Colocalization functors in derived categories and torsion theories

COLOCALIZA TION FUNCTORS IN DERIV E D CA TEGORIES AND TORSION THEORIE S SHOHAM SHAMIR Abstract. Let R b e a r ing and let A b e a heredita ry torsion class of R -mo dules. The inclusion of the lo calizing sub catego ry generated b y A into th e derived category of R has a rig ht adjoint, denoted Ce ll A . In [2 ], Benson shows how to c o mpute Cell A R when R is a gro up ring of a finite gro up o ver a prime field a nd A is the hereditar y torsion class genera ted b y a simple mo dule. W e generalize Benso n’s construction to the case where A is an y hereditary torsio n class on R . It is shown that for every R - module M there exists an injective R -mo dule E s uc h that: H n (Cell A M ) ∼ = Ext n − 1 End R ( E ) (Hom R ( M , E ) , E ) for n ≥ 2 1. Introd uction Let R b e a ring and let D R b e the (un b ounded) deriv ed category of c hain complexes of left R -mo dules. Fix a class A of ob jects of D R . W e recall some definitions o f Dwy er and Gr eenlees from [3]. An ob ject N of D R is A -nul l if Ext ∗ R ( A, N ) = 0 for ev ery A ∈ A . An ob ject C of D R is A -c el lular if Ext ∗ R ( C , N ) = 0 for ev ery A -null N . An A -cellular ob ject C is an A -c el lular app r oximation of X ∈ D R if there is a map µ : C → X suc h that Ext ∗ R ( A, µ ) is an isomorphism for all A ∈ A . Finally , a n A -n ull ob ject N is an A -nul lific ation of X if there is a map ν : X → N whic h is univ ersal among maps in D R from X to A -n ull ob j ects. Denote an A - cellular appro ximation of X b y Cell A X and an A -n ullification of X b y Null A X . The follo wing prop erties are easy to chec k. A map µ : C → X is an A -cellular appro ximation of X if and only if it is univ ersal among a ll maps fro m A - cellular ob jects to X . There is an exact triangle Cell A X → X → Null A X whenev er Cell A X o r Null A X exists. An A - cellular approxim ation of some ob ject X is unique up to isomorphism and the same go es for an A -nullification of X . No w supp ose A is a set, then it turns out that the full sub category of A -cellular ob jects is the lo calizing sub category generated by A ( see [4 ] and [7 , 5.1.5]). Moreov er, when A is a set the inclusion functor o f the full sub categor y of A -cellular ob jects into D R has a righ t adjoint, whic h is Cell A X for ev ery X ∈ D R , see [7] o r [6]. Hence Cell A can b e constructed as a c olo c alization functor (the righ t adjoint of an inclusion functor), and it follo ws that A -cellular appro ximation and A -nullification exist for an y ob ject of D R . Similarly , when A is a set t here exists a left adjoint t o the inclusion of the full sub- category of A -n ull ob jects, see for example Neeman’s b o ok [10, Section 9]. This functor is in fact A -nullification and it is a lo c alization functor (the left a djoin t to an inclusion functor). One metho d for calculating A -cellular appro ximations is the formula giv en b y Dwy er and Greenlees in [3 ], whic h holds whenev er A = { A } and A is a p erfect complex. This w as later g eneralized by Dwye r, G reenlees and Iy engar in [4]. A new metho d for calculat- ing the A -cellular approx imation for R -mo dules has b een constructed b y Benson in [2], dubb ed k -sque eze d r esolutions . This metho d can b e a pplied whenev er A is a set of simple Date : June 1 1 , 2 0 18. 1 mo dules and R is an Artinian ring. O ne ma jor b enefit of Benson’s construction is that it allow s for explicit calculations. As w e will see, it is more natural to use Benson’s metho d to construct the A -n ullification of a mo dule, ra t her than its A -cellular appro ximation. W e generalize Benson’s construc- tion so that it applies whenev er A is a hereditary torsion class of mo dules. A her e ditary torsion class of mo dules is a class o f mo dules that is closed under submo dules, quotien t mo dules, copro ducts and extensions. The main result of this pap er is the follow ing. Theorem 1.1. L et T b e a her e ditary torsion class on left R -mo dules. F or every left R -mo dule M ther e exists an inje ctive left R -m o dule E such that the c om p lex R Hom End R ( E ) (Hom R ( M , E ) , E ) is a T -nul lific ation of M . In p articular, the diff e r ential gr ade d alge b r a R End End R ( E ) ( E ) is a T -nul li fic ation of R . The form ula giv en in the abstract follo ws immediately from t he distinguished triangle Cell T M → M → Null T M men tioned a b o v e. The la y out of this pap er is as fo llo ws. The necessary bac kground on hereditary tor- sion classes and the ba c kground on cellular appro ximations and n ullifications is giv en in Section 2. In Section 3 w e describe the construction of n ullification with resp ect to a hereditary tor sion class and prov e Theorem 1.1. W e study the case where R is an Ar- tinian ring in Section 4 . This section offers a different pro of t o a result of Benson ([2, Theorem 5.1]). Finally , Section 5 pro vides sev eral examples. 