homomorphisms. Endow al l the r elevant tensor pr o ducts with the tensor pr o duct top olo gies. Then the natur al ring isomorphism ( T ⊗ R U ) ⊗ S V ≃ T ⊗ R ( U ⊗ S V ) is an isomorphism of top olo gic al rings. Pr o of. Part (a) Let J ⊂ S b e an ideal of deﬁnition. By Lemma 2.1.3(3), there exists an ideal of deﬁnition I ⊂ R such that f ( I ) ⊂ J . No w the isomorphism R ⊗ R S ≃ S transforms the ideal I ⊗ R S + R ⊗ R J = R ⊗ R J ⊂ R ⊗ R S in to the ideal J ⊂ S . 26 By Lemma 2.7.1(d), I ⊗ R S + R ⊗ R J is an ideal of deﬁnition in R ⊗ R S . Thus R ⊗ R S ≃ S is an isomorphism of top ological rings. Part (b) is ob vious. □ Lemma 2.7.3. In the notation ab ove, endow the ring W = T ⊗ R S with the tensor pr o duct top olo gy. Assume that the ring R is endowe d with an adic top olo gy such that f : R − → S is a tight c ontinuous ring map and g : R − → T is a c ontinuous ring map. Then (a) the c ontinuous ring map f ′ : T − → W is also tight; (b) if the tight c ontinuous ring map f is ﬂat, then the tight c ontinuous ring map f ′ is ﬂat, to o; (c) if the ring map Λ( f ) : Λ( R ) − → Λ( S ) is a (top olo gic al) isomorphism, then the ring map Λ( f ′ ) : Λ( T ) − → Λ( W ) is an isomorphism as wel l. Pr o of. Part (a): b y Lemma 2.1.3(2), there exist an ideal of deﬁnition I ⊂ R and an ideal of deﬁnition K ⊂ T such that g ( I ) ⊂ K . By Lemma 2.1.6(5), the ideal J = S f ( I ) ⊂ S is an ideal of deﬁnition in S . By Lemma 2.7.1(4), the ideal L = W ( f ′ ( K ) + g ′ ( J )) ⊂ W is an ideal of deﬁnition in W . N o w we hav e W ( f ′ ( K ) + g ′ ( J )) = W ( f ′ ( K ) + g ′ f ( I )) = W ( f ′ ( K ) + f ′ g ( I )) = W f ′ ( K ). By Lemma 2.1.4(1) or 2.1.6(4), it follows that the map f ′ is tigh t. P art (b): no w w e apply Lemma 2.1.3(3) to the eﬀect that, for ev ery ideal of deﬁnition K ⊂ T , there exists an ideal of deﬁnition I ⊂ R such that f ( I ) ⊂ K . Then, in the notation of the previous paragraph, we hav e W /L ≃ T /K ⊗ R/I S/J . The ring S/J is ﬂat as an R /I -mo dule by Lemma 2.1.8(2), hence the ring W /L is ﬂat as a T /K -mo dule. P art (c): ﬁrst of all, for a tight contin uous map f , the map Λ( f ) is also tigh t (and con tin uous) by Lemma 2.3.1(a). Hence Λ( f ) is an isomorphism of abstract rings if and only if Λ( f ) is an isomorphism of top ological rings. No w let K ⊂ T b e an ideal of deﬁnition. Once again, in the notation ab ov e, w e ha v e W /L ≃ T /K ⊗ R/I S/J . If Λ( f ) is an isomorphism, then so is the map R/I − → S/J . Hence the map T /K − → W /L is an isomorphism, to o. As this holds for all ideals of deﬁnition K ⊂ T , w e can pass to the pro jectiv e limit and conclude that the map Λ( f ′ ) is an isomorphism. □ No w consider the follo wing commutativ e diagram of homomorphisms of comm uta- tiv e rings: (1) e S e T S h S O O e R ˜ f ^ ^ ˜ g @ @ T h T O O R f ^ ^ h R O O g ? ? Assume that adic top ologies are given on the rings S , e S , T , and e T such that h S and h T are con tin uous ring maps. Consider the tensor pro duct rings T ⊗ R S and e T ⊗ e R e S , and 27 endo w them with the tensor pro duct top ologies. Then the homomorphism of tensor pro ducts h : T ⊗ R S − → e T ⊗ e R e S induced by the ring homomorphisms h S , h T , and h R is a con tin uous ring map. The follo wing lemma is a generalization of Lemma 2.7.3(c). Lemma 2.7.4. L et S , T , and R b e adic top olo gic al rings, and let f : R − → S and g : R − → T b e c ontinuous ring maps. Consider the induc e d c ontinuous maps of the c omplete d rings Λ( f ) : Λ( R ) − → Λ( S ) and Λ( g ) : Λ( R ) − → Λ( T ) . F urthermor e, c onsider the induc e d c ontinuous homomorphism of the tensor pr o duct rings h : T ⊗ R S − − → Λ( T ) ⊗ Λ( R ) Λ( S ) . Then the induc e d homomorphism of the c ompletions of the tensor pr o ducts Λ( h ) : Λ( T ⊗ R S ) − − → Λ(Λ( T ) ⊗ Λ( R ) Λ( S )) . is an isomorphism of top olo gic al rings. Pr o of. Put W = T ⊗ R S . Let J ⊂ S be an ideal of deﬁnition and K ⊂ T b e an ideal of deﬁnition. Since the in tersection of an y tw o ideals of deﬁnition in R is also an ideal of deﬁnition in R , Lemma 2.1.3(3) implies existence of an ideal of deﬁnition I ⊂ R suc h that f ( I ) ⊂ J and g ( I ) ⊂ K . Then, according to Lemma 2.7.1(d), L = W ( f ′ ( K ) + g ′ ( J )) ⊂ W is an ideal of deﬁnition in W . W e ha ve W /L ≃ T /K ⊗ R/I S/J . Put R = Λ( R ), S = Λ( S ), T = Λ( T ), and f W = T ⊗ R S (the latter notation is in tended to emphasize the fact that the top ological ring T ⊗ R S ne e d not b e separated or complete). Consider the ideals I = R λ R ( I ) ⊂ R , J = S λ S ( J ) ⊂ S , and K = T λ S ( K ) ⊂ T corresp onding to the ideals I , J , and K under the bijection describ ed in Section 2.3. Put f = Λ( f ) and g = Λ( g ), and denote b y f ′ : T − → f W and g ′ : S − → f W the induced homomorphisms of top ological rings. Then e L = f W ( f ′ ( K ) + g ′ ( J )) ⊂ f W is an ideal of deﬁnition in f W , and w e ha ve f W / e L ≃ T / K ⊗ R / I S / J . No w the ring homomorphisms R/I − → R / I , S/J − → S / J , and T /K − → T / K are isomorphisms. Hence so is the ring homomorphism W /L − → f W / e L . It remains to p oint out that one has Λ( W ) = lim ← − J,K W /L and Λ( f W ) = lim ← − J,K f W / e L , where J and K range ov er the directed p osets of all ideals of deﬁnition in S and T . The p oin t is that the ideals of deﬁnition of the form L = W ( f ′ ( K ) + g ′ ( J )) form a base of neighborho o ds of zero in W , and similarly for the ideals e L ⊂ f W . F urthermore, the adic top ology on Λ( W ) is the top ology of pro jectiv e lim t of discrete rings W /L , while the top ology on Λ( f W ) is the top ology of pro jectiv e limit of discrete rings f W / e L . Th us the map Λ( h ) : Λ( W ) − → Λ( f W ) is an isomorphism of topological rings. □ Corollary 2.7.5. In the c ontext of the c ommutative diagr am of c ommutative ring homomorphisms (1) , supp ose that al l the six rings ar e endowe d with adic top olo gies and al l the seven maps ar e c ontinuous ring homomorphisms. Assume further that the induc e d maps of the c ompletions Λ( h S ) : Λ( S ) − → Λ( e S ) , Λ( h T ) : Λ( T ) − → Λ( e T ) , and Λ( h R ) : Λ( R ) − → Λ( e R ) ar e isomorphisms of top olo gic al rings. Then the induc e d 28 map of the c ompletions of the tensor pr o ducts Λ( h ) : Λ( T ⊗ R S ) − − → Λ( e T ⊗ e R e S ) is also an isomorphism of top olo gic al rings. Pr o of. F ollows from Lemma 2.7.4. □ 2.8. F ormal op en immersions and formal op en cov erings. Let us sa y that a homomorphism of (discrete) comm utativ e rings f : R − → S is an op en immer- sion if the induced map of the sp ectra Sp ec S − → Sp ec R is an op en immersion of aﬃne sc hemes. Equiv alently , f is an op en immersion if and only if it is a ﬂat ring epimorphism of ﬁnite presen tation [39, Section XI.2], [10, Theoreme IV.17.9.1]. In particular, we will sa y that a homomorphism of commutativ e rings R − → S is a princip al op en immersion if there exists an elemen t r ∈ R such that S is isomorphic to R [ r − 1 ] as a commutativ e R -algebra. Similarly , let us sa y that a collection of homomorphisms of commutativ e rings f α : R − → S α is a ( princip al ) op en c overing if every map f α is a (principal) op en immersion and the collection of the induced maps of the sp ectra Sp ec S α − → Sp ec R is an aﬃne op en co vering of the aﬃne sc heme Sp ec R . Equiv alen tly , the latter condition means that, for any ﬁeld k and an y ring homomorphism R − → k , there exists an index α suc h that S α ⊗ R k  = 0. The follo wing criterion is a basic result concerning aﬃne op en subsc hemes in aﬃne schemes. Lemma 2.8.1. A homomorphism of c ommutative rings f : R − → S is an op en immersion if and only if ther e exists a ﬁnite c ol le ction of elements r 1 , . . . , r m ∈ R such that the induc e d ring maps R [ r − 1 j ] − → S [ f ( r j ) − 1 ] ar e isomorphisms for al l 1 ≤ j ≤ m and the c ol le ction of princip al op en immersions S − → S [ f ( r j ) − 1 ] , 1 ≤ j ≤ m , is an op en c overing. □ Lemma 2.8.2. L et f : R − → S b e a homomorphism of c ommutative rings such that S is a ﬂat R -mo dule, and let I ⊂ R b e a nilp otent ide al. Then (a) f : R − → S is an isomorphism if and only if ¯ f : R /I − → S/S f ( I ) is an isomorphism; (b) f : R − → S is a princip al op en immersion if and only if ¯ f : R /I − → S/S f ( I ) is a princip al op en immersion; (c) f : R − → S is an op en immersion if and only if ¯ f : R /I − → S/S f ( I ) is an op en immersion. Pr o of. Part (a): more generally , for an y nilp otent ideal I in a commutativ e ring R , a homomorphism of ﬂat R -mo dules f : F − → G is an isomorphism if and only if the induced map ¯ f : F /I F − → G/I G is an isomorphism. This is pro v able using the Nak ay ama lemma for nilp otent ideals. P art (b): the “only if ” assertion is ob vious. T o pro v e the “if ”, let ¯ r ∈ R/I b e an elemen t suc h that the R /I -algebra S/S f ( I ) is isomorphic to ( R/I )[ ¯ r − 1 ]. Let r ∈ R b e any element suc h that ¯ r = r + I . Applying part (a) to the R -algebra morphisms R [ r − 1 ] − → S [ f ( r ) − 1 ] ← − S and the nilp oten t ideals I [ r − 1 ] ⊂ R [ r − 1 ] and S I ⊂ S , one concludes that b oth of these R -algebra morphisms are isomorphisms. 