Non-left-complete derived categories

NON-LEFT -COMPLETE DERIVED CA TEGORIES AMNON NEEMAN Abstra ct. W e gi ve some examples of ab elian categories A for which the d erived cat- egory D ( A ) is not left-complete. P erh aps the most natural of these is where A is th e category of represen tations of the additive group G a o ver a field k of characteristic p > 0. Contents 0. Assumed backg round 1 1. The counterexample 1 2. The p ro of 4 References 6 0. Assumed back ground In this article w e assume the reader is familiar with derived cate gories and with t – structures on them. See V e rdier [5] for the theory of derive d categories, and Beilinson, Bernstein and Deligne [1 , Chap ter 1] for an int ro duction to t –structures. 1. The c ounterex ample Supp ose A is an ab elian category and D ( A ) is its d eriv ed category . F or any ob ject x ∈ D ( A ), w e write x ≥ n for the truncation of x w ith resp ect to the standard t –structure. W e ha v e canonical maps x ≥ n − → x ≥ n +1 , and a (non-canonical) map ϕ x : x − − − − → Holim ✛ x ≥ n . The category D ( A ) is said to b e left-c omp lete if, for eve ry ob ject x ∈ D ( A ), any map ϕ x as ab ov e is an isomorphism. Ev en though th e m ap ϕ x is not canonical, it can b e shown that, for giv en x , if one ϕ x is an isomorphism then they al l are. The reader can find m u c h more ab ou t left-complete categories in Lurie [3, Section 7] or [4, Subsection 1.2.1, more p recisely starting from Prop ositio n 1.2.1.1 7]. See also Drinfeld an d Gaitsg ory [2]. 2000 Mathemat ics Subje ct Cl assific ation. Primary 18E30, secondary 18G55. Key wor ds and phr ases. Deriv ed categories, t –structures, homotopy limits. The researc h wa s partly supp orted by th e Australian Research Council. 1 2 AMNON NEEMAN In th is note we will see how to pro duce man y A for which D ( A ) is not left-complete . Our counte rexamples will b e of a v ery sp ecial form, w hic h allo ws us to easily compute the homotopy in v erse limit Holim ✛ x ≥ n . Let us no w sk etc h what we will do. W e will s upp ose that the ab elian categ ory A satisfies the axiom [AB4], that is copro d- ucts are exact; this mak es it easy to co mpute copro ducts in the derived category D ( A ), just form the copro d uct as complexes. Supp ose A is an ob ject in our [AB4] ab elian catego ry A , and let x = ∞ a i =0 A [ i ] . It is clea r that, for n > 0, w e hav e x ≥− n = n a i =0 A [ i ] = n Y i =0 A [ i ] , where th e last equalit y is b ecause finite copro d ucts agree with fi nite pro ducts. No w the homotop y inv erse limit of the p ro du cts is a gen uin e inv erse limit, and we ha v e Holim ✛ x ≥ n = ∞ Y i =0 A [ i ] . Th us our problem b ecomes to decide whether th e map ∞ a i =0 A [ i ] ϕ − − − − → ∞ Y i =0 A [ i ] is an isomorph ism . No te that in this case the map is canonical; our h omotopy inv erse limit happ en s to b e a gen uine inv erse limit, removing th e arbitrarin ess. The left hand side is easy to w ork with; its cohomology is A in eac h degree n ≤ 0. What w e will sho w is ho w to pro du ce examples where the right hand side has lots more cohomology . More precisely , w e h a v e ∞ Y i =0 A [ i ] = A [0] ⊕ ∞ Y i =1 A [ i ] ! and the expectation w ould b e for th e