1.A. Notation and T erminology. By a r ing we alw a ys mean an asso ciativ e ring with a unit, no t necessarily comm utativ e. Unless otherwise noted all mo dules considered are left mo dules. A triangle alwa ys means an exact (distinguished) triangle in the un b o unded deriv ed category of left R -mo dules, denoted D R . A complex is alw ays a c hain- complex of R - mo dules. F or complexes we use the standard conv en tion that subscript grading is the negativ e of the superscript grading, i.e.  − i =  i . It is take n for grante d that ev ery R -mo dule is a complex concen trated in degree 0 and with zero differen tial. A complex X is b ounde d-ab ove if for some n and fo r all i > n , H i ( X ) = 0. F or complexes X and Y the notation Hom R ( X , Y ) stands for the usual chain complex of homomorphisms. The notation R Hom R ( − , − ) stands for the deriv ed functor o f the Hom R ( − , − ) functor . By End R ( M ) w e mean the endomorphisms ring of a n R -mo dule M . The sym b ol ≃ stands for quasi-isomorphism of complexes. 2. Back ground on Heredit ar y Torsion Theories and Cellular-Appro xima tion, Nullifica tion and Completion 2.A. Hereditary T or sion Theories. Belo w is a recollection of the definition and main prop erties of hereditary torsion theories. A thoro ug h review of this material can b e found in [11]. Definition 2.1. A he r e ditary torsion clas s T is a class of R -mo dules that is closed under submo dules, quotien t mo dules, copro ducts and extensions. Closure under extensions means that if 0 → M 1 → M 2 → M 3 → 0 is a short exact sequence with M 1 and M 3 in T , then so is M 2 . The mo dules in T will b e called T -torsion mo dules (or just torsion mo dules when the torsion theory is clear from the con text). The class of torsion-fr e e mo dules F is the class of a ll mo dules F s atisfying Hom R ( C , F ) = 0 fo r ev ery C ∈ T . The pair ( T , F ) is referred to as a her e ditary torsion the ory . T o ev ery hereditary torsion theory there is an asso ciated radical t , where t ( M ) is the maximal t o rsion submo dule of M . Note that M /t ( M ) is therefore torsion-free. 2 Ev ery hereditary torsion class T has an inje ctive c o gener ator (see [11 , VI.3.7]). T his means there exists an injectiv e mo dule E suc h that a mo dule M is torsion if and only if Hom R ( M , E ) = 0. It is also imp orta nt to note that in a ny hereditary torsion theory , the class of torsion-free mo dules is closed under injectiv e h ulls (see [11, VI.3.2 ]) . Th us, if F is a torsion-free mo dule then the injectiv e h ull of F is also to rsion-free. Definition 2.2. Let ( T , F ) b e a hereditary torsion theory and let t b e the a sso ciat ed radical. An R -mo dule M is called F -close d if, for ev ery left ideal a ⊂ R suc h tha t R/ a ∈ T , the induced map M = Ho m R ( R, M ) → Ho m R ( a , M ) is an isomorphism. The inclusion of the full sub category of F -closed mo dules has a left adjoint M 7→ M F . The mo dule M F is called the m o dule of q uotients o f M (see [11, IX.1]). The unit of this adjunction has the followin g prop erties: the kerne l of the map M → M F is t ( M ), M F is torsion-free and the cok ernel of this map is a torsion mo dule. 2.B. Cellular-Appro ximation, Nullification and Completion. The follo wing re- calls the basic prop erties of cellular approximation, as well as the definition of completion giv en b y Dwy er and Greenlees in [3]. Definition 2.3. Let R b e a ring and let A b e a class of R -complexes. W e sa y an R - complex X is A -c omplete if Ext ∗ R ( N , X ) = 0 fo r any A -n ull ob ject N . An R -complex C is an A -c ompletion o f X if C is A -complete and there is an A -equiv alence X → C . It is easy to see that an A - completion of a complex X is unique up to an isomorphism in D R . As in [3], w e denote an A -completion of X b y X ∧ A . The following criterion fo r n ullification is usually easier to che c k than the orig inal definition. Its pro of is easy a nd therefore omitted. Lemma 2.4. L et R b e a ring and let A b e a class of R -c ompl e xes. A c omplex N is an A -nu llification of X if ther e is a triangle C → X → N such that C is A -c el lular and N is A -nul l. I n this c ase it also fol lows that C an A -c el lular appr oximation of X . Recall that when A is a set, the full sub category of A -cellular ob jects o f D R is the lo calizing sub category of generated by A . The lo c alizing c ate gory gener ate d by A , denoted hAi , is the smallest full triangulated sub category of D R that is closed under triangles, direct sums and retracts. C losure under triangles means that f o r ev ery distinguished triangle in D R , if t w o of the ob jects are in t he lo calizing sub category , so is the third. The pro of of the follow ing lemma is clear. Lemma 2.5. L et A b e a class of R -c om p lexes, then every obje ct of hAi is A -c el lular. If B is another class of R -c omplexes such that hAi = hB i , then A -c el lular appr oximation is the same as B -c el lular appr oximation. Remark 2.6. In [3] A -cellular complexes w ere called A - torsio n while the t erm A -c el lular w as reserv ed for complexes in hAi . When T is a hereditary torsion theory , the t w o terms agree (b y Lemma 2 .8 b elow). 