29 P art (c): once again, the “only if ” assertion is obvious. T o pro ve the “if ”, let ¯ r 1 , . . . , ¯ r m ∈ R/I b e a ﬁnite collection of elemen ts satisfying the conditions of Lemma 2.8.1 for the ring homomorphism R/I − → S/S f ( I ). Let r j ∈ R , 1 ≤ j ≤ m , b e any element suc h that ¯ r j = r j + I . Applying part (a) to the ring homomorphisms R [ r − 1 j ] − → S [ f ( r j ) − 1 ] and the nilp oten t ideals I [ r − 1 j ] ⊂ R [ r − 1 j ], one concludes that these ring homomorphisms are isomorphisms. T o prov e that the collection of principal op en immersions S − → S [ f ( r j ) − 1 ], 1 ≤ j ≤ m , is an op en cov ering, it suﬃces to observe that an y homomorphism from S to a ﬁeld k factorizes through S/S f ( I ). □ Lemma 2.8.3. L et f α : R − → S α b e a c ol le ction of homomorphisms of c ommuta- tive rings such that the rings S α ar e ﬂat R -mo dules, and let I ⊂ R b e a nilp otent ide al. Consider the induc e d ring homomorphisms ¯ f α : R/I − → S α /S α f α ( I ) . Then the c ol le ction ( f α ) is an op en c overing if and only if the c ol le ction ( ¯ f α ) is an op en c overing. Pr o of. One needs to use Lemma 2.8.2(c). On top of that, the ﬁnal argumen t from the pro of of the same lemma w orks: Any ring homomorphism from R to a ﬁeld k factorizes through R/I . □ Lemma 2.8.4. A homomorphism of c ommutative rings f : R − → S is an op en immersion if and only if ther e exists an op en c overing g α : S − → T α of the ring S such that the c omp ositions g α f : R − → T α ar e op en immersions. If this is the c ase, then one c an cho ose ( g α ) to b e a princip al op en c overing for which al l the c omp ositions g α f ar e also princip al op en immersions. Pr o of. The ﬁrst assertion expresses lo cality of the notion of an op en immersion of sc hemes. It is immediate from the deﬁnitions. The second assertion is a restatement of the fact that principal aﬃne op en subsc hemes form a top ology base in Sp ec R . □ Theorem 2.8.5. L et R b e a c ommutative ring and I ⊂ R b e a nilp otent ide al. Then the functor S 7− → S/S f ( I ) pr ovides an e quivalenc e b etwe en the c ate gory of c ommutative R -algebr as f : R − → S such that f is an op en immersion and the c ate- gory of c ommutative R/I -algebr as ¯ f : R /I − → S such that ¯ f is an op en immersion. In other wor ds, the map Sp ec S 7− → Sp ec S/S f ( I ) is an isomorphism b etwe en the p oset of aﬃne op en subschemes in Sp ec R and the p oset of aﬃne op en subschemes in Sp ec R/I (with r esp e ct to the inclusion or der). An op en immersion f is princip al if and only if the op en immersion ¯ f is. Pr o of. W e are not a w are of an y argumen t within the realm of comm utativ e ring theory . The following pro of is, basically , a collection of references. By [10, The- orem IV.18.1.2], the functor X 7− → Sp ec R/I × Spec R X is an equiv alence b etw een the category of ´ etale sc hemes ov er Sp ec R and the category of ´ etale sc hemes ov er Sp ec R/I . By [10, Theorem IV.17.9.1], an ´ etale morphism is an op en immersion if and only if it is radicial (in the sense of [11, Section I.3.7]). It follo ws easily that an ´ etale morphism X − → Sp ec R is an op en immersion if and only if the corre- sp onding morphism Sp ec R /I × Spec R X − → Sp ec R /I is an op en immersion. So the map Sp ec R/I × Spec R − is an equiv alence b et w een the category of op en subschemes 30 in Sp ec R and the category of op en subsc hemes in Sp ec R /I . The same conclusion can b e simply obtained from the facts that op en subschemes in schemes correspond bijectiv ely to open subsets in the underlying topology , and the underlying topological spaces of Spec R and Sp ec R/I coincide. The k ey step is to prov e that an op en subscheme Y ⊂ Sp ec R is aﬃne if and only if the scheme Sp ec R /I × Spec R Y is aﬃne. The “only if ” assertion is ob vious. T o pro v e the “if ”, notice that the underlying top ological spaces of Y and Sp ec R /I × Spec R Y coincide. Quasi-compactness and quasi-separatedness are prop erties of the under- lying top ological space of a scheme; and all aﬃne schemes are quasi-compact and quasi-separated. So the scheme Y is quasi-compact and quasi-separated (in fact, separated, as an op en subsc heme of an aﬃne sc heme). By [8, Th´ eor ` eme I I.5.2.1(d)], [9, IV.1.7.17], [13, Lemmas T ags 01XB and 01XF or 01X G], a quasi-compact scheme Y is aﬃne if and only if H 1 ( Y , F ) = 0 for ev ery quasi-coherent sheaf F on Y . Here H ∗ ( Y , − ) denotes the cohomology groups of sheav es of ab elian groups on Y , which only dep end on the underlying top ological space of Y . It remains to p oint out that ev ery quasi-coherent sheaf F on Y has a ﬁnite decreasing ﬁltration by quasi-coherent subshea v es F ⊃ I F ⊃ I 2 F ⊃ · · · ⊃ I n F = 0, where n ≥ 1 is an integer such that I n = 0 in R , and the successiv e quotient shea v es I m F /I m +1 F are quasi-coheren t shea v es on Sp ec R/I × Spec R Y . Finally , if f : R − → S is an open immersion and the R/I -algebra S/S f ( I ) is isomorphic to ( R/I )[ ¯ r − 1 ] for some element ¯ r ∈ R /I , then the R -algebra S is isomor- phic to R [ r − 1 ] for any element r ∈ R such that ¯ r = r + I . This assertion follo ws immediately from the bijection estalished ab ov e. □ Lemma 2.8.6. L et f : R − → S b e a tight c ontinuous map of adic top olo gic al rings. Then the fol lowing thr e e c onditions ar e e quivalent: (1) for every ide al of deﬁnition I ⊂ R , the induc e d ring map ¯ f : R /I − → S/S f ( I ) is an op en immersion; (2) ther e exists an ide al of deﬁnition I ⊂ R such that the induc e d ring map ¯ f : R /I n − → S /S f ( I n ) is an op en immersion for every n ≥ 1 ; (3) f is ﬂat (as a map of adic top olo gic al rings) and ther e exists an ide al of deﬁnition I ⊂ R such that the induc e d ring map ¯ f : R/I − → S/S f ( I ) is an op en immersion. Pr o of. (1) = ⇒ (2) Holds b ecause I n ⊂ R is an ideal of deﬁnition for any ideal of deﬁnition I ⊂ R . (2) = ⇒ (1) The p oin t is that if I ⊂ I ′ ⊂ R are tw o ideals and the ring map R/I − → S/S f ( I ) is an op en immersion, then the ring map R /I ′ − → S /S f ( I ′ ) is an op en immersion, too. (1) = ⇒ (3) Holds b ecause, for all op en immersions of commutativ e rings ¯ f : R − → S , the ring S is a ﬂat R -mo dule. (3) = ⇒ (2) F ollo ws from Lemma 2.8.2(c). □ W e will sa y that a tight con tin uous map of adic top ological rings f : R − → S is a formal op en immersion if it satisﬁes the equiv alen t conditions of Lemma 2.8.6. Using 31 the criterion of Lemma 2.8.6(1), one can easily c heck that the comp osition of an y t w o formal op en immersions is a formal op en immersion. Lemma 2.8.7. L et f : R − → S b e a c ontinuous map of adic top olo gic al rings. Con- sider the ring S ⊗ R S , and endow it with the tensor pr o duct top olo gy. In this c ontext: (a) the induc e d map of top olo gic al rings S ⊗ R S − → S is c ontinuous and tight; (b) the induc e d maps of top olo gic al rings S ⇒ S ⊗ R S ar e c ontinuous; (c) if the map f is tight, then so ar e the maps S ⇒ S ⊗ R S ; (d) if the map f is a formal op en immersion, then al l the thr e e induc e d maps of the c ompletions Λ( S ) ⇒ Λ( S ⊗ R S ) − → Λ( S ) ar e isomorphisms of top olo gic al rings. Pr o of. Parts (a) and (b) do not dep end on the top ology on R (one can use the discrete top ology on R , for example). P art (a): denote the map in question b y p : S ⊗ R S − → S . Let J ⊂ S b e an ideal of deﬁnition. Then, b y Lemma 2.7.1(d), J ⊗ R S + S ⊗ R J is an ideal of deﬁnition in S ⊗ R S . Now we hav e J = p ( J ⊗ R S + S ⊗ R J ), so p is a tight contin uous map by Lemma 2.1.6(4). P art (b) follo ws from the discussion in Section 2.7 (including Lemma 2.7.2(a)). P art (c): denote the maps in question by i 1 and i 2 : S ⇒ S ⊗ R S . Let I ⊂ R b e an ideal of deﬁnition. By Lemma 2.1.6(5), the ideal J = S f ( I ) ⊂ S is an ideal of deﬁnition in S . By Lemma 2.7.1(d), the ideal J ⊗ R S + S ⊗ R J ⊂ S ⊗ R S is an ideal of deﬁnition in S ⊗ R S . No w we ha v e S f ( I ) ⊗ R S = S ⊗ R S f ( I ), hence J ⊗ R S + S ⊗ R J = J ⊗ R S = S ⊗ R J ⊂ S ⊗ R S . By Lemma 2.1.6(4), it follows that i 1 and i 2 are tigh t ring maps. P art (d): in the notation of the previous paragraph, let I range ov er