second term to ha v e a v anishing H 0 ; what w e will sho w is ho w to pro duce non-zero classes in H 0 ∞ Y i =1 A [ i ] ! . It is time to disclose what will b e our c h oice for the cat egory A and for the ob ject A ∈ A . Construction 1.1. Let k b e a field, let R 1 b e a finitely generated k algebra, and let m b e a k –p oin t of Sp ec( R 1 ). In other w ords, m ⊂ R is a maximal id eal with R 1 / m ∼ = k . W e mak e a string of definitions: (i) R n = ⊗ n i =1 R 1 , where the tensor is ov er the field k . COGENERA TORS IN K ( R –Pro j) 3 (ii) The inclusion R n − → R n +1 is the inclusion of the tensor pro d uct of the first n terms. (iii) R = colim − → R n . (iv) The map Φ i : R 1 − → R is the inclusion of the i th factor. (v) The category A will b e the category of all those R –mo d ules, on which Φ i ( m ) acts trivially for all b ut fi nitely man y i . The ob ject A ∈ A will b e the colimit o v er n of the R n –mo dules k = ⊗ n i =1 [ R 1 / m ]. The m ain result is Theorem 1.2. Assume that k = R 1 / m is not pr oje ctive over the lo c alization ( R 1 ) m of the ring R 1 at the maximal ide al m . With the c ate gory A and the obje ct A ∈ A as in Construction 1.1, ther e is a non-zer o element in H 0 ∞ Y i =1 A ! . Remark 1.3. The case where R 1 = k [ x ] / ( x p ) is of particular inte rest. If the fi eld k is of c haracteristic p then the category A happ ens to b e the cate gory of representa tions of the additive group G a , and w e learn that its deriv ed category is not left-c omplete. Remark 1.4. W e trivially ha v e ∞ Y i =1 A [ i ] = n Y i =1 A [ i ] ! ⊕ ∞ Y i = n +1 A [ i ] ! , and hence H 0 ∞ Y i =1 A [ i ] ! = H 0 n Y i =1 A [ i ] ! ⊕ H 0 ∞ Y i = n +1 A [ i ] ! . On the other hand , with the finite pro duct we hav e no p roblem computing H 0 n Y i =1 A [ i ] ! = H 0 n a i =1 A [ i ] ! = 0 , and Theorem 1.2 now allo w s u s to deduce that H 0 ∞ Y i = n +1 A [ i ] ! 6 = 0 . T ranslati ng w e ha v e that H n ∞ Y i =1 A [ i ] ! 6 = 0 for all n ≥ 0. The complexes A [ i ] , i > 0 all b elong to D ( A ) < 0 , bu t the pr o duct Q ∞ i =1 A [ i ] is not b ound ed ab o v e. Ac kno wledgements. The author would lik e to thank Drinfeld and Gait sgory for asking the question that led to these coun terexamples. 4 AMNON NEEMAN 2. The p roof W e b egin with a little le mma. Lemma 2.1. L et k b e a field, and let R and S b e finitely gener ate d k –algebr as. Supp ose further that we ar e given k –p oints of Sp ec ( R ) and Sp ec( S ) ; that is m ⊂ R and n ⊂ S ar e maximal ide als, with R/ m ∼ = k ∼ = S/ n . L et E b e an inje ctive envelop e of k = R / m over the ring R , and F an inje ctive envelop e of k = S/ n over the ring S . Then E ⊗ k F is an inje ctive envelop e of k over the ring R ⊗ k S . Pr o of. W e will first prov e the case where R and S are p olynomial rings. Let R ′ = k [ x 1 , x 2 , . . . , x m ] b e a p olynomial ring, and let m b e the maximal ideal generated by { x 1 , x 2 , . . . , x m } . Then we kno w the injectiv e env elop e E ′ of k = R ′ / m explicitly: it is the qu otien t of S = k [ x 1 , x − 1 1 , x 2 , x − 1 2 , . . . , x m , x − 1 m ] by the R ′ –submo du le generated b y al l monomials x i 1 1 x i 2 2 · · · x i m m with at least one of the i j > 0. As