2.C. Cellular-Appro ximation with r esp ec t t o a Hereditary T orsion Theory . Let T b e a hereditary torsion class. It is not immediately apparent t hat T -cellular appro x- imation exists. Below, in Lemma 2.8, w e sho w that hT i is the same as the lo calizing sub category generated b y a set A T . This immediately implies that T -cellular approxi- mation and T -nu llification exist for any R - complex, see Corollary 2.9. Definition 2.7. Giv en a hereditary torsion class of R -mo dules T , w e denote b y A T the set of all cyclic T -torsion mo dules. 3 Lemma 2.8. L et T b e a her e ditary torsion class, then every T -torsion mo dule is A T - c el lular and hen c e hT i = hA T i . Pr o of. Clearly , ev ery cyclic T -torsion mo dule is A T -cellular. Therefore ev ery direct sum of cyclic T -torsion mo dules is A T -cellular. Let M b e a T -torsion mo dule, then there is a surjection C ( M ) = ⊕ m ∈ M R/ ann( m ) ։ M . Since eve ry hereditary to r sion theory is closed under submo dules, R / ann( m ) is T -torsion for ev ery m ∈ M . Clearly C ( M ) is A T - cellular and T -torsion. Next w e build a resolution X o f M using A T -cellular modules. Let X 0 = C ( M ), and let d 0 : X 0 ։ M the map defined ab o v e. The ke rnel of d 0 is T - torsion, so there is an epimorphism C ( ker( d 0 )) ։ k er( d 0 ). Let X 1 = C (k er ( d 0 )) and let d 1 b e the comp osition X 1 ։ k er ( d 0 ) ֒ → X 0 . In this w ay X is built inductiv ely and it is clear that X is quasi-isomorphic to M . By construction, X is in the lo calizing sub category generated b y A T .  Corollary 2.9. L et T b e a her e ditary torsion cla s s , then T -c el l ular a ppr oximation, T - nul lific ation and T -c ompletion ex ist for every c omplex. Mor e over, a c omplex X is T - c el lular if and only if X ∈ h T i . Pr o of. Lemma 2.8 implies that T -cellular approximation is the same as A T -cellular ap- pro ximation. As men tioned in Section 1, A T -cellular appro ximation exists for ev ery complex. The pro of of the other claims is similar.  Lemma 2.10. L et T b e a h e r e ditary torsion class. (1) If X is a T -c el lular c om p lex then the homolo gy gr oups o f X ar e T -torsion R - mo dules. (2) If X is a b ounde d-ab ove c omplex such that the homolo gy gr oups of X ar e T -torsion then X is T -c el lular. Pr o of. Let C b e the full sub category of D R con taining all ob jects whose homology groups are T - torsion R -mo dules. The prop erties of a hereditary torsion theory show that C is lo calizing sub category . Since C con tains T , then C also contains hT i . This pro ves the first statemen t. No w suppo se X is a b o unded-ab o v e complex and that H i X ∈ T for all i . Because X is b ounded-ab ov e, X b elongs to the lo calizing sub category generated by the homology groups of X (see for example [3, 5.2 ]). Since the homology groups of X all b elong to hT i , so do es X .  Remark 2.11. If R is a comm utative No etherian ring, then a complex X is T -cellular if and only if all the homology groups of X are T -torsion. This easily follow s from a result of Neeman [9, Theorem 2 .8]. How ev er, Example 5.2 sho ws a noncomm utative ring R a nd a complex X suc h that H i ( X ) is T -tor sion for all i but X is not T - cellular. 3. Nullifica tion Constr uction In [2], Benson giv es a construction called k -sque eze d r esolution which yields k -cellular appro ximations o v er the ring k G , where k is a prime field and G is a finite group. W e generalize Benson’s construction so as to pro duce T -cellular approx imations ov er a n y ring R , where T is a hereditary torsion class. In fact, w e giv e tw o isomorphic constructions. Nullification C onstr uction 3.1. Let ( T , F ) be a hereditary torsion theory with radical t . F or an R - mo dule M we construct the T -nullific ation of M as a co chain complex I 0 d − → I 1 d − → I 2 d − → · · · inductiv ely . 