all the ideals of deﬁnition in R . Then the ideals J = S f ( I ) form a base of neighborho o ds of zero in S , while the ideals J ⊗ R S = S ⊗ R J form a base of neighborho o ds of zero in S ⊗ R S . Now we ha v e S/ ( J ⊗ R S + S ⊗ R J ) ≃ S/J ⊗ R/I S/J . The natural maps S/J ⇒ S/J ⊗ R/I S/J − → S/J are ring isomorphisms, since R/I − → S/J is an op en immersion (of abstract commutativ e rings) by Lemma 2.8.6(1). Passing to the pro jectiv e limit ov er I , we obtain the desired isomorphisms of complete adic top ological rings. □ Lemma 2.8.8. L et f : R − → S b e a tight c ontinuous map of adic top olo gic al rings. Then the fol lowing four c onditions ar e e quivalent: (1) for every ide al of deﬁnition I ⊂ R , the induc e d ring map ¯ f : R /I − → S/S f ( I ) is a princip al op en immersion; (2) ther e exists an ide al of deﬁnition I ⊂ R such that the induc e d ring map ¯ f : R /I n − → S /S f ( I n ) is a princip al op en immersion for every n ≥ 1 ; (3) f is ﬂat (as a map of adic top olo gic al rings) and ther e exists an ide al of deﬁnition I ⊂ R such that the induc e d ring map ¯ f : R /I − → S/S f ( I ) is a princip al op en immersion; (4) ther e exists an element r ∈ R such that, endowing the ring R [ r − 1 ] with the adic top olo gy for which the natur al ring map l : R − → R [ r − 1 ] is c ontinuous and 32 tight, ther e exists a unique isomorphism of top olo gic al rings Λ( S ) ≃ Λ( R [ r − 1 ]) making the fol lowing diagr am c ommutative: (2) R [ r − 1 ] λ R [ r − 1 ] / / Λ( R [ r − 1 ]) Λ( S ) S λ S o o R λ R / / l O O Λ( R ) Λ( l ) O O id O O Λ( R ) Λ( f ) O O R λ R o o f O O Pr o of. (1) = ⇒ (2) and (1) = ⇒ (3) Similar to the pro of of Lemma 2.8.6. (2) = ⇒ (1) The p oint is that if I ⊂ I ′ ⊂ R are tw o ideals and the ring map R/I − → S/S f ( I ) is a principal op en immersion, then the ring map R/I ′ − → S/S f ( I ′ ) is a principal op en immersion, to o. (3) = ⇒ (2) F ollo ws from Lemma 2.8.2(b). (4) = ⇒ (1) Obvious. (4) = ⇒ (3) F ollo ws from Lemma 2.3.1(b). (3) = ⇒ (4) Let ¯ r ∈ R/I b e an elemen t suc h that the R/I -algebra S/S f ( I ) is isomorphic to ( R/I )[ ¯ r − 1 ]. Pick any elemen t r ∈ R for which ¯ r = r + I . Let n ≥ 1 b e an in teger; put ˜ r = r + I n ∈ R/I n . By the proof of Lemma 2.8.2(b), the R/I n -algebra S/S f ( I n ) is isomorphic to ( R/I n )[ ˜ r − 1 ]. As the category of op en subschemes in Spec R is a poset, an isomorphism of R /I n -algebras ( R/I n )[ ˜ r − 1 ] ≃ S/S f ( I n ) is unique. Hence, as n v aries, such isomorphisms form a pro jectiv e system. P assing to the pro jectiv e limit o v er n ≥ 1, we obtain the desired isomorphism of adic top ological rings and Λ( R )-algebras Λ( R [ r − 1 ]) ≃ Λ( S ). □ W e will sa y that a tigh t contin uous map of adic top ological rings f : R − → S is a princip al formal op en immersion if it satisﬁes the equiv alent conditions of Lemma 2.8.8. Using the criterion of Lemma 2.8.8(1), one can easily chec k that the comp osition of an y tw o principal formal op en immersions is a principal formal op en immersion. Corollary 2.8.9. L et R b e an adic top olo gic al ring and I ⊂ R b e an ide al of deﬁni- tion. Then the functor S 7− → S/S f ( I ) pr ovides an e quivalenc e b etwe en the c ate gory of c ommutative R -algebr as f : R − → S with an adic top olo gy on S such that f is a formal op en immersion and the c ate gory of c ommutative R/I -algebr as ¯ f : R /I − → S such that ¯ f is an op en immersion. A formal op en immersion f is princip al if and only if the op en immersion ¯ f is princip al. Pr o of. This is a corollary of Theorem 2.8.5. Let us just explain ho w to construct the in verse functor. Supp ose giv en an op en immersion ¯ f : R/I − → S . By Theo- rem 2.8.5, for ev ery n ≥ 1 there exists an op en immersion f n : R/I n − → S n to- gether with a R /I -algebra isomorphism R/I ⊗ R/I n S n ≃ S . The uniqueness assertion of Theorem 2.8.5 implies existence and uniqueness of R/I m -algebra isomorphisms R/I m ⊗ R/I n S n ≃ S m for all n ≥ m ≥ 1. It remains to set S = lim ← − n ≥ 1 S n . The result of [41, Theorem 1.2 or 2.8] or [17, Lemma E.1.3] is helpful here. T o pro v e the ﬁnal assertion of the corollary , one needs to use Lemma 2.8.8(1). □ 33 Lemma 2.8.10. L et f α : R − → S α b e a c ol le ction of tight c ontinuous maps of adic top olo gic al rings. Then the fol lowing thr e e c onditions ar e e quivalent: (1) for every ide al of deﬁnition I ⊂ R , the c ol le ction of induc e d ring maps ¯ f α : R/I − → S α /S α f α ( I ) is an op en c overing; (2) ther e exists an ide al of deﬁnition I ⊂ R such that the c ol le ction of induc e d ring maps ¯ f α : R/I n − → S α /S α f α ( I n ) is an op en c overing for every n ≥ 1 ; (3) the maps f α ar e ﬂat (as maps of adic top olo gic al rings) and ther e exists an ide al of deﬁnition I ⊂ R such that the c ol le ction of induc e d ring maps ¯ f α : R/I − → S α /S α f α ( I ) is an op en c overing. Pr o of. F ollows from Lemmas 2.8.3 and 2.8.6. □ W e will sa y that a collection of tigh t con tinuous maps of adic top ological rings f α : R − → S α is a formal op en c overing if it satisﬁes the equiv alent conditions of Lemma 2.8.10. Clearly , any formal op en cov ering of an adic top ological ring has a ﬁnite sub cov ering. A princip al formal op en c overing is a formal op en cov ering ( f α ) suc h that ev ery map f α is a principal formal op en immersion. Lemma 2.8.11. A tight c ontinuous map of adic top olo gic al rings f : R − → S is an formal op en immersion if and only if ther e exists a formal op en c overing g α : S − → T α of the adic top olo gic al ring S such that the c omp ositions g α f : R − → T α ar e formal op en immersions. If this is the c ase, then one c an cho ose ( g α ) to b e a princip al formal op en c overing for which al l the c omp ositions g α f ar e also princip al formal op en immersions. Pr o of. First assertion: to prov e the “only if ”, it suﬃces to take the set of indices { α } to b e the singleton { 0 } , and the trivial formal op en co vering with T 0 = S and g 0 = id S . T o pro v e the “if ”, let I ⊂ R b e an ideal of deﬁnition. By Lemma 2.1.6(5), the ideal S f ( I ) ⊂ S is an ideal of deﬁnition in S , while the ideals T α g α f ( I ) ⊂ T α are ideals of deﬁnition in T α . By Lemma 2.8.10(1), the collection of induced ring maps ¯ g α : S/S f ( I ) − → T α /T α g α f ( I ) is an op en co v ering. By Lemma 2.8.6(1), the comp ositions ¯ g α ¯ f : R /I − → T α /T α g α f ( I ) are op en immersions. By the ﬁrst assertion of Lemma 2.8.4, it follows that the induced ring map ¯ f : R /I − → S/S f ( I ) is an op en immersion. Using Lemma 2.8.6(1) again, w e conclude that f : R − → S is a formal op en immersion. Second assertion: let I ⊂ R b e an ideal of deﬁnition. By Lemma 2.8.6(1), the induced ring map ¯ f : R /I − → S/S f ( I ) is an op en immersion. By the sectond as- sertion of Lemma 2.8.4, there exists a principal op en cov ering ¯ g α : S /S f ( I ) − → T α suc h that the comp ositions ¯ g α ¯ f : R/I − → T α are principal op en immersions. By Corollary 2.8.9, there exist principal formal op en immersions g α : S − → T α together with S /S f ( I )-algebra isomorphisms T α /T α g α f ( I ) ≃ T α . By Lemma 2.8.10(1), the collection ( g α ) is a formal op en cov ering. □ Lemma 2.8.12. L et f α : R − → S α b e a formal op en c overing of an adic top olo gic al ring R . Then a quotsep ar ate d c ontr amo dule R -mo dule F is ﬂat if and only if the quot- sep ar ate d c ontr amo dule S α -mo dule f ♯ α ( F ) is ﬂat for every index α . A c ontr amo dule 34 R -mo dule F is ﬂat if and only if the c ontr amo dule S α -mo dule f # α ( F ) is ﬂat for every index α . Pr o of. Let us prov e the quotseparated version. Let I ⊂ R b e an ideal of deﬁnition in R . Then J α = S α f α ( I ) ⊂ S α is an ideal of deﬁnition in S α for every index α . By Lemma 2.8.10(1), the collection of induced ring maps ¯ f α : R/I − → S α /J α is an open co v ering. By Lemma 2.6.3(c), w e ha v e f ♯ α ( F ) /J f ♯ α ( F ) ≃ S α /J α ⊗ R/I F /I F . No w it remains to p oint out that the R/I -mo dule F /I F is ﬂat if and only if the S α /J α -mo dule S α /J α ⊗ R/I F /I F is ﬂat for ev ery index α . The pro of of the nonquotseparated case is similar and based on Lemma 2.6.3(b). □ Lemma 2.8.13. L et R , S , and T b e adic top olo gic al rings, let f : R − → S b e a tight c ontinuous ring map, and let g : R − → T b e a c ontinuous ring map. Put W = T ⊗ R S , and denote by f ′ : T − → W and g ′ : S − → W the induc e d ring maps. Endow the c ommutative ring W with the tensor pr o duct top olo gy, as in Se ction 2.7. In this setting: (a) if the tight c ontinuous ring map f is a formal op en immersion, then the tight c