a k –ve ctor space E ′ = k [ x − 1 1 , x − 1 2 , . . . , x − 1 m ], and the R ′ –mo dule stru cture is ob vious w h en w e declare x i 1 1 x i 2 2 · · · x i m m = 0 if s ome i j > 0. If S ′ = k [ y 1 , y 2 , . . . , y n ] and n ⊂ S ′ is the ideal generated b y { y 1 , y 2 , . . . , y n } , then the fac t that E ′ ⊗ k F ′ = k [ x − 1 1 , x − 1 2 , . . . , x − 1 m ] ⊗ k k [ y − 1 1 , y − 1 2 , . . . , y − 1 n ] is the injectiv e hull of k o v er R ′ ⊗ S ′ is b y insp ection. No w for the general case: assume R = R ′ /I and S = S ′ /J where R ′ and S ′ are p olynomial rings, and I ⊂ R ′ and J ⊂ S ′ are ideals con ta ined in the m and n ab o v e. Then the injectiv e h ull E of k = R/ m o v er the ring R is the largest R –sub mo dule of the R ′ –mo dule E ′ , that is the R ′ –submo du le E ⊂ E ′ of all elemen ts annihilated by the ideal I . The lemma therefore comes do wn to the fact that th e submo dule of E ′ ⊗ k F ′ annihilated by the ideal I ⊗ k S ′ + R ′ ⊗ k J is precisely E ⊗ k F .  Pro of of Theorem 1.2. Let R b e the lo calization of R 1 at the m aximal ideal m . W e are assu ming that k is not pro jectiv e o v er R , that is th e pro j ectiv e dimension of k is at least one. Cho ose and fix a minimal free resolution of k = R/ m R as an R –mo d ule. Let us write this resolution as − − − − → P 2 − − − − → P 1 − − − − → P 0 − − − − → k − − − − → 0 . Then the mo dules P i are all finite and fr ee ov er the ring R , the differentials are all matrices o v er R , and the minimalit y guarantee s that the en tries in th ese matrices all b elong to the ideal m = m R ⊂ R . Now let E b e the R –injectiv e env elop e of the m o dule k ; applying th e functor Hom R ( − , E ) to the p ro jectiv e resolution ab o v e, w e pro du ce an injectiv e r esolution I ∗ of k , whic h we write out as 0 − − − − → k − − − − → I 0 − − − − → I 1 − − − − → I 2 − − − − → COGENERA TORS IN K ( R –Pro j) 5 W e know that eac h I j = Hom( P j , E ) is a fi nite copro du ct of copies of E , and that the differen tials I j − → I j +1 are matrices whose en tries b elong to the id eal m . The fact th at the pro jectiv e dimension of k is at least one tells us that P 1 6 = 0, and therefore I 1 6 = 0. Note that an injectiv e env elop e E of k o v er the lo calized ring R = ( R 1 ) m is also an injectiv e env elop e of k o v er the ring R 1 , hence w e hav e pro duced an in j ectiv e resolution of k o v er R 1 . Next we (i) Cho ose a non-zero elemen t a in the image of the map k − → I 0 . (ii) Cho ose a non-zero elemen t b ∈ I 1 , with m b = 0. If we view k as a m o dule ov er th e ring R n = ⊗ n i =1 R 1 , then the tensor pro duct J ∗ n = ⊗ n i =1 I ∗ is certainly a resolution of k as an R n mo dule, and Lemm a 2.1 guarantees f urther that (iii) Eac h J i n is injectiv e as a mo dule o ve r R n . (iv) Let the inclusion J ∗ n − → J ∗ n +1 b e the map taking x ∈ J ∗ n to x ⊗ a ∈ J ∗ n ⊗ I 0 ⊂ J ∗ n ⊗ I ∗ = J ∗ n +1 , where a ∈ I 0 is as in (i) ab o v e. W e defin e J ∗ to b e J ∗ = colim − → J ∗ n ; then J ∗ is an injectiv e r esolution of k in the category A . T o pro v e the theorem we need to find a non-zero elemen t in H 0  Q i> 0 k [ i ]  , and our next observ ation is that th e pro du ct in the deriv ed category Q i> 0 k [ i ] is obtained as th e the ord in ary p ro du ct of injectiv e resolutions. The complex J ∗ [ i ] is an injectiv e resolution of k [ i ], and hence th e deriv ed pro duct Q i> 0 k [ i ] is just the usual p r o duct Q i> 0 J ∗ [ i ]. No w for every i ≥ 1 let S i = { i 2 + 1 , . . . , i 2 + i } , and observe that the sets S i are d isjoin t. In the injectiv e R i 2 + i –mo dule J i i 2 + i = a P ℓ m = i I ℓ 1 ⊗ I ℓ 2 ⊗ · · · ⊗ I ℓ i 2 + i or more sp ecifically in the su mmand ( I 0 ) ⊗ i 2 ⊗ ( I 1 ) ⊗ i w e tak e th e te rm λ i = a ⊗ i 2 ⊗ b ⊗ i , where a ∈ I 0 and b ∈ I 1 are as in (i) and (ii) ab ov e. The emb edding J ∗ i 2 + i − → J ∗ of (iv) giv es us an elemen t whic h w e w ill denote λ i ∈ J i . The ele men ts λ i ha v e the prop erties (v) Eac h λ i is a cycle; the differen tial J i − → J i +1 kills λ i . (vi) Φ j ( m ) λ i = 0 for all i and j . 6 AMNON NEEMAN W e are assuming i > 0, so eac h λ i m ust b e a b oundary b ecause H i ( J ∗ ) = 0. But if µ i ∈ J i − 1 maps to λ i , then there m ust exist an in teg er j ∈ S i so that Φ j ( m ) do es not kill µ i . No w form the element ∞ Y i =1 λ i ∈ ∞ Y i =1 J i , where the pro duct is in the categ ory of all R –mo dules. Caution 2.2. The reader is remind ed that the category A is a sub catego ry of the catego ry of R –mo dules. Both categories hav e infin ite p ro ducts; th e pro ducts in th e catego ry of R –mo dules are just the usual cartesian p ro ducts, w hile the pro ducts in A are subtler. T o form the pro duct in A of a bun c h of ob jects in A , one first forms the usual cartesian pro duct, and th en consider inside it the largest ob ject b elonging to A , that is the collection of all elemen ts satisfying part (v) of Constru ction 1.1. The elemen t Q ∞ i =1 λ i is a degree 0 cycle in th e complex Q i ≥ 1 J ∗ [ i ], and it is annihilated b y Φ j ( m ) for al l j . By Caution 2. 2 w e h a v e that Q ∞ i =1 λ i b elongs to Q ∞ i =1 J i ev en when the pr o duct is understo od in A . Ho wev er , it is not a b oundary in A . If we try to express Q ∞ i =1 λ i as th e bou n dary of ∞ Y i =1 µ i ∈ ∞ Y i =1 J i − 1 , then we disco ver that ea c h µ i fails to b e annihilated by some Φ j ( m ) with j ∈ S i . As the S i are disjoin t, this pro du ces infinitely many Φ j ( m ) not annihilating Q ∞ i =1 µ i , meaning it do es n ot b elong to A . Referen ces 1. Alexander A. Beilinson, Joseph Bernstein, and Pierre Deligne, Analys e et top olo gie sur les ´ esp ac es singuliers , Ast´ erisque, vol. 100, S oc. Math. F rance, 1982 (F renc h). 2. Vladimir Drinfeld and Dennis Gaitsgory , On some finiteness questions for algebr aic stacks , in p rep a- ration. 3. Jacob Lurie, Derive d al gebr aic ge ometry I : stable ∞ -c at e gories , arXiv:math/0608228v5. 4. , Higher Algebr a , Prepint, av ailable from h ttp://www.math.harv ard.edu/ ∼ lu rie/. 5. Jean-Louis V erdier, Des c at ´ ego ries d ´ eriv ´ ees des c at´ egories ab eliennes , Asterisque, vo l. 239, So ci´ et ´ e Math´ ematique de F rance, 1996 (F rench). Centre f or Ma thema tics and its Applica tions, Ma the ma tical Sciences I nstitute, John Dedman Bui lding, The Australian Na tional Un iversity, Canberra, A CT 0200, AUSTRALIA E-mail addr ess : Amnon. Neeman@an u.edu.au