4 Let M 0 = M , let F 0 = M 0 /t ( M 0 ) and let I 0 b e the injectiv e hu ll of F 0 . Note that since F 0 is torsion-free, so is I 0 . W e pro ceed b y induction, set M n +1 = I n /F n , F n +1 = M n +1 /t ( M n +1 ) and let I n +1 b e the injectiv e h ull of F n +1 . Aga in I n +1 is tor sion-free b ecause F n +1 is. The differen tia l d : I n → I n +1 is the comp osition I n → M n +1 → F n +1 → I n +1 . The image o f d : I n → I n +1 is F n +1 and therefore d ◦ d = 0. D enote the resulting complex by I . The natura l map M → I 0 extends to a map of complexes M → I . Nullification Construction 3.2. F or an R - mo dule M w e construct a co chain complex J 0 d 0 − → J 1 d 1 − → J 2 d 2 − → · · · inductiv ely . Let Q 0 = M , let N 0 = ( Q 0 ) F and let J 0 b e the injectiv e h ull of N 0 . Denote by d − 1 the map M → J 0 . Now pro ceed b y induction, let Q n +1 = J n / im( d n − 1 ) , N n +1 = ( Q n +1 ) F and let J n +1 b e the injectiv e hull of J n . The differential d n : J n → J n +1 is the comp osition J n → Q n +1 → N n +1 → J n +1 . Clearly , d n +1 ◦ d n = 0. Denote the resulting complex b y J . The natura l map M → J 0 extends to a map of complexes M → J . Note that for ev ery n , J n is torsion-free b ecause N n is. Lemma 3.3. L et J b e the c omplex c onstructe d fr om M in 3.2, then H 0 ( J ) ∼ = M F . Pr o of. It easily follows from the definition of an F -closed mo dule that an y injective torsion-free mo dule is F -closed, therefore J 0 is F -closed. F or any F -closed mo dule K there is an isomorphism K ∼ = K F (see [11, pag e 198]), therefore ( J 0 ) F ∼ = J 0 and ( M F ) F ∼ = M F . The mo dule of quotien ts functor is left exact ( see [11, page 19 9]). Hence a pplying the mo dule of quotien ts functor to the sequence M F → J 0 → J 0 / M F yields an exact sequence : 0 → M F → J 0 → ( J 0 / M F ) F W e see that J 0 / M F is torsion-f ree, b ecause it is isomorphic to a submo dule of the torsion- free mo dule ( J 0 / M F ) F . No w consider t he short exact seque nce M F / im( M ) → Q 1 → J 0 / M F The mo dule M F / im( M ) is a torsion mo dule (see D efinition 2.2), while the mo dule J 0 / M F is torsion free. F rom the definition of the radical t it follo ws tha t M F / im( M ) ∼ = t ( Q 1 ). Therefore M F is the k ernel of J 0 → N 1 and the pro of is complete.  Lemma 3.4. L et M b e an R -mo dule, let I b e the c omp l e x c onstructe d fr om M in 3.1, let C b e a c omplex such that C → M → I is a distinguishe d triang l e and let J b e the c omplex c onstructe d fr om M in 3.2. Then C is a T -c el lular appr oxima tion of M and b oth I and J ar e T -nul lific ations of M . I n p articular, H 0 (Null T M ) ∼ = M F for any R -m o dule M . Pr o of. W e can choose C to b e the complex M → I 0 → I 1 → · · · with M in degree 0. The homolog y of C is easy to compute: H n ( C ) = t ( M n ), with M n as defined in the Nullification Construction 3.1 ab ov e. By Lemma 2.10, the complex C is T -cellular. The complex I is T -n ull, simply b ecause I is comp osed of torsion-free injectiv e mo dules. Th us, by Lemma 2.4, I is a T -n ullification of M and C is a T -cellular approx imation of M . Similar reasoning shows tha t J is a T -nullific ation of M . T he complex J is T -n ull, simply b ecause J is comp osed of torsion-free inj ectiv e mo dules. The homology of J is: 5 H n ( J ) = t ( Q n ) for n > 0 and H 0 ( J ) = M F . Let C ′ b e a complex suc h that there is a distinguished triangle C ′ → M → J . Th e long exact sequenc e in homology yields: H 0 ( C ′ ) = t ( M ) , H 1 ( C ′ ) = M F / M and H n ( C ′ ) = H n − 1 ( J ) for n > 1. Note that M F / M is a T -torsion mo dule. By Lemma 2.10, the complex C ′ is T -cellular and hence J is a T -nullific ation of M and C ′ is a T -cellular approximation of M . In particular, H 0 (Null T M ) = H 0 ( J ) ∼ = M F .  Remark 3.5. It follo ws that the complexes I and J in Lemma 3.4 a r e isomorphic in the derived category of R . In fa ct, they are isomorphic as complexes. T o construct this isomorphism one needs the following prop ert y: for an y R -mo dule L the injectiv e hull of L F and the injectiv e-hull o f L/t ( L ) are the same; this is b ecause ( L/t ( L )) F = L F and L/t ( L ) is an essen tial submodule of ( L/t ( L )) F (see [11, IX.2.4]) . Using the aforemen tioned prop ert y it is a simple exercise to construct the isomorphism inductiv ely . Using the construction a b o ve w e can giv e a differen t description of T -n ullificatio n, the one sho wn in Theorem 1.1 . Before proving Theorem 1.1 it is necessary t o note some prop erties of the functor Hom R ( − , E ). Let E b e a n R - mo dule a nd let E b e the endomorphism ring End R ( E ) = Hom R ( E , E ). The functor Hom R ( − , E ) is a con tr av arian t functor from left R - complexes to left E - complexes. This left E - action is simply comp osition on the left with the morphisms in E . In other w ords, the left E -action on Hom R ( − , E ) is induced by the left E -action on E itself. Moreo v er, the functor Hom E ( − , E ) is a con trav arian t functor, this time from left E -complexes to left R -complexes. Here the left R -action on Hom E ( − , E ) comes from the left R -action on E (whic h commutes