ontinuous ring map f ′ is a formal op en immersion, to o; (b) if the tight c ontinuous ring map f is a princip al formal op en immersion, then the tight c ontinuous ring map f ′ is a princip al formal op en immersion, to o. Pr o of. Similar to the pro of of Lemma 2.7.3(b). □ 2.9. V ery ﬂat and con traadjusted con tramo dules. In the sequel, we will use a shorthand notation R – Contra = R – Mo d qs ctra for an y adic top ological ring R . In the case of a complete, separated adic top ological ring R , a con tramo dule ov er R as a top ological ring is essen tially the same thing as a quotseparated con tramo dule R -mo dule (see the end of Section 2.4), so this notation is unambiguous. Let us also recall a similar notation R – T ors = R – Mo d tors from Section 2.2. W e start with a recollection of the case of an abstract commutativ e ring R (with- out any adic or other top ology). An R -mo dule P is said to b e c ontr aadjuste d [17, Section 1.1], [38, Section 5], [20, Sections 2 and 8], [34, Section 0.5], [26, Section 2 and Example 3.2], [28, Section 4.3] if Ext 1 R ( R [ s − 1 ] , P ) = 0 for all elemen ts s ∈ R . An R -mo dule F is said to b e very ﬂat [17, Section 1.1], [38, Section 2], [20, Section 8], [34, Section 0.5], [26, Section 2 and Example 2.5], [28, Section 4.3] if Ext 1 R ( F , C ) = 0 for all con traadjusted R -mo dules C . The full sub category of contraadjusted R -mo dules is closed under extensions, quo- tien ts, and inﬁnite products in R – Mo d . The full sub category of very ﬂat R -mo dules is closed under extensions, kernels of epimorphisms, inﬁnite direct sums, and tensor pro ducts in R – Mo d . All very ﬂat R -mo dules hav e pro jectiv e dimensions at most 1. A homomorphism of comm utativ e rings f : R − → S is said to b e very ﬂat if, for every element s ∈ S , the R -mo dule S [ s − 1 ] is very ﬂat. If this is the case, the ring S is also said to b e a very ﬂat c ommutative R -algebr a . V ery ﬂatness of the R -mo dule S is not suﬃcient for the ring homomorphism R − → S to b e very ﬂat [34, Example 9.7], [28, Section 6.1]. The composition of an y t w o very ﬂat homomorphi sms 35 of comm utative rings is very ﬂat [17, Lemma 1.2.3(b)]. Ev ery op en immersion of comm utativ e rings is v ery ﬂat [17, Lemma 1.2.4]. Lemma 2.9.1. L et R b e an adic top olo gic al ring and F b e a c ontr amo dule R -mo dule. Then the fol lowing two c onditions ar e e quivalent: (1) F is a ﬂat c ontr amo dule R -mo dule and ther e exists an ide al of deﬁnition I ⊂ R for which F /I F is a very ﬂat R/I R -mo dule; (2) for every op en ide al I ⊂ R , the R /I -mo dule F /I F is very ﬂat. Pr o of. (2) = ⇒ (1) holds b ecause all v ery ﬂat R/I -mo dules are ﬂat. (1) = ⇒ (2) Let I ⊂ I ′ ⊂ R be t wo ideals of deﬁnition in R . Then I ′ /I is a ﬁnitely generated nilp oten t ideal in the ring R/I . Therefore, assuming that the R/I -mo dule F /I F is ﬂat, the R/I -mo dule F /I F is very ﬂat if and only if the R/I ′ -mo dule F /I ′ F is very ﬂat [17, Lemma 1.7.10(b)]. Now if I ⊂ I ′ ⊂ R are tw o ideals suc h that I is an ideal of deﬁnition, and the R/I -mo dule F /I F is very ﬂat, then the R /I ′ -mo dule F /I ′ F is v ery ﬂat b y [17, Lemma 1.2.2(b)]. □ Let R be an adic top ological ring. A contramodule R -mo dule F is said to b e very ﬂat if it satisﬁes the equiv alen t conditions of Lemma 2.9.1. All pro jectiv e con- tramo dule R -mo dules and all pro jectiv e quotseparated contramodule R -mo dules are v ery ﬂat (see the pro of of Lemma 2.5.5 (2) ⇒ (3)). It follo ws from Lemma 2.5.9 that the full sub category of very ﬂat quotseparated con tramo dule R -mo dules is closed under extensions and k ernels of epimorphisms in R – Mo d qs ctra . F urthermore, in view of [17, Corollary E.1.10(a)], all v ery ﬂat quotseparated con tramodule R -mo dules ha v e pro jectiv e dimensions at most 1 as ob jects of R – Mo d qs ctra . These assertions are sp ecial cases of [17, Corollary E.4.1]. Lemma 2.9.2. L et R b e an adic top olo gic al ring and F b e a ﬂat R -mo dule. Then, for any quotsep ar ate d c ontr amo dule R -mo dule P , ther e is a natur al Ext 1 -adjunction isomorphism of ab elian gr oups/ R -mo dules Ext 1 R ( F , P ) ≃ Ext 1 R – Contra (Λ( F ) , P ) . Pr o of. By [14, Lemma 3.5 or Prop osition 3.6], w e ha v e L i Λ( F ) = 0 for all i > 0. Therefore, for any short exact sequence of R -mo dules 0 − → M − → N − → F − → 0, the short sequence of quotseparated contramodule R -mo dules 0 − → L 0 Λ( M ) − → L 0 Λ( N ) − → L 0 Λ( F ) − → 0 is exact. No w w e ha ve a pair of adjoin t functors b et w een ab elian categories R – Mo d / / / / R – Contra , L 0 Λ o o where the righ t adjoin t functor R – Contra − → R – Mo d is the identit y inclusion. The righ t adjoin t functor is exact, while the ob ject F ∈ R – Mo d is adjusted to the left adjoin t functor L 0 Λ in the sense stated in the previous paragraph. By [26, Lemma 1.7(e)], the desired Ext 1 -adjunction isomorphism follows. Here we are also using the natural R -mo dule isomorphism L 0 Λ( F ) ≃ Λ( F ), whic h holds by the s ame results from [14] or by Corollary 2.5.7 ab ov e. □ 36 Lemma 2.9.3. L et R b e an adic top olo gic al ring, R b e the adic c ompletion of R , and P b e a quotsep ar ate d c ontr amo dule R -mo dule. Then the fol lowing seven c onditions ar e e quivalent: (1) the functor Hom R ( − , P ) takes short exact se quenc es of very ﬂat quotsep ar ate d c ontr amo dule R -mo dules to short exact se quenc es of ab elian gr oups/ R -mo d- ules/quotsep ar ate d c ontr amo dule R -mo dules; (2) one has Ext 1 R – Contra ( F , P ) = 0 for al l very ﬂat quotsep ar ate d c ontr amo d- ule R -mo dules F (wher e the Ext gr oup is c ompute d in the ab elian c ate gory R – Contra = R – Mo d qs ctra ); (3) one has Ext 1 R – Contra (Λ( R [ r − 1 ]) , P ) = 0 for al l elements r ∈ R ; (4) P is a c ontr aadjuste d R -mo dule; (5) P is a c ontr aadjuste d R -mo dule; (6) for every ide al I ⊂ R , the R/I -mo dule P /I P is c ontr aadjuste d; (7) for every ide al I ⊂ R , the R /I -mo dule P /I P is c ontr aadjuste d. F urthermor e, if P is a sep ar ate d c ontr amo dule R -mo dule, then c onditions (1–7) ar e also e quivalent to the fol lowing two c onditions: (8) ther e exists an ide al of deﬁnition I ⊂ R for which P /I P is a c ontr aadjuste d R/I -mo dule; (9) for every op en ide al I ⊂ R , the R /I -mo dule P /I P is c ontr aadjuste d. Pr o of. (1) ⇐ ⇒ (2) F ollows from the facts that there are enough v ery ﬂat quotsepa- rated con tramo dules in R – Mod qs ctra and the full subcategory of v ery ﬂat quotseparated con tramo dules is closed under kernels of epimorphisms. (3) ⇐ ⇒ (4) F ollo ws immediately from Lemma 2.9.2 (applied to the ﬂat R -mo dule F = R [ r − 1 ]). (1) ⇐ ⇒ (4) This is a sp ecial case of [17, ﬁrst assertion of Corollary E.4.8]. In the case of a No etherian ring R , one can also refer to [17, Corollary D.3.5(c,e)]. (1) ⇐ ⇒ (5) This is a particular case of (1) ⇐ ⇒ (4). (6) = ⇒ (4) or (7) = ⇒ (5) T ak e I = 0. (4) = ⇒ (6) or (5) = ⇒ (7) This is [17, Lemma 1.7.7(b)]. (6) = ⇒ (9) and (7) = ⇒ (9) = ⇒ (8) Obvious. (8) = ⇒ (9) This is [17, Lemma 1.7.10(a)]. (9) = ⇒ (2) This is a sp ecial case of [17, Lemma E.4.3]. □ A quotseparated contramodule R -mo dule P is said to b e c ontr aadjuste d if it satis- ﬁes any one of the equiv alen t conditions (1–7) of Lemma 2.9.3. The full sub category of con traadjusted quotseparated con tramo dule R -mo dules is closed under extensions, quotien ts, and inﬁnite pro ducts in R – Mo d qs ctra . Prop osition 2.9.4. L et R b e an adic top olo gic al ring and M b e a quotsep ar ate d c ontr amo dule R -mo dule. Then (a) ther e exists a short exact se quenc e of quotsep ar ate d c ontr amo dule R -mo dules 0 − → P − → F − → M − → 0 with a very ﬂat quotsep ar ate d c ontr amo dule R -mo dule F and a c ontr aadjuste d quotsep ar ate d c ontr amo dule R -mo dule P ; 37 (b) ther e exists a short exact se quenc e of quotsep ar ate d c ontr amo dule R -mo dules 0 − → M − → P − → F − → 0 with a c ontr aadjuste d quotsep ar ate d c ontr amo dule R -mo dule P and a very ﬂat quotsep ar ate d c ontr amo dule R -mo dule F . Pr o of. Part (a) is a special case of [17, Corollary E.4.5]. F or a No etherian ring R , one can also refer to [17, Corollary D.3.5(b,e)]. P art (b) is a sp ecial case of [17, Corol- lary E.4.6]. F or a No etherian ring R , one can also refer to [17, Corollary D.3.5(a,e)]. Another reference is [33, Example 7.12(3)]. □ Corollary 2.9.5. L et R b e an adic top olo gic al ring and F b e a quotsep ar ate d c on- tr amo dule R -mo dule. Then the fol lowing c onditions ar e e quivalent: (1) the functor Hom R ( F , − ) takes short exact se quenc es of c ontr aadjuste d quotsep