with the left E -action on E ) . In particular, there is a deriv ed v ersion of this functor: R Hom E ( − , E ) : D E → D R . Pr o of of The or em 1.1. Giv en an R -mo dule M , construct a T -n ullification of M in the w ay prescrib ed in 3 .1 . This construction results in a co c hain complex I , with I n b eing an injectiv e torsion-f r ee mo dule. Let E b e a torsion-free injectiv e R -mo dule suc h that for ev ery n , I n is a direct summand of a finite direct sum of copies of E . F or example, one can tak e E to b e the pro duct Q n I n . Denote b y E t he endomorphism ring End R ( E ). No w consider the triangle C → M → I . Since I is a T -n ullification of M , C is a T -cellular a ppro ximation of M . Applying the f unctor Hom R ( − , E ) to this triangle yields a triangle in D E : Hom R ( I , E ) → Hom R ( M , E ) → Hom R ( C , E ) Since E is injectiv e, H i (Hom R ( C , E )) ∼ = Hom R ( H i ( C ) , E ). Since the homology gr o ups of C are torsion, Hom R ( H i ( C ) , E ) = 0. Therefore the map Hom R ( I , E ) → Hom R ( M , E ) is a quasi-isomorphism of E -complexes. Because I n is a direct summand of a finite direct sum of copies of E , the E - mo dule Hom R ( I n , E ) is pro jectiv e. Th us the map Hom R ( I , E ) → Hom R ( M , E ) is a pro jectiv e resolution of Ho m R ( M , E ) in t he catego ry of E -mo dules. W e conclude that t he complex Hom E (Hom R ( I , E ) , E ) is the deriv ed functor R H om E (Hom R ( M , E ) , E ). Because I n is a direct summand of a finite direct sum of copies o f E , one readily sees that the R -mo dule Hom E (Hom R ( I n , E ) , E ) is naturally isomorphic to I n and therefore Hom E (Hom R ( I , E ) , E ) ∼ = I .  Remark 3.6. As noted in Theorem 1 .1, Null T R ≃ R End E ( E ) and therefore R F ∼ = H 0 (Null T R ) ∼ = H 0 ( R End E ( E )) = End E ( E ) 6 This isomorphism r ecov ers [11, IX.3.3], where it is stated that there is an injectiv e R - mo dule E suc h t ha t R F ∼ = End End R ( E ) ( E ). Also note that Null T R is quasi-isomorphic to a differen tia l gra ded algebra. This also follow s from a result of Dwy er [5, Prop osition 2.5], where it is show n that for an y set of complex es A , the complex Null A R is quasi-isomorphic to a differen tial g raded algebra. Remark 3.7. Let X = · · · → X n → X n − 1 → · · · b e a complex suc h tha t there exists some m for whic h X n = 0 for all n > m . Then it is p ossible to generalize the Nullification Construction 3.1 to giv e the T -n ullification of X . Moreo ve r, this generalized construction of Null T X can b e done in suc h a w ay that for n > m (Null T X ) n = 0, while for n ≤ m (Null T X ) n is a finite direct sum of to r sion- free inj ective mo dules. Therefore (Null T X ) n is itself a torsion-free injectiv e for n ≤ m . Now it is easy to see that the pro of of Theorem 1.1 w orks fo r Null T X as we ll and yields the same result. Namely , there exists an injectiv e R -mo dule E suc h that Null T X ≃ R Ho m End R ( E ) (Hom R ( X , E ) , E ) Clearly , this result carries ov er to any b ounded-ab ov e complex X . Sa y an injectiv e mo dule E is sufficient to c ompute the T -nul lific ation of M if Null T M ≃ R H om E (Hom R ( M , E ) , E ) where E = End R ( M ). Giv en an R -mo dule M one can use the pro of of Theorem 1.1 to construct an injective mo dule E whic h is sufficien t to compute the T -n ullification of M . Ho wev er there are other injectiv e mo dules sufficien t to compute the T - n ullification of M , as sho wn b y the following prop o sition. Prop osition 3.8. L et M b e an R -mo dule and let E b e an i n je ctive c o gener ator o f T . Denote by E the ring End R ( E ) . (1) If the E -mo dule Hom R ( M , E ) has a r esolution c omp ose d of finitely gener ate d pr o- je ctive mo dules in e ach de gr e e, then E is sufficient to c ompute the T -nul lific ation of M . (2) Ther e e x ists an or dinal α such that the mo dule E ′ = Q i<α E is sufficient to c ompute the T -nul lific ation of M . Pr o of. Let P b e a finitely generated pro jectiv e E -mo dule, then it is easy to see that Hom R (Hom E ( P , E ) , E ) is naturally isomorphic to P . No w let F b e a pro jectiv e resolution of Hom R ( M , E ) o v er E and assume F is comp osed of finitely generated pro jectiv e mo dules in eac h degree. Then Ho m R (Hom E ( F , E ) , E ) is naturally isomorphic to F . The quasi-isomorphism η : F → Hom R ( M , E ) induces a map Hom E (Hom R ( M , E ) , E ) → Hom E ( F , E ). Comp osing with the na t ur a l map M → Hom E (Hom R ( M , E ) , E ) yields a map µ : M → Hom E ( F , E ). It is easy to see that Hom R ( µ, E ) is the quasi-isomorphism η . Consider the triangle C → M µ − → Hom E ( F , E ). Clearly , Hom E ( F , E ) is T -null. Since Hom R ( µ, E ) is a quasi-isomorphism, Hom R ( C , E ) is quasi-isomorphic to zero. This im- plies Hom R ( H i ( C ) , E ) = 0 for all i . Since E is a n inj ective cogenerator for T , H i ( C ) is torsion for all i . Clearly C is b o unded-a b o ve a nd so, b y Lemma 2.10, C is T -cellular. W e conclude that Hom E ( F , E ) is a T -n ullfication of M and E is sufficien t to compute the T -n ullification of M . W e no w turn our atten tio n to the second item in the pro p osition. By [11, VI.3.9], ev ery torsion-fr ee mo dule has a monomorphism to some direct pro duct of copies of E . In particular, every torsion-free injectiv e is a isomorphic to a direct summand of some direct pro duct of copies of E . 7 Let I b e the complex describ ed in the Nullification Construction 3 .1 . Let E ′ b e a direct pro duct o f copies of E suc h that for ev ery n , I n is isomorphic to a direct summand of E ′ . Clearly the End R ( E ′ )-complex Hom R ( I , E ′ ) is a pro jectiv e resolution of Hom R ( M , E ′ ) whic h is comp osed of finitely generated pro jectiv e mo dules in ev ery degree. Hence E ′ is sufficien t to compute the T -n ullification o f M .  4. Torsion Theories and Cellular Appro xima tion in Ar tinian Rings Throughout this section R is an Artinian ring and S is a set of non-isomorphic simple mo dules of R . Define a class F of R -mo dules by F = { F | Hom R ( S, F ) = 0 fo r all S ∈ S } and define a class T by setting T = { M | Hom R ( M , F ) = 0 for all F ∈ F } . By [11, VI I I.3], the pair ( T , F ) forms a hereditary to r sion theory (a lt ernat iv ely , one can easily deduce this f rom Lemma 4.3 b elow ). Because R is Artinian, ev ery hereditary torsion theory of R -mo dules is generated b y a set of simple mo dules (see [11, VI I I]), so this con text co vers all hereditary torsion theories o v er R . In this section w e giv e sev eral results r ega rding T -n ullification. W e also giv e a differen t pro of for a result of Benson [2, Theorem 5.1] in Corollary 4.5. Let Ω b e the set o f isomorphism classes of simple mo dules of R and let S ′ b e the complemen t o f S in Ω. W e denote b y E the pro duct of the injectiv e h ulls of t he simple mo dules in S ′ and denote by P the direct sum of the pro jectiv e co v ers of t hose simple mo dules. W e sho w that E is an injectiv e cogenerator of T and tha t b eing T -cellular is the same as the b eing S -cellular. Lemma 4.1. L et C b e a cyclic R -mo dule such that Hom R ( C , E ) = 0 , then C is S -c el lular. Pr o of. Since R is Artinia n, C admits a comp osition series 0 = C 0 ⊂ C 1 ⊂ · · · ⊂ C m = C , where a ll the quotients C i /C i − 1 are simple mo dules. W e next show that C i /C i − 1 ∈ S for all i . Supp o se that for some i , C i /C i − 1 ∼ = S ′ for some S ′ ∈ S ′ . Let x ∈ C i r C i − 1 , then the cyclic mo dule generated b y x has S ′ as a quotien t. This implies the submodule Rx of C has a non-zero map to E ( S ′ ) - the injectiv e hull of S ′ . Clearly suc h a map can b e lifted to a non-zero map C → E , in con tradiction. Therefore C i /C i − 1 ∼ = S f or some S ∈ S . No w a simple inductiv e argumen t on i show s that C i ∈ h S i for ev ery i , and hence C is S -cellular.  Corollary 4.2. A c omplex X is T -c el lular if a n d only if X is S -c el lular. Pr o of. W e need to show that hT i = h S i . Since S ⊂ T , w e only need to sho w that T ⊂ h S i . By Lemma 2 .8 it is enough to show that ev ery cyclic R - mo dule is S -cellular, but that is immediate from Lemma 4.1.  Lemma 4.3. The mo dule E i s an inje ctive c o gener ator for T . Pr o of. Let U b e the class of mo dules M suc h tha t Hom R ( M , E ) = 0. The n U is a hereditary to r sion theory . Because Hom R ( S, S ′ ) = 0 for ev ery S ∈ S and S ′ ∈ S ′ , w e see that Hom R ( S, E ( S ′ )) = 0, where E ( S ′ ) is the injective en velope of S ′ . Hence E ∈ F and therefore U con tains T . Next, let M b e in U . T o show that M is a T -torsion mo dule it is enough to show that ev ery cyclic submo dule of M is a torsion mo dule, b ecause M is a quotien t of the direct sum of its cyclic submo dules. So let C b e a cyclic submodule of M . Since E is injectiv e, it follows that Ho m R ( C , E ) = 0. By Lemma 4 .1 C is S -cellular. Therefore C is T -cellular and b y Lemma 2 .1 0 C is T -to rsion.  8 Lemma 4.4. F or any c omplex X , Ext ∗ R ( P , X ) = 0 if and only if Ext ∗ R ( X , E ) = 0 . Pr o of. This is kno wn when X is a finitely generated R - mo dule, see Benson’s b o ok [1, 1.7.6 &1.7.7]. Now supp ose X is any R -mo dule. Since P is a finitely generated pro jectiv e mo dule, Hom R ( P , X ) = 0 if and only if Hom R ( P , X ′ ) = 0 for ev ery finitely generated submo dule X ′ of X . Similarly , b ecause E is injectiv e, Hom R ( X , E ) = 0 if and only if Hom R ( X ′ , E ) = 0 for ev ery finitely g enerated subm o dule X ′ of X . Hence the lemma holds for an y R -mo dule. Finally , let X b e an y complex, then Ext ∗ R ( P , X ) = Hom R ( P , H ∗ ( X )). Similarly Ext ∗ R ( X , E ) = Hom R ( H ∗ ( X ) , E ).  