ar ate d c ontr amo dule R -mo dules to short exact se quenc es of ab elian gr oups/ R -mo dules/quotsep ar ate d c ontr amo dule R -mo dules; (2) one has Ext 1 R – Contra ( F , P ) = 0 for al l c ontr aadjuste d quotsep ar ate d c ontr amo d- ule R -mo dules P ; (3) F is a very ﬂat quotsep ar ate d c ontr amo dule R -mo dule. Pr o of. (1) ⇐ ⇒ (2) F ollo ws from the facts that there are enough contraadjusted quot- separated contramodules in R – Mo d qs ctra (b y Proposition 2.9.4(b)) and the full sub- category of contraadjusted quotseparated contramodules is closed under cok ernels of monomorphisms. (3) = ⇒ (2) Holds by Lemma 2.9.3(2). (2) = ⇒ (3) This a standard argument. By Prop osition 2.9.4(a), there exists a short exact sequence 0 − → P − → G − → F − → 0 in R – Mo d qs ctra suc h that G is very ﬂat and P is con traadjusted. By (2), it follows that F is a direct summand of G . It remains to make the obvious observ ation that the class of v ery ﬂat quotseparated con tramo dule R -mo dules is closed under direct summands. (F or a No etherian ring R , the reference to [17, Corollary D.3.5(d,e)] is also applicable.) □ The follo wing lemma is the very ﬂat counterpart of Corollary 2.6.4. Lemma 2.9.6. L et f : R − → S b e a c ontinuous homomorphism of adic top olo gic al rings. Then, for any very ﬂat quotsep ar ate d c ontr amo dule R -mo dule F , the quot- sep ar ate d c ontr amo dule S -mo dule f ♯ ( F ) is very ﬂat. F or any very ﬂat c ontr amo dule R -mo dule F , the c ontr amo dule S -mo dule f # ( F ) is very ﬂat. Pr o of. The argumen t is similar to the pro of of Corollary 2.6.4 and based on Lemma 2.6.3(b–c). The p oin t is that, for an y homomorphism of (abstract, nontopo- logical) commutativ e rings ¯ f : R − → S , the functor ¯ f ∗ : R – Mo d − → S – Mo d tak es v ery ﬂat R -mo dules to v ery ﬂat S -modules [17, Lemma 1.2.2(b)]. □ Lemma 2.9.7. L et f : R − → S b e a c ontinuous homomorphism of adic top olo gic al rings. Then, for any c ontr aadjuste d quotsep ar ate d c ontr amo dule S -mo dule P , the quotsep ar ate d c ontr amo dule R -mo dule f ♯ ( P ) is c ontr aadjuste d. Pr o of. Using the criterion of Lemma 2.9.3(4), the question reduces to the case of a homomorphism of abstract (non topological) commutativ e rings, where w e 38 can refer to [17, Lemma 1.2.2(a)]. Alternativ ely , an argument pro ceeding en tirely within the realm of quotseparated con tramo dules is p ossible, using the criterion of Lemma 2.9.3(2) and the results of Corollary 2.6.5 and Lemma 2.9.6, and similar to the pro of of Lemma 2.10.4 sketc hed b elow. □ The next lemma is a very ﬂat complement to Lemma 2.1.8. Lemma 2.9.8. L et f : R − → S b e a tight c ontinuous map of adic top olo gic al rings. Then the fol lowing thr e e c onditions ar e e quivalent: (1) for every ide al of deﬁnition I ⊂ R , the induc e d ring map ¯ f : R /I − → S/S f ( I ) is very ﬂat; (2) ther e exists an ide al of deﬁnition I ⊂ R such that the induc e d ring map ¯ f : R /I n − → S /S f ( I n ) is very ﬂat for every n ≥ 1 ; (3) f is ﬂat (as a map of adic top olo gic al rings) and ther e exists an ide al of deﬁnition I ⊂ R such that the induc e d ring map ¯ f : R /I − → S/S f ( I ) is very ﬂat. Pr o of. All the assertions follow from [17, Lemma 1.7.10(b)]. □ W e will sa y that a tight con tin uous homomorphism of adic top ological rings f : R − → S is very ﬂat if it satisﬁes the equiv alen t conditions of Lemma 2.9.8. An y formal op en immersion of adic top ological rings in v ery ﬂat. Using the criterion of Lemma 2.9.8(1), one can easily chec k that the comp osition of any tw o v ery ﬂat tight con tin uous ring maps of adic topological rings is very ﬂat. The follo wing lemma is the very ﬂat version of Lemma 2.6.9(c–d). Lemma 2.9.9. L et f : R − → S b e a very ﬂat tight c ontinuous map of adic top olo gic al rings. Then (a) the functor f # : S – Mo d ctra − → R – Mo d ctra takes very ﬂat c ontr amo dule S -mo dules to very ﬂat c ontr amo dule R -mo dules; (b) the functor f ♯ : S – Mo d qs ctra − → R – Mo d qs ctra takes very ﬂat quotsep ar ate d c on- tr amo dule S -mo dules to very ﬂat quotsep ar ate d c ontr amo dule R -mo dules. Pr o of. The argumen t is similar to the pro of of Lemma 2.6.9(c–d). The p oint is that, for an y v ery ﬂat homomorphism of (abstract, nontopological) commutativ e rings ¯ f : R − → S , the functor ¯ f ∗ : S – Mo d − → R – Mo d tak es very ﬂat S -mo dules to very ﬂat R -mo dules [17, Lemma 1.2.3(b)], [34, Lemma 9.3(a)], [28, Lemma 6.1]. □ The next lemma is the very ﬂat version of Lemma 2.8.12. Lemma 2.9.10. L et f α : R − → S α b e a formal op en c overing of an adic top olo gic al ring R (in the sense of Se ction 2.8). Then a quotsep ar ate d c ontr amo dule R -mo dule F is very ﬂat if and only if the quotsep ar ate d c ontr amo dule S α -mo dule f ♯ α ( F ) is very ﬂat for every index α . A c ontr amo dule R -mo dule F is very ﬂat if and only if the c ontr amo dule S α -mo dule f # α ( F ) is very ﬂat for every index α . Pr o of. The pro of is similar to that of Lemma 2.8.12 and based on [17, Lemma 1.2.6(a)] together with Lemma 2.6.3(b–c) ab ov e. □ 39 Lemma 2.9.11. L et R b e an adic top olo gic al ring with an ide al of deﬁnition I ⊂ R , and let F b e an R -mo dule such that the R/I n -mo dule F /I n F is very ﬂat for al l n ≥ 1 . L et 0 − → P − → M − → N − → 0 b e a short exact se quenc e of quot- sep ar ate d c ontr amo dule R -mo dules with a c ontr aadjuste d quotsep ar ate d c ontr amo d- ule R -mo dule P . Then the short se quenc e of quotsep ar ate d c ontr amo dule R -mo dules 0 − → Hom R ( F , P ) − → Hom R ( F , M ) − → Hom R ( F , N ) − → 0 is exact. Pr o of. First of all, the Hom R -mo dules in question are quotseparated contramod- ules b y Lemma 2.4.2(c). In the sp ecial case when F is a ﬂat R -mo dule, the ex- actness of the short sequence in question follo ws from the fact that Ext 1 R ( F , P ) = 0, see [17, Corollary E.4.8]. In the general case, we notice that Hom R ( F , N ) ≃ Hom R (Λ( F ) , N ) and similarly for M and P , b y Corollary 2.5.7. By Lemma 2.5.3, Λ( F ) is a very ﬂat (quot)separated con tramo dule R -mo dule under our current as- sumptions. No w the exactness of the short sequence of Hom R ( F , − ) follo ws from the fact that Ext 1 R – Contra (Λ( F ) , P ) = 0 by Lemma 2.9.3(2). □ 2.10. Cotorsion con tramo dules. Once again, w e start with a reminder of the case of an abstract asso ciative ring R . A left R -mo dule P is said to b e c otorsion (in the sense of Eno chs [5]) if Ext 1 R ( F , P ) = 0 for all ﬂat left R -mo dules P . The full sub cate- gory of cotorsion R -mo dules is closed under extensions, cok ernels of monomorphisms, and inﬁnite products in R – Mod . Lemma 2.10.1. L et R b e an adic top olo gic al ring and P b e a quotsep ar ate d c on- tr amo dule R -mo dule. Then the fol lowing thr e e c onditions ar e e quivalent: (1) the functor Hom R ( − , P ) takes short exact se quenc es of ﬂat quotsep ar ate d c ontr amo dule R -mo dules to short exact se quenc es of ab elian gr oups/ R -mo d- ules/quotsep ar ate d c ontr amo dule R -mo dules; (2) one has Ext 1 R – Contra ( F , P ) = 0 for al l ﬂat quotsep ar ate d c ontr amo dule R -mo d- ules F ; (3) one has Ext n R – Contra ( F , P ) = 0 for al l ﬂat quotsep ar ate d c ontr amo dule R -mo d- ules F and al l inte gers n ≥ 1 . Pr o of. All the equiv alences follo w from the facts that there are enough ﬂat quotsep- arated con tramo dules in R – Mo d qs ctra and the full sub category of ﬂat quotseparated con tramo dules is closed under kernels of epimorphisms (see Lemma 2.5.9(a)). □ Let R b e an adic top ological ring. A quotseparated con tramo dule R -mo dule P is said to b e c otorsion if it satisﬁes the equiv alent conditions of Lemma 2.10.1. The full sub category of cotorsion quotseparated con tramo dule R -mo dules is closed under extensions, cok ernels of monomorphisms, and inﬁnite pro ducts in R – Mo d qs ctra . Prop osition 2.10.2. L et R b e an adic top olo gic al ring and M b e a quotsep ar ate d c ontr amo dule R -mo dule. Then (a) ther e exists a short exact se quenc e of quotsep ar ate d c ontr amo dule R -mo dules 0 − → P − → F − → M − → 0 with a ﬂat quotsep ar ate d c ontr amo dule R -mo dule F and a c otorsion quotsep ar ate d c ontr amo dule R -mo dule P ; 40 (b) ther e exists a short exact se quenc e of quotsep ar ate d c ontr amo dule R -mo dules 0 − → M − → P − → F − → 0 with a c otorsion quotsep ar ate d c ontr amo dule R -mo dule P and a ﬂat quotsep ar ate d c ontr amo dule R -mo dule F . Pr o of. This is a sp ecial case of [33, Corollary 7.8]. F or a No etherian ring R , one can also refer to [17, Corollary D.2.8(a–b)]. □ Corollary 2.10.3. L et R b e an adic top olo gic al ring and F b e a quotsep ar ate d c on- tr amo dule R -mo dule. Then the fol lowing c onditions ar e e quivalent: (1) the functor Hom R ( F , − ) takes short exact se quenc es of c otorsion quotsep ar ate d c ontr amo dule R -mo dules to short exact se quenc es of ab elian gr oups/ R -mo d- ules/quotsep ar ate d c ontr amo dule R -mo dules; (2) one has Ext 1 R – Contra ( F , P ) = 0 for al l c otorsion quotsep ar ate d c ontr amo dule R -mo dules P ; (3) one has Ext n R – Contra ( F , P ) = 0 for al l c otorsion quotsep ar ate d c ontr amo dule R -mo dules F and al l inte gers n ≥ 1 ; (4) F is a ﬂat quotsep ar ate d c ontr amo dule R -mo dule. Pr o of. The argument is similar to the pro of of Corollary 2.9.5 and based on Prop osi- tion 2.10.2 together with Lemma 2.10.1(2–3). F or a No etherian ring R , one can also refer to [17, Corollary D.2.8(d)]. □ Lemma 2.10.4. L et f : R − → S b e a c ontinuous homomorphism of adic top olo gic al rings. Then, for any c otorsion quotsep ar ate d c ontr amo dule S -mo dule P , the quotsep- ar ate d c ontr amo dule R -mo dule f ♯ ( P ) is c otorsion. Pr o of. This is a sp ecial case of [32, Corollary 6.3]. The assertion is pro v able using the criterion of Lemma 2.10.1(2), by an argumen t based on the Ext 1 -adjunction lemma from [26, Lemma 1.7(e)] together with the results of Corollaries 2.6.4 and 2.6.5. □ The follo wing lemma is a cotorsion version of Lemma 2.9.11. Lemma 2.10.5. L et R b e an adic top olo gic al ring with an ide al of deﬁnition I ⊂ R , and let F b e an R -mo dule such that the R /I n -mo dule F /I n F is ﬂat for al l n ≥ 1 . L et 0 − → P − → M − → N − → 0 b e a short exact se quenc e of quotsep ar ate d c ontr amo d- ule R -mo dules with a c otorsion quotsep ar ate d c ontr amo dule R -mo dule P . Then the short se quenc e of quotsep ar ate d c ontr amo dule R -mo dules 0 − → Hom R ( F , P ) − → Hom R ( F , M ) − → Hom R ( F , N ) − → 0 is exact. Pr o of. The argumen t is similar to the pro of of Lemma 2.9.11. One notices that L 0 Λ( F ) ≃ Λ( F ) is a ﬂat (quot)separated contramodule R -mo dule and uses the fact that Ext 1 R – Contra (Λ( F ) , P ) = 0 by Lemma 2.10.1(2). □ Lemma 2.10.6. L et R b e a No etherian c ommutative ring endowe d with an adic top olo gy, R b e the adic c ompletion of R , and P b e a c ontr amo dule R -mo dule. Then the fol lowing c onditions ar e e quivalent: (1) P is a c otorsion (quotsep ar ate d) c ontr amo dule R -mo dule; (2) P is a c otorsion R -mo dule; 41 (3) P is a c otorsion R -mo dule. Pr o of. Recall that a con tramo dule R -mo dule F is ﬂat if and only if it is ﬂat as an R -mo dule (or as an R -mo dule), b y Lemma 2.5.10. F urthermore, the inclusion of ab elian categories R – Contra − → R – Mo d induces isomorphisms of all the Ext groups/mo dules, b y Prop osition 2.5.12. In view of these observ ations, the equiv alence (1) ⇐ ⇒ (2) is pro vided by [17, Corollary D.2.8(c)]. The equiv alence (1) ⇐ ⇒ (3) is a particular case of (1) ⇐ ⇒ (2). □ 2.11. The co extension of scalars is quotseparated. The aim of this section is to formulate and pro ve quotseparated v ersions of Lemmas 2.6.6(b) and 2.6.7(b). Our exp osition is based on the ideas and tec hniques from the new preprin t [32]. F or any adic topological ring R with an ideal of deﬁnition I ⊂ R and an y R -mo dule M , w e will denote the kernel of the I -adic completion map λ I ,M : M − → Λ I ( M ) b y Ω R ( M ) = T n ≥ 1 I n M ⊂ M . Lemma 2.11.1. L et R and S b e adic top olo gic al rings, and let f : R − → S b e a tight c ontinuous ring map. Then an S -mo dule Q is sep ar ate d and c omplete if and only if the R -mo dule f ∗ ( Q ) is sep ar ate d and c omplete. Mor e over, for any S -mo dule N , one has Ω R ( N ) = Ω S ( N ) ⊂ N . Denoting by I ⊂ R and J ⊂ S any ide als of deﬁnition in R and S , we also have a natur al isomorphism of R -mo dules Λ I ( N ) ≃ Λ J ( N ) forming a c ommutative triangular diagr am with the c ompletion maps λ I ,N : N − → Λ I ( N ) and λ J,N : N − → Λ J ( N ) . Pr o of. All the assertions follow immediately from Lemma 2.1.6(4). □ Let f : R − → S be a tigh t contin uous map of adic top ological rings. With Lemma 2.11.1 in mind, w e will write simply Ω( N ) instead of Ω R ( N ) = Ω S ( N ), and Λ( N ) instead of Λ I ( N ) ≃ Λ J ( N ), for an y S -mo dule N . The completion map N − → Λ( N ) will b e denoted, as usual, by λ N . Theorem 2.11.2. L et R and S b e adic top olo gic al rings, and let f : R − → S b e a ﬂat tight c ontinuous ring map. Then an S -mo dule Q is a quotsep ar ate d c ontr amo dule if and only if the R -mo dule f ∗ ( Q ) is a quotsep ar ate d c ontr amo dule. Pr o of. This is our version of [32, Theorem 9.5]. The “only if ” assertion holds b y Lemma 2.6.1(d); the nontrivial implication is the “if ”. So let Q b e an S -mo dule whose underlying R -module is a quotseparated con- tramo dule. Then Q is a con tramo dule S -module by Lemma 2.6.6(b). W e need to p o v e that Q is a quotseparated contramodule S -mo dule. An y contramodule R -mo dule is complete, as men tioned in Section 2.4. Hence the completion map λ Q : Q − → Λ( Q ) is surjective, and we hav e a short exact sequence of S -modules 0 − → Ω( Q ) − → Q − → Λ( Q ) − → 0. Pic k a ﬂat quotseparated con tramo dule S -mo dule F together with a surjectiv e S -module map F − → Λ( Q ). F or example, it suﬃces to take a free S -mo dule F together with a surjective S -mo dule morphism F − → Λ( Q ), and put F = Λ( F ). 42 Consider the pullbac k diagram in the category of S -mo dules (3) 0 / / Ω( Q ) / / / / Q / / / / Λ( Q ) / / 0 0 / / Ω( Q ) / / / / Q 1 O O O O / / / / F O O O O / / 0 Here the Q 1 is the pullback of the pair of surjectiv e maps λ Q : Q − → Λ( Q ) adn F − → Λ( Q ). Both the top and b ottom rows in (3) are short exact sequences. Since the underlying R -mo dules of Q , Λ( Q ), and F are quotseparated contramod- ule R -mo dules, it follo ws that the underlying R -mo dule of Q 1 is a quotseparated con tramo dule R -mo dule, to o. By Lemma 2.6.9(d), the underlying R -mo dule of F is a ﬂat quotseparated con- tramo dule R -mo dule. By Lemma 2.5.6, F is a separated con tramo dule R -mo dule (as w ell as a separated con tramo dule S -module). Applying the functor Λ and the natural transformation λ to the surjectiv e S -mo dule map Q 1 − → F , w e obtain a factorization of that map as the comp osition of surjectiv e S -module maps Q 1 / / / / Λ( Q 1 ) / / / / F . It follows, in view of (3), that the surjectiv e S -module map Q 1 − → Q induces an injectiv e S -module map Ω( Q 1 ) − → Ω( Q ). Let us pro v e that the map Ω( Q 1 ) − → Ω( Q ) c annot b e an isomorphism if Ω( Q )  = 0. Indeed, if the map Ω( Q 1 ) − → Ω( Q ) is an isomorphism, then Λ( Q 1 ) ≃ F . So Λ( Q 1 ) is a ﬂat (quot)separated contramodule R -mo dule. Since Q 1 is a quotseparated con tramo dule R -mo dule, Corollary 2.5.8 is applicable, and it follows that Ω( Q 1 ) = 0. Hence Ω( Q ) = 0. No w we pro ceed by transﬁnite induction, iterating the construction ab o ve at the successor steps, and passing to the pro jectiv e limit at the limit steps. Let ℵ b e a cardinal greater than the cardinalit y of Q . Viewing ℵ as an ordinal, we will construct a pro jectiv e system of S -mo dules Q i , 0 ≤ i < ℵ , and surjective S -module morphisms Q j − → Q i for all ordinals 0 ≤ i ≤ j < ℵ . The underlying R -mo dule of Q i will b e a quotseparated contramodule R -mo dule for every 0 ≤ i < ℵ . The induced maps Ω( Q j ) − → Q i will be injectiv e for all 0 ≤ i ≤ j < ℵ , and the map Ω( Q i +1 ) − → Ω( Q i ) will b e a prop er embedding (i. e., not an isomorphism) whenev er Ω( Q i )  = 0. Put Q 0 = Q . F or every successor ordinal j = i + 1 < ℵ , pick a ﬂat quotseparated con tramo dule S -mo dule F i together with a surjective S -mo dule map F i − → Λ( Q i ). Set Q j to b e the pullback of the t wo surjective S -mo dule maps Q i − → Λ( Q i ) and F i − → Λ( Q i ). F or ev ery limit ordinal j < ℵ , put Q j = lim ← − i 0 ; (3) P is a (glob al ly) c ontr aher ent c oshe af of c ontr amo dules on U . Pr o of. (2) = ⇒ (1) Ob vious. (3) = ⇒ (2) The coagumented homological ˇ Cec h complex C • ( { W α } , P ) − → P [ U ] is acyclic b y Lemma 2.12.11 (applied to the con traadjusted contramodule P [ U ] ov er the adic topological ring R = O U ( U )). (1) = ⇒ (3) This is dual-analogous to (but more complicated than) the pro of of Prop osition 3.4.2. Let V ⊂ U ⊂ U b e t wo aﬃne op en formal subschemes. W e need to c hec k the con traherence axiom (iv) for P with resp ect to V ⊂ U . Notice that V = S N α =1 ( V ∩ W α ) is an ﬁnite open cov ering of V b y aﬃne open formal subsc hemes sub ordinate to W . By the cosheaf axiom (12) and the