Corollary 4.5. F or any R -mo dule M , a T -nul lific ation of M is als o a P -c ompletion of M and is ther efor e given by Null T M ≃ R Hom End R ( P ) (Hom R ( P , R ) , Hom R ( P , M )) Pr o of. Consider the triang le Cell T M → M ν − → Null T M . Lemma 4.3 implies that E is T -n ull and therefore Ext ∗ R (Cell T M , E ) = 0. By Lemma 4 .4, Cell T M is P - n ull and ν is a P -equiv alence. It remains t o sho w t ha t Null T M is P -complete. Let I b e t he T -n ullificatio n of M described in 3.1. The full sub category of P -complete ob jects in D R is closed under isomorphisms, completion of tr ia ngles, pro ducts and retracts. Denote this sub category b y C . F rom Lemma 4.4, w e see that E ∈ C and therefore ev ery pro duct of E is also in C . Lemma 4.3 and [11, VI.3.9] imply that ev ery torsion-free mo dule is a submo dule o f a pro duct of copies of E . Since I n is injective , it is a direct summand of some pro duct of copies of E , hence I n is also an ob ject of C . Let I ( n ) denote the co ch ain complex I 0 → I 1 → · · · → I n . An inductiv e argumen t sho ws that I ( n ) ∈ C . There is a t riangle I → Q n I ( n ) φ − 1 − − → Q n I ( n ), where the map φ is induced b y the maps I ( n + 1) → I ( n ). Hence I is P -complete. By Dwy er and Greenlees [3, Theorem 2.1], the P - completion of an R -mo dule M is giv en b y M ∧ P ≃ R Hom End R ( P ) (Hom R ( P , R ) , Hom R ( P , M ))  Corollary 4.5 ab ov e implies Benson’s formula f o r T -cellular approx imation given in [2, Theorem 5.1]. This corollary also explains the connection b et w een Benson’s f o rm ula and Dwy er and G r eenlees fo rm ula fo r P -completion from [3, Theorem 2.1]. 5. Examples Example 5.1. Let I b e a tw o-sided ideal of R suc h that I is finitely generated as a left R -mo dule. An R -mo dule M will b e called I -torsion if fo r ev ery m ∈ M there exists some n suc h that I n m = 0 . It is not difficult to show that the class of I -torsion mo dules forms a hereditary torsion class T (see [11, VI.6.10]). Us ing Lemma 2.8 it is easy to conclude that hT i = h R/I i . Hence T -cellular appro ximation is the same as R /I -cellular appro ximation and the same go es for nullification. Note tha t in this case the radical t asso ciated with T ha s a simple description: for an y R -mo dule M t ( M ) = colim n →∞ Hom R ( R/I n , M ) No w supp ose R is a comm utative No etherian ring. D wy er and Greenlees ha v e show n in [3] that R/I -cellular approximation computes I -lo cal cohomology , namely that there is a natural isomorphism H ∗ I ( M ) ∼ = H ∗ (Cell R/I M ). Recall there is a n isomorphism: H ∗ I ( M ) ∼ = colim n →∞ Ext ∗ R ( R/I n , M ) 9 These facts sho w that T -cellular approximation is the deriv ed functor of the radical t . Moreo ve r, in this case an ob ject X ∈ D R is T -cellular if and o nly H n ( X ) is T -t o rsion for all n , see [3, 6.12]. Example 5.2. Here is an example of a case where T -cellular appro ximation is not the deriv ed functor of the asso ciated radical. Let G b e the symme tric gro up on 3 elemen ts, let k b e the field Z / 3 Z a nd let R b e the group ring k [ G ]. There is an a ug men tation map R → k , where k has the trivial G -action. Let I b e the augmentation ideal. As b efore, denote t he class of I -torsion mo dules b y T and t he asso ciated radical b y t . Since R is an Artinian ring, the sequenc e I ⊇ I 2 ⊇ I 3 ⊇ · · · stabilizes. So there is a fixed index m suc h that t ( M ) = Hom R ( R/I m , M ) fo r ev ery R - mo dule M . Therefore, the deriv ed f unctors of the torsion radical t are t he functors Ext ∗ R ( R/I m , − ). In particular, Ext i R ( R/I m , R ) = 0 for all i > 0, b ecause R is injectiv e. On the other hand, a calculation using Benson’s metho ds from [2] sho ws that H n (Cell T R ) is non- zero for infinitely man y v alues of n ; thereb y sho wing that Cell T is not the deriv ed functor of t . W e describ e this calculatio n next. F rom the surjection G → Z / 2 one sees that R has tw o simple mo dules, the trivial mo dule k and a o ne dimensional simple mo dule ω . As a left mo dule, R ∼ = E k ⊕ E ω where E k and E ω are the injectiv e hulls of k and ω r esp ectiv ely . The mo dule E k has a comp osition series k ⊂ B ⊂ E k , where B /k ∼ = ω and E k /B ∼ = k The comp osition series for E ω is ω ⊂ B ′ ⊂ E ω , where B ′ /ω ∼ = k and E ω /B ′ ∼ = ω In addition E ω /ω ∼ = B and E k /k ∼ = B ′ . Since E ω is k -null (see Lemma 4.3), then Null T R ≃ Null T E ω ⊕ Null T E k ∼ = E ω ⊕ Null T E k So w e need only compute Null T E k . Applying Construction 3.1 to the mo dule E k w e get the complex I whic h is E ω d − → E ω d − → E ω d − → · · · where d is the comp osition E ω ։ ω ֒ → E ω . Hence H n (Null T E k ) = k for n > 1 and therefore H n (Cell T R ) is non- zero for infinitely man y v alues of n . In f a ct H n (Cell T R ) =    k , n = 0; 0 , n = 1; k , n > 1 . It is imp ortant to note that in this case, a complex X suc h tha t H n ( X ) is T -torsion for all n need not b e T -cellular. Consider, for example, the complex R ∧ T . As we explain b elo w, the homology groups H n ( R ∧ T ) a r e T -torsion for all n . On the o ther hand, the T -equiv alences Cell T R → R and R → R ∧ T sho w that Cell T R is T -equiv alen t to R ∧ T . If R ∧ T w as T -cellular, then R ∧ T w ould hav e b een quasi-isomorphic t o Cell T R , b ecause a T -equiv alence b et we en T -cellular complexes is a quasi-isomorphism. As w e sho w next, the complex R ∧ T has no homology in negativ e degrees and so cannot b e quasi-isomorphic to Cell T R . It remains to explain the prop erties o f R ∧ T used ab ov e. F rom Corollary 4.2 w e learn that R ∧ T ≃ R ∧ k and Cell T R ≃ Cell k R . Without going in to details, com bining [4, 5.9] with [3, 4.3] sho ws that R ∧ k ≃ R Hom R (Cell k R, R ) This immediately implies t ha t R ∧ T has no homology in negat ive degrees. W e next sho w that H n ( R ∧ T ) is T -torsion for all n . Since E ω is a T -n ull mo dule, Ext ∗ R ( E ω , R ∧ T ) = 0. 10 Recall tha t R is a g roup-algebra a nd therefore E ω is also the pro jectiv e co v er of ω . Because E ω is pro jectiv e w e hav e Ext − n R ( E ω , R ∧ T ) ∼ = Hom R ( E ω , H n ( R ∧ T )) Hence, by Lemma 4.4, Ext ∗ R ( H n ( R ∧ T ) , E ω ) = 0. Lemma 4.3 sho ws E ω is an injectiv e cogenerator for T , therefore H n ( R ∧ T ) is T -torsion. Example 5.3. This example relates T -n ullification with Cohn lo calization. W e b egin by recalling the definition of Cohn lo calization. Let S = { f α : P α → Q α } b e a set of maps b et wee n finitely generated pro jectiv e R -mo dules. Say a ring map R → R ′ is S -inverting if Hom R ( f , R ′ ) is an isomorphism for eve ry f ∈ S . A Cohn lo c aliz a tion of R with resp ect to S is a ring map R → S − 1 R whic h is initial among all S -in v erting ring maps. Note that the definition giv en here is no t the standard definition (see e.g. [5 ]), but it is equiv alen t to the standard one. Let C S b e the set of cones of the maps f α . In [5], Dwy er considers C S -n ullificatio n and sho ws that H 0 (Null C S R ) = S − 1 R (see [5, 3.2]). Combining Dwye r’s results with Theorem 1.1 yields the follow ing prop osition. Prop osition 5.4. L et T b e a her e ditary torsion-class of R -mo dules. I f hT i = hC S i for some set of maps S b etwe en finitely gener ate d pr oje ctive R -mo dules, then (1) Null T ( − ) ≃ ( − ) F , (2) the mo dule of quotients functor ( − ) F is exa c t and (3) ther e is an i s o morphism S − 1 R ⊗ R M ∼ = M F for every mo dule M . Pr o of. By Theorem 1.1, f or ev ery R -mo dule M the complex Null T M has no homology in p ositiv e degrees. By [5, Prop osition 3.1], Null T R has no homology in negativ e degrees. Moreo ve r, a result of Miller [8] (see also [5, Prop osition 2.10]) sho ws that for ev ery R - mo dule M , Null T M ≃ Null T R ⊗ L R M . This implies that Null T M has no homology in negativ e degrees. W e conclude that for ev ery R - mo dule M , Null T M has homology only in degree zero and therefore, b y Lemma 3 .4, Null T M is quasi-isomorphic to M F . Since the f unctor Null T is exact, so is ( − ) F . Since Null T R ≃ Null C S R , D wy er’s result [5, 3.2] sho ws that R F ∼ = S − 1 R . Finally , the quasi-isomorphism Null T M ≃ Null T R ⊗ L R M implies S − 1 R ⊗ R M ∼ = M F .  Example 5.5. This example is o f a top olo g ical nature. Let M b e a discrete monoid and let k = Z /p Z for some pr ime p . The ring R w e consider is the monoid ring R = k [ M ], it has a natural augmen tatio n R → k with augmen tatio n ideal I . W e also mak e t he follo wing assumptions: (1) The classifying space B M o f M has a finite f undamental group. (2) The aug men tation ideal I is finitely generated as a left R -mo dule. (3) There is a pro jectiv e resolution P = · · · P 2 → P 1 → P 0 of k ov er R suc h that ev ery P n is finitely generated as an R - mo dule. Let T b e the hereditary to rsion class of I -torsion R - mo dules, then hT i = h k i and hence Cell T ≃ Cell k . Denote by R ∨ the left R -mo dule Hom k ( R, k ). F rom the results of D wy er, Greenlees and Iy engar [4, 6.15 and 7.5] it is easy to conclude t ha t Cell k R ∨ is quasi- isomorphic to the co chain complex (with co efficien ts in k ) of a certain space w e de- scrib e next. Let (B M ) ∧ p b e the Bousfield-Kan p -completion of the classifying space of M . The space Ω(B M ) ∧ p is the lo op- space of (B M ) ∧ p . So, Cell k R ∨ is quasi-isomorphic to C ∗ (Ω(B M ) ∧ p ; k ) - the singular co chain complex of Ω(B M ) ∧ p with co efficien ts in k . 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