assumption (1), w e ha ve a four-term righ t exact sequence of con tramo dule O U ( U )-mo dules (19) M 1 ≤ α<β <γ ≤ N P [ W α ∩ W β ∩ W γ ] − − → M 1 ≤ α<β ≤ N P [ W α ∩ W β ] − − → M N α =1 P [ W α ] − − → P [ U ] − − → 0 . By the cosheaf axiom (12), we also ha v e a right exact sequence of contramodule O U ( V )-mo dules (20) M 1 ≤ α<β ≤ N P [ V ∩ W α ∩ W β ] − − → M N α =1 P [ V ∩ W α ] − − → P [ V ] − − → 0 . All the terms of (19) are contraadjusted contramodule O U ( U )-mo dules b y Lemma 2.9.7. Since the class of contraadjusted contramodule R -mo dules is closed 66 under quotien ts in R – Contra for any adic top ological ring R , we obtain a four-term exact sequence of contramodule O U ( U )-mo dules (21) 0 − − → Q − − → M 1 ≤ α<β ≤ N P [ W α ∩ W β ] − − → M N α =1 P [ W α ] − − → P [ U ] − − → 0 . with a contraadjusted con tramo dule O U ( U )-mo dule Q . The contramodule O U ( U )-mo dule O U ( V ) is very ﬂat, so applying the functor Hom O U ( U ) ( O U ( V ) , − ) to the sequence (21), we obtain a four-term exact sequence of O U ( V )-mo dules 0 − − → Hom O U ( U ) ( O U ( V ) , Q ) − − → M 1 ≤ α<β ≤ N Hom O U ( U ) ( O U ( V ) , P [ W α ∩ W β ]) − − → M N α =1 Hom O U ( U ) ( O U ( V ) , P [ W α ]) − − → Hom O U ( U ) ( O U ( V ) , P [ U ]) − − → 0 . So, in particular, the sequence (22) − − → M 1 ≤ α<β ≤ N Hom O U ( U ) ( O U ( V ) , P [ W α ∩ W β ]) − − → M N α =1 Hom O U ( U ) ( O U ( V ) , P [ W α ]) − − → Hom O U ( U ) ( O U ( V ) , P [ U ]) − − → 0 . is righ t exact. The corestriction maps in the cosheaf P induce a natural morphism from the righ t exact sequence (20) to the right exact sequence (22). It remains to refer to the next Lemma 3.6.2 to the eﬀect that the map from (20) to (22) is an isomorphism on the middle and leftmost terms. Hence the natural map P [ V ] − → Hom O U ( U ) ( O U ( V ) , P [ U ]) on the righ tmost terms is an isomorphism, to o. □ Lemma 3.6.2. L et U b e an aﬃne No etherian formal scheme with an op en c overing W , let P b e a W -lo c al ly c ontr aher ent c oshe af of c ontr amo dules on U , let V ⊂ U b e an aﬃne op en formal subscheme, and let W ⊂ U b e an aﬃne op en formal subscheme sub- or dinate to W . Then the c or estriction map of O U ( W ) -mo dules P [ V ∩ W ] − → P [ W ] induc es an isomorphism of O U ( V ) -mo dules P [ V ∩ W ] − → Hom O U ( U ) ( O U ( V ) , P [ W ]) . Pr o of. This is the dual-analogous v ersion of Lemma 3.4.3. By the con traherence axiom (iv), the restriction map P [ V ∩ W ] − → P [ W ] induces an isomorphism of O U ( V ∩ W )-mo dules P [ V ∩ W ] ≃ Hom O U ( W ) ( O U ( V ∩ W ) , P [ W ]) . There is also an obvious isomorphism Hom O U ( U ) ( O U ( V ) , P [ W ]) ≃ Hom O U ( W ) ( O U ( V ) ⊗ O U ( U ) O U ( W ) , P [ W ]) . It remains to show that the map Hom O U ( W ) ( O U ( V ∩ W ) , P [ W ]) − − → Hom O U ( W ) ( O U ( V ) ⊗ O U ( U ) O U ( W ) , P [ W ]) induced b y the m ultiplication map of rings (23) O U ( V ) ⊗ O U ( U ) O U ( W ) − − → O U ( V ∩ W ) 67 is an isomorphism. Finally , the map (23) becomes an isomorphism after the adic completion functor Λ is applied, while the O U ( W )-mo dule P [ W ] is a (quotseparated) con tramo dule, and it remains to refer to Corollary 2.5.7 (whose ﬂatness assumption is ob viously satisﬁed for the O U ( W )-mo dule F = O U ( V ) ⊗ O U ( U ) O U ( W )). □ W e will denote the category of W -lo cally con traherent coshea v es of contramodules on X by X – Lcth W . The category of contraheren t cosheav es of contramodules on X will b e denoted by X – Ctrh = X – Lcth { X } . A cosheaf of con tramo dule O X -mo dules on X is said to b e lo c al ly c ontr aher ent if it is W -lo cally con traherent with resp ect to some op en cov ering W of the formal scheme X . The category of lo cally contraheren t coshea v es of con tramo dules on X will be denoted by X – Lcth = S W X – Lcth W . All inﬁnite direct pro ducts exists in the category X – Lcth W (in particular, in X – Ctrh ). The functors of cosections P 7− → P [ U ] ov er aﬃne op en formal subschemes U ⊂ X sub ordinate to W preserv e inﬁnite pro ducts. In fact, the condition that U is subordinate to W is not needed here; and the same assertion holds for all quasi-compact, quasi-separated open formal subschemes U ⊂ X . A short sequence of W -lo cally con traheren t coshea v es of con tramo dules 0 − → L − → M − → N − → 0 on X is said to b e ( admissible ) exact if the short sequence of (contramodule) O X ( U )-mo dules 0 − → L [ U ] − → M [ U ] − → N [ U ] − → 0 is exact for ev ery aﬃne op en formal subscheme U ⊂ X sub ordinate to W . By Lemma 2.13.10(a), it suﬃces to chec k this condition for aﬃne op en subschemes U b elonging to an y chosen aﬃne op en co v ering of X subordinate to W . Hence a short sequence of contraheren t coshea ves of con tramo dues on X is exact in X – Ctrh if and only if it is exact in X – Lcth W . More generally , let W ′ b e an op en co v ering of X sub ordinate to W . Then it follows from Lemma 2.13.10(a) that a short sequence in X – Lcth W is exact in X – Lcth W if and only if it is exact in X – Lcth W ′ . A short sequence is said to b e exact in X – Lcth if it is exact in X – Lcth W for some op en co v ering W of X . Lemma 3.6.3. (a) L et X b e a lo c al ly No etherian formal scheme with an op en c ov- ering W . Then the c ate gory X – Lcth W of W -lo c al ly c ontr aher ent c oshe aves of c on- tr amo dules on X , endowe d with the class of (admissible) short exact se quenc es sp e ci- ﬁe d ab ove, is an exact c ate gory (in the sense of Quil len [4] ). (b) L et U b e an aﬃne No etherian formal scheme. Then the functor of glob al c o- se ctions P 7− → P [ U ] establishes an e quivalenc e b etwe en the exact c ate gory U – Ctrh of c ontr aher ent c oshe aves of c ontr amo dules on U and the exact c ate gory O U ( U )– Contra cta of c ontr aadjuste d c ontr amo dules over the adic No etherian ring O U ( U ) . Pr o of. Both the assertions follow from Corollary 2.12.1 and Prop osition 2.12.3. □ It follows from Prop osition 3.6.1 that the full sub category X – Ctrh is closed under extensions in the exact category X – Lcth W . More generally , the full sub category X – Lcth W is closed under extensions in the exact category X – Lcth W ′ . F urthermore, it follo ws from Lemma 2.13.10(b) that the full sub category X – Ctrh is closed under k ernels of admissible epimorphisms in X – Lcth W . More generally , the full sub category X – Lcth W is closed under kernels of admissible epimorphisms in 68 X – Lcth W ′ . So a morphism in X – Lcth W is an admissible epimorphism in X – Lcth W if and only if it is an admissible epimorphism in X – Lcth W ′ . The similar assersions for admissible monomorphisms are not true [17, Example 3.2.1]. 3.7. Lo cally cotorsion lo cally con traheren t cosheav es. A W -lo cally contraher- en t cosheaf of contramodules P on X is said to b e lo c al ly c otorsion if the contramodule O X ( U )-mo dule P [ U ] is cotorsion for every aﬃne op en subsc heme U ⊂ X sub ordinate to W . By Prop osition 2.12.4 and Corollary 2.12.12, it suﬃces to chec k that condi- tion for aﬃne op en subsc hemes U b elonging to any chosen aﬃne op en co v ering of X sub ordinate to W . W e will denote the full sub category of lo cally cotorsion W -lo cally con traheren t coshea v es of con tramo dules on X b y X – Lcth lct W ⊂ X – Lcth W . The category of lo- cally cotorsion contraheren t coshea v es of contramodules on X will b e denoted by X – Ctrh lct = X – Lcth lct { X } . It follo ws from the previous paragraph that X – Ctrh lct = X – Ctrh ∩ X – Lcth lct W and X – Lcth lct W = X – Lcth W ∩ X – Lcth lct W ′ . The category of lo cally cotorsion lo cally contraheren t coshea ves of contramodules on X will b e denoted b y X – Lcth lct = S W X – Lcth lct W . Clearly , the full subcategory X – Lcth lct W is closed under extensions, cok ernels of admissible monomorphisms, and inﬁnite direct pro ducts in X – Lcth W . So the full sub categories X – Ctrh lct , X – Lcth lct W , and X – Lcth lct inherit exact category structures from the am bien t exact categories X – Ctrh , X – Lcth W , and X – Lcth . A morphism in X – Lcth lct W is an admissible monomorphism in X – Lcth lct W if and only if it is an admissible monomorphism in X – Lcth W . Lemma 3.7.1. L et U b e an aﬃne No etherian formal scheme. Then the e quivalenc e of exact c ate gories U – Ctrh ≃ O U ( U )– Contra cta fr om L emma 3.6.3(b) r estricts to an e quivalenc e U – Ctrh lct ≃ O U ( U )– Contra cot b etwe en the exact c ate gory of lo c al ly c otor- sion c ontr aher ent c oshe aves of c ontr amo dules on U and the exact c ate gory of c otorsion c ontr amo dules over the adic No etherian ring O U ( U ) . Pr o of. F ollows from Prop osition 2.12.4. □ Lemma 3.7.2. L et X b e a lo c al ly No etherian formal scheme with an op en c over- ing W . In this c ontext: (a) A c omplex P • in the exact c ate gory of W -lo c al ly c ontr aher ent c oshe aves of c ontr amo dules X – Lcth W is exact if and only if, for every aﬃne op en subscheme U ⊂ X sub or dinate to W , the c omplex of O X ( U ) -mo dules P • [ U ] is exact. It suﬃc es to che ck this c ondition for aﬃne op en subschemes U ⊂ X b elonging to any chosen op en c overing of X sub or dinate to W . (a) A c omplex P • in the exact c ate gory of lo c al ly c otorsion W -lo c al ly c ontr aher ent c oshe aves of c ontr amo dules X – Lcth lct W is exact if and only if, for every aﬃne op en subscheme U ⊂ X sub or dinate to W , the c omplex of O X ( U ) -mo dules P • [ U ] is exact. It suﬃc es to che ck this c ondition for aﬃne op en subschemes U ⊂ X b elonging to any chosen op en c overing of X sub or dinate to W . 69 Pr o of. This is similar to [17, Lemma 3.1.2]. The results of Lemmas 2.13.10(c) and 2.13.11(c) are relev an t, as w ell as the results of Prop osition 2.13.12. □ References [1] L. Alonso T arr ´ ıo, A. Jerem ´ ıas L´ opez, J. Lipman. Duality and ﬂat base change on formal sc hemes. In: Studies in dualit y on No etherian formal sc hemes and non-Noetherian ordi- nary schemes, p. 3–90, Contemp or ary Math. 244 , American Math. Society , Pro vidence, 1999. arXiv:alg-geom/9708006 Correction, Pr o c e e dings of the Americ an Math. So ciety 131 , #2, p. 351–357, 2003. arXiv:math.AG/0106239 [2] L. Alonso T arr ´ ıo, A. Jerem ´ ıas L´ opez, M. P´ erez Ro dr ´ ıguez, M. J. V ale Gonsalves. On the existence of a compact generator on the derived category of a No etherian formal scheme. Appl. Cate goric al Struct. 19 , #6, p. 865–877, 2011. arXiv:0905.2063 [math.AG] [3] A. Beilinson. Remarks on top ological algebras. Mosc ow Math. Journ. 8 , #1, p. 1–20, 2008. arXiv:0711.2527 [math.QA] [4] T. B ¨ uhler. Exact categories. Exp ositiones Math. 28 , #1, p. 1–69, 2010. [math.HO] [5] E. Eno c hs. Flat cov ers and ﬂat cotorsion mo dules. Pr o c. of the Amer. Math. So c. 92 , #2, p. 179–184, 1984. [6] E. Eno chs, S. Estrada. Relativ e homological algebra in the category of quasi-coherent sheav es. A dvanc es in Math. 194 , #2, p. 284–295, 2005. [7] W. Geigle, H. Lenzing. P erp endicular categories with applications to representations and shea ves. Journ. of Algebr a 144 , #2, p. 273–343, 1991. [8] A. Grothendiec k, J. Dieudonn ´ e. ´ El ´ emen ts de g ´ eom´ etrie alg ´ ebrique: II. ´ Etude globale ´ el ´ ementaire de quelques classes de morphismes. Public ations Mathematiques de l’IH ´ ES 8 , p. 5–222, 1961. [9] A. Grothendiec k, J. Dieudonn ´ e. ´ El ´ emen ts de g ´ eom´ etrie alg´ ebrique: IV. ´ Etude locale des sch ´ emas et des morphismes des sc h´ emas, Premi` ere partie. Public ations Mathematiques de l’IH ´ ES 20 , p. 5–259, 1964. [10] A. Grothendiec k, J. Dieudonn ´ e. ´ El ´ emen ts de g ´ eom´ etrie alg´ ebrique: IV. ´ Etude locale des sch ´ emas et des morphismes des sch ´ emas, Quatri ` eme partie. Public ations Mathematiques de l’IH ´ ES 32 , p. 5–361, 1967. [11] A. Grothendieck, J. A. Dieudonn ´ e. ´ El ´ emen ts de g ´ eom ´ etrie alg ´ ebrique. I. Grundlehren der math- ematisc hen Wissenschaften, 166. Springer-V erlag, Berlin–Heidelb erg–New Y ork, 1971. [12] R. Hartshorne. Algebraic geometry . Graduate T exts in Math., 52, Springer-V erlag, New Y ork– Heidelb erg, 1977. [13] A. J. de Jong et al. The Stac ks Pro ject. Av ailable from https://stacks.math.columbia.edu/ [14] M. Porta, L. Shaul, A. Y ekutieli. On the homology of completion and torsion. Algebr as and R epr esent. The ory 17 , #1, p. 31–67, 2014. arXiv:1010.4386 [math.AC] . Erratum in Algebr as and R epr esent. The ory 18 , #5, p. 1401–1405, 2015. arXiv:1506.07765 [math.AC] [15] L. Positselski. Homological algebra of semimodules and semicon tramodules: Semi-inﬁnite homological algebra of associative algebraic structures. Appendix C in collab oration with D. Rum ynin; Appendix D in collab oration with S. Arkhip o v. Monograﬁe Matemat yczne v ol. 70, Birkh¨ auser/Springer Basel, 2010. xxiv+349 pp. arXiv:0708.3398 [math.CT] [16] L. Positselski. W eakly curv ed A ∞ -algebras o ver a topological lo cal ring. M´ emoir es de la So ci ´ et´ e Math ´ ematique de F r anc e 159 , 2018. vi+206 pp. arXiv:1202.2697 [math.CT] [17] L. Positselski. Con traherent coshea ves on schemes. Electronic preprint [math.CT] . [18] L. Positselski. Contramodules. Conﬂuentes Math. 13 , #2, p. 93–182, 2021. [math.CT] 70 [19] L. Positselski. Dedualizing complexes and MGM duality . Journ. of Pur e and Appl. Algebr a 220 , #12, p. 3866–3909, 2016. arXiv:1503.05523 [math.CT] [20] L. Positselski. Con traadjusted mo dules, contramodules, and reduced cotorsion mo dules. Mosc ow Math. Journ. 17 , #3, p. 385–455, 2017. arXiv:1605.03934 [math.CT] [21] L. Positselski. Ab elian righ t perp endicular subcategories in module categories. Electronic preprin t arXiv:1705.04960 [math.CT] . [22] L. Positselski. Flat ring epimorphisms of countable t yp e. Glasgow Math. Journ. 62 , #2, p. 383– 439, 2020. arXiv:1808.00937 [math.RA] [23] L. P ositselski. Con tramo dules o ver pro-perfect topological rings. F orum Mathematicum 34 , #1, p. 1–39, 2022. arXiv:1807.10671 [math.CT] [24] L. Positselski. Remarks on derived complete mo dules and complexes. Math. Nachrichten 296 , #2, p. 811–839, 2023. arXiv:2002.12331 [math.AC] [25] L. Positselski. Semi-inﬁnite algebraic geometry of quasi-coherent shea v es on ind-sc hemes: Quasi-coheren t torsion shea v es, the semideriv ed category , and the semitensor product. Birkh¨ auser/Springer Nature, Cham, Switzerland, 2023. xix+216 pp. [math.AG] [26] L. Positselski. Lo cal, colo cal, and antilocal prop erties of mo dules and complexes o ver commu- tativ e rings. Journ. of Algebr a 646 , p. 100–155, 2024. arXiv:2212.10163 [math.AC] [27] L. Positselski. Flat como dules and con tramo dules as directed colimits, and cotorsion p erio d- icit y . Journ. of Homotopy and R elate d Struct. 19 , #4, p. 635–678, 2024. [math.RA] [28] L. Positselski. Philosophy of contraheren t cosheav es. Electronic preprint [math.AG] . [29] L. Positselski. D -Ω duality on the contra side. Electronic preprin t [math.AG] . [30] L. P ositselski. T orsion modules and diﬀeren tial operators in inﬁnitely man y v ariables. Electronic preprin t arXiv:2505.07739 [math.AC] . [31] L. P ositselski. Pseudo-dualizing complexes of torsion mo dules and semi-inﬁnite MGM dualit y . Electronic preprin t arXiv:2511.04571 [math.AC] . [32] L. Positselski. Homomorphisms of top ological rings and change-of-scalar functors. Electronic preprin t arXiv:2603.15048 [math.RA] . [33] L. P ositselski, J. Rosic k ´ y. Cov ers, en v elop es, and cotorsion theories in locally presen table ab elian categories and con tramo dule categories. Journ. of Algebr a 483 , p. 83–128, 2017. arXiv:1512.08119 [math.CT] [34] L. Positselski, A. Sl´ avik. Flat morphisms of ﬁnite presentation are very ﬂat. Annali di Matem- atic a Pur a e d Applic ata 199 , #3, p. 875–924, 2020. arXiv:1708.00846 [math.AC] [35] L. Positselski, J. ˇ St ’o v ´ ıˇ cek. The tilting-cotilting corresp ondence. Internat. Math. R ese ar ch No- tic es 2021 , #1, p. 189–274, 2021. arXiv:1710.02230 [math.CT] [36] L. Positselski, J. ˇ St ’o v ´ ıˇ cek. T op ologically semisimple and top ologically perfect top ological rings. Public acions Matem` atiques 66 , #2, p. 457–540, 2022. arXiv:1909.12203 [math.CT] [37] A.-M. Simon. Approximations of complete mo dules b y complete big Cohen–Macaulay mo dules o ver a Cohen–Macaula y local ring. A lgebr as and R epr esent. The ory 12 , #2–5, p. 385–400, 2009. [38] A. Sl´ avik, J. T rlifa j. V ery ﬂat, lo cally v ery ﬂat, and contraadjusted mo dules. Journ. of Pur e and Appl. Algebr a 220 , #12, p. 3910–3926, 2016. arXiv:1601.00783 [math.AC] [39] B. Stenstr¨ om. Rings of quotien ts. An In tro duction to Methods of Ring Theory . Die Grundlehren der Mathematisc hen Wissenschaften, Band 217. Springer-V erlag, New Y ork, 1975. [40] A. Y ekutieli. On ﬂatness and completion for inﬁnitely generated mo dules o v er no etherian rings. Communic at. in Algebr a 39 , #11, p. 4221–4245, 2011. arXiv:0902.4378 [math.AC] [41] A. Y ekutieli. Flatness and completion revisited. A lgebr as and R epr esent. The ory 21 , #4, p. 717– 736, 2018. arXiv:1606.01832 [math.AC] 71 Institute of Ma thema tics, Czech Academy of Sciences, ˇ Zitn ´ a 25, 115 67 Prague 1, Czech Republic Email addr ess : positselski@math.cas.cz 72

Contraherent cosheaves of contramodules on Noetherian formal schemes

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment