On the Ext-computability of Serre quotient categories
aa r X i v : . [ m a t h . K T ] S e p ON THE
Ext -COMPUTABILITY OF SERRE QUOTIENT CATEGORIES
MOHAMED BARAKAT AND MARKUS LANGE-HEGERMANNA
BSTRACT . To develop a constructive description of
Ext in categories of coherent sheaves overcertain schemes, we establish a binatural isomorphism between the
Ext -groups in Serre quotientcategories A / C and a direct limit of Ext -groups in the ambient Abelian category A . For Ext the isomorphism follows if the thick subcategory C ⊂ A is localizing. For the higher extensiongroups we need further assumptions on C . With these categories in mind we cannot assume A / C to have enough projectives or injectives and therefore use Yoneda’s description of Ext .
1. I
NTRODUCTION
Our original motivation is to develop a constructive and computer-friendly description ofAbelian categories of coherent sheaves
Coh X on various classes of Noetherian schemes X . Inthis setup the functors Hom and
Ext c are ubiquitous, and any constructive approach needs toincorporate these functors. For example, the global section functor on Coh X can be definedas Γ := Hom( O X , − ) , i.e., in terms of the Hom functor and the structure sheaf O X . Thehigher sheaf cohomology H i is usually defined in the nonconstructive larger category of quasi-coherent sheaves on X as H i = R i Γ = Ext i ( O X , − ) .In this paper we deal with computing the bivariate Ext i ( − , − ) , where for the special uni-variate case of sheaf cohomology H i = Ext i ( O X , − ) there often exist good algorithms. Ourminimal assumption on X is that the category Coh X is equivalent to a Serre quotient category A / C ≃
Coh X where A is a computable category (in the sense of Appendix A) of finitelypresented graded modules and C ⊂ A is its thick subcategory of all modules with zero sheafi-fication. The canonical functor Q : A → A / C then plays the role of the exact sheafificationfunctor Sh :
A →
Coh
X, M f M . Some classes of schemes for which this holds are listed in[BLH14b, Section 4], including projective and toric schemes.The computability of Ext c would usually follow from that of Hom in case the underlying cat-egory is computable and has constructively enough projectives or enough injectives. However,as categories of coherent sheaves do not in general admit enough injectives or projectives wecannot assume this for the computation of
Ext c in an abstract Serre quotient A / C . Hence, Ext c in such an A / C cannot even be defined constructively as a derived functor using projective orinjective resolutions and we are left over with Yoneda’s description of Ext c [Oor64]. AlthoughYoneda’s description does not a priori provide an algorithm to compute Ext c , it is sufficientto prove our main result: Under certain assumptions on C the computability of Ext c in A / C Mathematics Subject Classification.
Key words and phrases.
Serre quotients, reflective localizations of Abelian categories, Yoneda
Ext , Ext -computability. can be reduced to the computability of
Ext c in A , provided a certain (infinite) direct limit isconstructive. More precisely: Theorem 1.1. If C is an almost split localizing subcategory of an Abelian category A thenthe binatural transformation Q Ext : lim −→ M ′ ≤ M,M/M ′ ∈C Ext c A ( M ′ , N ) → Ext c A / C ( M, N ) is an isomorphism (of Abelian groups) for all C -torsion-free M ∈ A and C -saturated N ∈ A . For applications to coherent sheaves A / C ≃
Coh X (for X as above) we need to prove thatthe thick subcategory C of modules with zero sheafification is almost split localizing. This isfor example the case if X is a projective space and A is suitably chosen (see Section 3 andExample 6.5). However, it is worth mentioning that Theorem 1.1 cannot cover the case ofcoherent sheaves on nonsmooth toric varieties. One can see this easily since the Cox ring andhence the category A of finitely presented graded modules over this ring is of finite globaldimension while one can easily construct coherent sheaves on a nonsmooth toric variety withnon-vanishing Ext c for arbitrarily high c (see Example 6.6).The theorem suggests an algorithmic approach to the computability of Ext c in A / C . Tocompute the left hand side lim −→ M ′ ≤ M,M/M ′ ∈C Ext c A ( M ′ , N ) we need to be able to compute Ext c A anda direct limit of Abelian groups. For categories of graded modules A there are well-knownalgorithms to compute Ext c A . Proving that the (infinite) direct limit can be computed in finiteterms depends on A and C . For example, in the category A / C ≃
Coh X of coherent sheaveson a projective space the finiteness of this direct limit follows from the Castelnuovo-Mumfordregularity. Thus, Theorem 1.1 is an abstract form of [Smi00, Theorem 1] and [Smi13], withoutthe context-specific convergence analysis. If A is the category of graded modules and if thelimit is reached for a certain M ′ ≤ M then one can use a graded free resolution of M ′ in A tocompute Ext c A / C ( M, N ) . In this case this graded free resolution of M ′ in A corresponds, underthe canonical functor Q : A → A / C , to a locally free resolution of M in A / C = Coh X satisfying some regularity bounds. We believe that the multigraded Castelnuovo-Mumford[MS04, MS05, HSS06] can be used to prove the finiteness of the limit in the case of smoothprojective toric varieties. We leave this for future work.We briefly recall the language of Serre quotient categories in Section 2 and deal with the c = 0 case of Theorem 1.1 in Section 3. In Section 4 we recall Yoneda’s description of Ext c and in Section 5 we define the binatural transformation Q Ext . In the main Section 6 we definethe above mentioned condition which C needs to satisfy and prove Theorem 1.1. There it isalso proved that the theorem is valid if c = 1 under the weaker condition that C is a localizingsubcategory of A (cf. Theorem 6.2). Finally, in Appendix A we briefly sketch a constructivecontext for this paper. Cf. Definition 6.3. We drop the canonical functor Q in Ext c A / C ( Q ( M ) , Q ( N )) since Q is the identity on objects. contrary to Theorem 6.2. N THE
Ext -COMPUTABILITY OF SERRE QUOTIENT CATEGORIES 3
2. P
RELIMINARIES ON S ERRE QUOTIENTS
In this section we recall some results about Serre quotients [Gab62]. From now on A is anAbelian category.A non-empty full subcategory C of an Abelian category A is called thick if it is closed underpassing to subobjects, factor objects, and extensions. In this case the (Serre) quotient category A / C is a category with the same objects as A and Hom -groups
Hom A / C ( M, N ) := lim −→ M ′ ≤ M,N ′ ≤ N,M/M ′ ,N ′ ∈C Hom A ( M ′ , N/N ′ ) .The canonical functor Q : A → A / C is defined to be the identity on objects and maps amorphism ϕ ∈ Hom A ( M, N ) to its class in the direct limit Hom A / C ( M, N ) . The category A / C is Abelian and the canonical functor Q : A → A / C is exact.Let C ⊂ A be thick. An object M ∈ A is called C -torsion-free if M has no nonzerosubobjects in C . We will need the following simple lemma. Lemma 2.1.
An extension of C -torsion-free objects is again C -torsion-free.Proof. Let E be an object in A with C -torsion-free subobject L and C -torsion-free factor object B = E/L . Assume that E has a nontrivial C -subobject T . Since L ∩ T = 0 we conclude that T is isomorphic to the nontrivial C -subobject ( T + L ) /L of B , a contradiction. (cid:3) If every object M ∈ A has a maximal C -subobject H C ( M ) then we call C a thick torsion subcategory. An object M ∈ A is called C -saturated if it is C -torsion-free and every extensionof M by an object C ∈ C is trivial. Denote by Sat C ( A ) ⊂ A the full subcategory of C -saturated objects with embedding functor ι : Sat C ( A ) ֒ → A . The thick subcategory C ⊂ A iscalled a localizing subcategory if the canonical functor Q : A → A / C admits a right adjoint S : A / C → A , called the section functor of Q . The section functor S : A / C → A is left exact and preserves products, the counit of the adjunction δ : Q ◦ S ∼ −→ Id A / C is anatural isomorphism. Let η : Id A → S ◦ Q denote the unit of the adjunction. The kernel ker ( η M : M → ( S ◦ Q )( M )) is then the maximal C -subobject H C ( M ) of M . The cokernel of η M lies in C . We call ( S ◦ Q )( M ) the C -saturation of M . An object M in A is C -saturated ifand only if η M is an isomorphism. The image S ( A / C ) of S is a subcategory of Sat C ( A ) andthe inclusion functor S ( A / C ) ֒ → Sat C ( A ) is an equivalence of categories with the restricted-corestricted monad S ◦ Q : Sat C ( A ) → S ( A / C ) as a quasi-inverse. The restricted canonicalfunctor Q : Sat C ( A ) → A / C and the corestricted section functor S : A / C →
Sat C ( A ) arequasi-inverse equivalences of categories. In particular, Sat C ( A ) ≃ S ( A / C ) ≃ A / C is anAbelian category. Define the reflector b Q := co-res Sat C ( A ) ( S ◦ Q ) : A →
Sat C ( A ) . Theadjunction b Q ⊣ ( ι : Sat C ( A ) ֒ → A ) corresponds under the above equivalence to the adjunction Q ⊣ ( S : A / C → A ) . They both share the same adjunction monad S ◦ Q = ι ◦ b Q : A → A .In particular, the reflector b Q is exact and ι is left exact. S ( A / C ) ≃ Sat C ( A ) are not in generalAbelian sub categories of A , as short exact sequences in Sat C ( A ) are not necessarily exact in A . For more details and for a characterization of the monad S ◦ Q see [BLH13]. A functor is called a reflector if it has a fully faithful right adjoint.
MOHAMED BARAKAT AND MARKUS LANGE-HEGERMANN
3. T HE c = 0 CASE
If the thick subcategory
C ⊂ A is torsion then the double direct limit in the definition of the
Hom -groups in A / C simplifies to a single direct limit Hom A / C ( M, N ) = lim −→ M ′ ≤ MM/M ′ ∈C Hom A ( M ′ , N/H C ( N )) .If furthermore C ⊂ A is localizing then the
Hom -adjunction between Q and S yields(Hom) Hom A ( M, ( S ◦ Q )( N )) ∼ = Hom A / C ( M, N ) ,for all M, N ∈ A , avoiding the direct limit completely. Theorem 1.1 generalizes this lastformula, being the c = 0 case.The monad S ◦ Q together with its unit are constructive in the case A / C ≃
Coh P nk , i.e.,of coherent sheaves on the projective space X = P nk over a field k . Hence, the above men-tioned Hom -adjunction can be used to compute (global)
Hom -groups. More precisely, let A be the category of finitely presented Z -graded k [ x , . . . , x n ] -modules generated in degree ≥ and C be the thick subcategory of finite length modules. The C -saturation of an N ∈ A isthe truncated module of twisted global sections, i.e., ( S ◦ Q )( N ) = L i ≥ Γ( e N ( i )) , where e N ∈ Coh P nk is the sheafifications of N . For X = P nk , and hence for any projective scheme,there are already several algorithms to compute the monad S ◦ Q ; e.g., as an ideal transform[BS98, Theorem 20.3.15], or by the Beilinson monad [Be˘ı78, EFS03, DE02], or by the BGG-correspondence.Recently, Perling [Per14] described the section functor S and hence the monad S ◦ Q for alarger class of schemes, but in a (yet) nonconstructive way. A constructive description is highlydesirable as it would widen the applicability of Theorem 1.1 as an algorithm to compute Ext ’sfor further classes of smooth schemes.4. Y
ONEDA ’ S DESCRIPTION OF
Ext
Since in applications to coherent sheaves the quotient category A / C does not have enoughprojectives or injectives we use Yoneda’s description of Ext c (cf. [ML95, Section III.5]).So let B be an Abelian category. A c -cocycle in the Ext c B ( M, N ) group ( c > ) is an equiva-lence class of c -extensions of M by N , i.e., exact B -sequences e : 0 ←− M ←− G c ←− · · · ←− G ←− N ←− of length c + 2 . Two c -extensions e, e ′ of M by N are in directed relation if there exists a chainmorphism of the form M G c · · · G N M G ′ c · · · G ′ N e : e ′ : Again, we drop the canonical functor Q in Hom A / C ( Q ( M ) , Q ( N )) since Q is the identity on objects. N THE
Ext -COMPUTABILITY OF SERRE QUOTIENT CATEGORIES 5
For c > this directed relation is not symmetric. A c -cocycle is an equivalence class of theequivalence relation generated by this directed relation. Abusing the notation we will denoteby e the c -cocycle in Ext c B ( M, N ) of a c -extension e of M by N .We now recall the definition of the Yoneda composition Ext c B ( M, N ) × Ext c ′ B ( N, L ) → Ext c + c ′ B ( M, L ) . We start with the case c, c ′ > . For e MN ∈ Ext c B ( M, N ) and e NL ∈ Ext c ′ B ( N, L ) represented by the extensions e MN : 0 ←− M ←− G c ←− · · · ←− G ←− N ←− and e NL : 0 ←− N ←− G ′ c ′ ←− · · · ←− G ′ ←− L ←− ,respectively. The Yoneda composite e ML = e MN e NL ∈ Ext c + c ′ B ( M, L ) is the ( c + c ′ ) -cocyclerepresented by the ( c + c ′ ) -extension e ML : 0 ←− M ←− G c ←− · · · ←− G ←− G ′ c ′ ←− · · · ←− G ′ ←− L ←− of M by L , where the morphism G ←− G ′ c ′ is the composition G ←− N ←− G ′ c ′ .For c = 0 and c ′ > let ϕ MN ∈ Hom B ( M, N ) and e NL ∈ Ext c ′ B ( N, L ) as above. The Yonedacomposite ϕ MN e NL ∈ Ext c ′ B ( M, L ) is given by the pullback c ′ -extension N G ′ c ′ G ′ c ′ − · · · G ′ L M G c ′ G ′ c ′ − · · · G ′ L ϕ MN e NL : ϕ MN e NL : For c > and c ′ = 0 let e MN ∈ Ext c B ( M, N ) as above and ψ NL ∈ Hom B ( N, L ) . The Yonedacomposite e MN ψ NL ∈ Ext c B ( M, L ) is given by the pushout c -extension M G c · · · G G N M G c · · · G G ′ L ψ NL e MN : e MN ψ NL : For more details see, e.g., [HS97, Section IV.9], [BB08, Appendix B].5. T
HE BINATURAL TRANSFORMATION
Let A is an Abelian category and C ⊂ A a thick subcategory. Applying the exact canon-ical functor Q : A → A / C to a cocycle e ∈ Ext c A ( M, N ) we obtain a cocycle Q ( e ) in Ext c A / C ( M, N ) . In other words, the canonical functor Q : A → A / C induces maps Ext c A ( M ′ , N/N ′ ) → Ext c A / C ( M, N ) We write
Ext c A / C ( M, N ) for Ext c A / C ( Q ( M ) , Q ( N )) since Q is the identity on objects. By this we mean the composition
Ext c A ( M ′ , N/N ′ ) → Ext c A / C ( M ′ , N/N ′ ) ∼ = −→ Ext c A / C ( M, N ) , where thelast isomorphism is the inverse of the one induced by the A -mono M ′ ֒ → M and the A -epi N ։ N/N ′ , as bothbecome isomorphisms in A / C . MOHAMED BARAKAT AND MARKUS LANGE-HEGERMANN for all
M, N ∈ A , M ′ ≤ M , N ′ ≤ N with M/M ′ ∈ C and N ′ ∈ C . For M ′′ ≤ M ′ with M ′ /M ′′ ∈ C and N ′′ ≥ N ′ with N ′′ /N ′ ∈ C the cocycle e ′ : 0 ←− M ′ ←− G ′ c ←− G ′ c − ←− · · · ←− G ′ ←− G ′ ←− N/N ′ ←− ∈ Ext c A ( M ′ , N/N ′ ) induces a cocycle e ′′ : 0 ←− M ′′ ←− G ′′ c ←− G ′ c − ←− · · · ←− G ′ ←− G ′′ ←− N/N ′′ ←− ∈ Ext c A ( M ′′ , N/N ′′ ) , as the Yoneda composite e ′′ = ( M ′′ ֒ → M ′ ) e ′ ( N/N ′ ։ N/N ′′ ) . Hence, Q induces a map Q Ext : lim −→ M ′ ≤ M,N ′ ≤ N,M/M ′ ∈C ,N ′ ∈C Ext c A ( M ′ , N/N ′ ) → Ext c A / C ( M, N ) for all M, N ∈ A .If N ∈ A is C -saturated, then the above double limit simplifies to Q Ext : lim −→ M ′ ≤ M,M/M ′ ∈C Ext c A ( M ′ , N ) → Ext c A / C ( M, N ) for all M ∈ A , as there are no nontrivial C -subobjects N ′ ≤ N .Now we consider the functoriality of the left hand side F : ( M, N ) lim −→ M ′ ≤ M,M/M ′ ∈C Ext c A ( M ′ , N ) . To describe the functoriality in the first argument let ϕ : M → L be an A -morphism, N ∈ A ( C -saturated), G L = G LN ∈ Ext c A ( L, N ) , and G M = G MN = ϕG L ∈ Ext c A ( M, N ) , the Yonedacomposition of ϕ and G L (by construction we have that G Mc ′ = G Lc ′ for c ′ ≤ c − ). Taking thepullback of a subobject ι L ′ : L ′ ֒ → L with L/L ′ ∈ C we obtain a subobject ι ′ M : M ′ ֒ → M with M/M ′ ∈ C . Sending the cocycle ι L ′ G L ∈ Ext c A ( L ′ , N ) to ι M ′ G M ∈ Ext c A ( M ′ , L ) defines thefirst argument action of F on ϕ . The proof of functoriality in the first argument follows fromthe identity ι M ′ G M = ι M ′ ϕG L = ϕ | M ′ ι L ′ G L = ϕ | M ′ G L ′ . ML M ′ L ′ G Mc G Lc G M ′ c G L ′ c G Mc − G Lc − G Mc − G Lc −
00 00 ι M ′ ι L ′ ϕ ϕ | M ′ For the functoriality in the second argument consider a morphism ψ : N → L and take the col-imit lim −→ M ′ ≤ M,M/M ′ ∈C of the maps Ext c A ( M ′ , N ) Ext c A ( M ′ ,ψ ) −−−−−−−→ Ext c A ( M ′ , L ) given by the usual functo-riality of Ext c A in the second argument. Finally, the exact functor Q commutes with pullbacks,implying the binaturality of Q Ext .As Q is exact, the map Q Ext respects Baer sums.
N THE
Ext -COMPUTABILITY OF SERRE QUOTIENT CATEGORIES 7
6. T
HE PROOF
Our goal is to give sufficient conditions for the binatural transformation Q Ext to be an iso-morphism. For this we assume that
C ⊂ A is a localizing subcategory of the Abelian category A . Then the restricted canonical functor Q : Sat C ( A ) → A / C and the corestricted sectionfunctor S : A / C →
Sat C ( A ) are adjoint equivalences of categories. Remark . We will use this equivalence to replace
Ext A / C by the isomorphic Ext
Sat C ( A ) , thefunctor Q : A → A / C by b Q := co-res Sat C ( A ) ( S ◦ Q ) : A →
Sat C ( A ) , and finally Q Ext by b Q Ext : lim −→ M ′ ≤ M,M/M ′ ∈C Ext c A ( M ′ , N ) → Ext c Sat C ( A ) ( M, N ) .For simplicity we write Ext c Sat C ( A ) ( M, N ) for Ext c Sat C ( A ) ( b Q ( M ) , b Q ( N )) . Recall that in The-orem 1.1 we require M to be C -torsion-free and N to be C -saturated. Since the cokernel ( S ◦ Q )( M ) /M of η M lies in C we can, without loss of generality, as well assume M tobe C -saturated as the limit does not distinguish between M and its saturation ( S ◦ Q )( M ) .6.1. The proof for
Ext . For c = 1 it turns out that assuming C ⊂ A to be localizing is alreadysufficient for b Q Ext to be an isomorphism.
Theorem 6.2. If C is a localizing subcategory of the Abelian category A then b Q Ext : lim −→ M ′ ≤ M,M/M ′ ∈C Ext A ( M ′ , N ) → Ext C ( A ) ( M, N ) ∼ = Ext A / C ( M, N ) is an isomorphism (of Abelian groups) for all C -saturated M, N ∈ A .Proof.
Recall that a short exact sequence e : 0 ←− M π ←− E ←− N ←− in Sat C ( A ) is in generalonly left exact in A , since the embedding functor ι : Sat C ( A ) ֒ → A is in general only left exact.The A -cokernel of π lies in C , i.e., for M ′ := im π the sequence ←− M ′ π ←− E ←− N ←− is exact in A and M/M ′ = coker π ∈ C . This yields the preimage of e under b Q Ext and showssurjectivity.For the injectivity take an exact A -sequence e : 0 ←− M ′ ←− E ϕ ←− N ←− such that thecorresponding exact Sat C ( A ) -sequence b Q Ext ( e ) : 0 ←− b Q ( M ′ ) ←− b Q ( E ) b Q ( ϕ ) ←−−− b Q ( N ) ←− is split, i.e., e is in the kernel of b Q Ext . By definition of split short exact sequences, thereis a b ψ : b Q ( E ) → b Q ( N ) such that b ψ ◦ b Q ( ϕ ) = Id b Q ( N ) . Since N is C -saturated the unit η N : N → ι ( b Q ( N )) is an isomorphism and we can define ψ := η − N ◦ ι ( b ψ ) ◦ η E . Notethat η E ◦ ϕ = ι ( b Q ( ϕ )) ◦ η N , by the naturality of η . Then ψ ◦ ϕ = η − N ◦ ι ( b ψ ) ◦ η E ◦ ϕ = η − N ◦ ι ( b ψ ) ◦ ι ( b Q ( ϕ )) ◦ η N = η − N ◦ ι (Id b Q ( N ) ) ◦ η N = η − N ◦ Id ι ( b Q ( N )) ◦ η N = Id N implies that e is split, i.e., zero in Ext A ( M ′ , N ) . (cid:3) MOHAMED BARAKAT AND MARKUS LANGE-HEGERMANN
The proof of surjectivity for higher
Ext ’s.
For c ≥ we need further conditions on thecategories A and C . Definition 6.3.
Let A be an Abelian category and C ⊂ A a thick subcategory. For an object a ∈ A we call a subobject a ⊥ ≤ a an almost C -complement if a ⊥ is C -torsion-free and a/a ⊥ ∈ C . We call C an almost split (thick) subcategory if for each object a ∈ A there existsan almost C -complement a ⊥ . Remark . Let A be an Abelian Noetherian category and C ⊂ A a thick subcategory. Thefollowing two properties are equivalent:(a) C is almost split.(b) For each object a ∈ A which does not lie in C there exists a nontrivial C -torsion-freesubobject of a . Proof.
We only discuss the nontrivial direction. Start with a nontrivial C -torsion-free subobject a ≤ a . If a/a lies in C we are done. Otherwise define a to be the preimage in a of anontrivial C -torsion-free subobject in a/a . By Lemma 2.1, a is C -torsion-free. Iterating theprocess yields a strictly ascending chain of C -torsion-free subobjects of a . Due to Noetherianitythis iteration has to stop, say at a n , and it can only stop at a n if a/a n lies in C . (cid:3) Example 6.5.
Let S be the polynomial ring k [ x , . . . , x n ] graded by total degree and A thecategory of f.g. graded S -module. Consider the thick subcategory C of -dimensional gradedmodules. These are the modules living in a finite degree interval. For any M ∈ A there existsa maximal submodule N ∈ C , and let d be the smallest integer with N ≥ d = 0 . Then M ≥ d is anontrivial C -torsion-free submodule of M . Recall that A / C ≃
Coh P nk . Example 6.6.
Consider the cone σ = Cone((1 , , (1 , ⊂ R and the nonsmooth affinetoric variety U σ with Cox ring S = k [ x, y ] , deg x = deg y = 1 ∈ Cl( U σ ) = Z / Z , and affinecoordinate ring S = k [ x , xy, y ] . The kernel of the sheafification from the category A off.g. graded S -modules to Coh U σ is the thick subcategory C of graded modules with M = 0 (cf. [CLS11, Proposition 5.3.3]). The graded S -module M := S/ h x , xy, y i violates condition(b) of the previous remark. Hence, C ⊂ A is not almost split.Now we show that b Q Ext is not surjective. Note that all
Ext c A vanish for c > , where is the global dimension of S . However, for the S -module k = S / h x , xy, y i the group Ext c A / C ( k, k ) = 0 for all c ∈ Z ≥ (in fact, Ext c A / C ( k, k ) ∼ = k for all c > ). The sheafificationof k is the skyscraper sheaf in Coh U σ ≃ A / C on the singular point of Spec( S ) .One can replace C -torsion-free A -complexes having defects in C with exact A -complexes,which are equivalent in the following sense: Definition 6.7.
Let C be a thick subcategory of the Abelian category A and e an A -complex.We say a subcomplex e ′ equals e up to C -factors if e/e ′ is a complex in C . Lemma 6.8.
Let C be an almost split thick subcategory of the Abelian category A and e : 0 ←− M ←− G c ←− · · · ←− G ←− N ←− . N THE
Ext -COMPUTABILITY OF SERRE QUOTIENT CATEGORIES 9 a C -torsion-free A -complex which is exact up to C -defects. Then there exists an exact ( C -torsion-free) A -subcomplex e ⊥ of ee ⊥ : 0 ←− M ⊥ ←− G ⊥ c ←− · · · ←− G ⊥ ←− N ←− which equals e up to C -factors. NG ⊥ G G ⊥ G G ⊥ c G c M ⊥ f MM . . .
Proof.
Define G := G / im A ( G ← ֓ N ) . Construct the subobject G ⊥ ≤ G as the preimagein G of an almost C -complement in G . Since G ⊥ is the preimage of an almost C -complementit follows that G /G ⊥ ∈ C .For i > we assume to have constructed G ⊥ i − ≤ G i − with G i − /G ⊥ i − ∈ C . We proceedinductively and consider G i := G i / im A ( G i ←− G i − ← ֓ G ⊥ i − ) . Again construct the subobject G ⊥ i ≤ G i as the preimage in G i of an almost C -complement in G i . As above G i /G ⊥ i ∈ C .Finally define the subobject M ⊥ ≤ M as the A -image im A ( M ←− G c − ← ֓ G ⊥ c − ) . Let f M := im A ( M ←− G c − ) . Then f M /M ⊥ ∈ C as an epimorphic image of G c − /G ⊥ c − under M ←− G c − . Since also M/ f M ∈ C it follows that M/M ⊥ ∈ C as an extension of two objectsin C . The whole argument is visualized in the diagram below, where the dotted lines stand for(factor) objects in C . (cid:3) The above lemma yields the preimages needed to prove the surjectivity of b Q Ext . Proposition 6.9.
Let C be an almost split localizing subcategory of the Abelian category A .Then b Q Ext : lim −→ M ′ ≤ M,M/M ′ ∈C Ext c A ( M ′ , N ) → Ext c Sat C ( A ) ( M, N ) is an epimorphism (of Abelian groups) for all C -saturated M, N ∈ A .Proof.
For the surjectivity consider a c -extension b e ∈ Ext c Sat C ( A ) ( M, N ) for c > , representedby an exact Sat C ( A ) -complex b e : 0 ←− M ←− G c ←− · · · ←− G ←− N ←− . Cf. [Bar09] for the use of Hasse diagrams to prove statements in Abelian categories.
Lemma 6.8 applied to the A -complex e = ι ( b e ) which is exact up to C -defects yields a preimage e ⊥ of b e . (cid:3) Due to the left exactness of ι we can even choose G ⊥ := G when applying Lemma 6.8 inthe proof of Proposition 6.9. This is illustrated by the diagram below. NG = G ⊥ G ⊥ G G ⊥ G . . . The proof of injectivity for higher
Ext ’s.
To prove the injectivity we show that almost C -complements exist on the level of exact A -complexes. Definition 6.10.
Let C be a thick subcategory of the Abelian category A and e : 0 ←− M ←− G c ←− · · · ←− G ←− N ←− an A -complex where M, N are C -torsion-free. We call a C -torsion-free A -subcomplex e e ≤ e e e : 0 ←− f M ←− e G c ←− · · · ←− e G ←− N ←− ,which equals e up to C -factors an almost C -complement in e . Proposition 6.11.
Let C be an almost split thick subcategory of the Abelian category A and e : 0 ←− M ←− G c ←− · · · ←− G ←− N ←− an exact A -complex where M, N are C -torsion-free. Then there exists an exact almost C -complement e e ∈ Ext c A ( f M , N ) in e . The value of this proposition lies in the following fact:
Lemma 6.12.
The A -subcomplex e e ≤ e in the previous proposition represents in the colimit lim −→ M ′ ≤ M,M/M ′ ∈C Ext c A ( M ′ , N ) the same c -cocycle as e .Proof. The cocycle e ∈ Ext c A ( M, N ) is identical to the Yoneda product e ′ := ( f M ֒ → M ) e : 0 ←− f M ←− G ′ c ←− G c − ←− · · · ←− G ←− N ←− ∈ Ext c A ( f M , N ) , in the colimit lim −→ M ′ ≤ M,M/M ′ ∈C Ext c A ( M ′ , N ) . N THE
Ext -COMPUTABILITY OF SERRE QUOTIENT CATEGORIES 11 e : 0 M G c G c − · · · G G N e ′ : 0 f M G ′ c G c − · · · G G N e e : 0 f M e G c e G c − · · · e G e G N By definition f M և G ′ c ֒ → G c is the pullback of f M ֒ → M և G c . The two morphisms f M և e G c ֒ → G c form a commutative square with f M ֒ → M և G c . The universal property ofthe pullback yields a mono e G c ֒ → G ′ c and we get an embedding of e e in e ′ yielding the same c -cocycle in Ext c A ( f M , N ) . (cid:3) For the induction proof of Proposition 6.11 we need the next lemma, which shows how toreplace short exact sequences in A by short exact sequences of C -torsion-free objects. Lemma 6.13.
Let C be an almost split thick subcategory of the Abelian category A and e : 0 ←− M ←− G ←− L ←− a short exact A -sequence with C -torsion-free M . Then there existsa short exact A -subsequence e ′ : 0 ←− M ′ ←− G ′ ←− L ′ ←− with C -torsion-free objects whichequals e up to C -factors. G G ′ L L ′ M ′ M M ′′ Proof.
We interpret L as a subobject of G with factor object M . Let G ′ bean almost C -complement in G . Hence G/G ′ lies in C . Now define L ′ = L ∩ G ′ and M ′ := G ′ /L ′ . The object L ′ is C -torsion-free as a subobject of G ′ and M ′ is C -torsion-free since it is isomorphic to the subobject M ′′ =( G ′ + L ) /L of M = G/L . Finally
M/M ′′ lies in C as a factor of G/G ′ and L/L ′ lies in C since it is isomorphic to the subobject ( G ′ + L ) /G ′ of G/G ′ . (cid:3) Proof of Proposition 6.11.
We will construct e e by induction on c . For c = 1 take e e = e byLemma 2.1. Now assume the statement is true for c − (i.e., for complexes of length c + 1 ).Define L := im( G c ←− G c − ) . Write e as the Yoneda product (i.e., concatenation) e e ofthe short exact A -sequence e : 0 ←− M ←− G c ←− L ←− and the exact A -complex e :0 ←− L ←− G c − ←− · · · ←− G ←− N ←− . First apply Lemma 6.13 to e and obtain theshort exact C -torsion-free A -sequence e ′ : 0 ←− M ′ ←− G ′ c ←− L ′ ←− . Now define the exact A -subcomplex e ′ := ( L ′ ֒ → L ) e : 0 ←− L ′ ←− G ′ c − ←− G c − ←− · · · ←− G ←− N ←− of e . By the induction hypothesis there exists an exact C -torsion-free A -subcomplex e e : 0 ←− e L ←− e G c − ←− e G c − ←− · · · ←− e G ←− N ←− of e ′ . Replacing L ′ by its subobject e L in e ′ we obtain the C -torsion-free A -complex e e ′ : 0 ←− M ′ ←− G ′ c ←− e L ←− which is exact up toa C -defect. Now apply Lemma 6.8 to e e ′ and obtain the short exact C -torsion-free A -sequence e e : 0 ←− f M ←− e G c ←− e L ←− . Finally define e e := e e e e .All constructions in this proof yield subcomplexes equal to their super-complexes up to C -factors. Thus, we conclude the e e equals e up to C -factors. (cid:3) By Remark 6.1 the following theorem is the equivalent “saturated form” of Theorem 1.1.
Theorem 6.14. If C is an almost split localizing subcategory of the Abelian category A then b Q Ext : lim −→ M ′ ≤ M,M/M ′ ∈C Ext c A ( M ′ , N ) → Ext c Sat C ( A ) ( M, N ) is an isomorphism (of Abelian groups) for all C -saturated M, N ∈ A .Proof.
Let e ′ ∈ Ext c A ( M ′ , N ) and e ′′ ∈ Ext c A ( M ′′ , N ) be two cocycles which map to the sameelement b e := b Q ( e ′ ) = b Q ( e ′′ ) ∈ Ext c Sat C ( A ) ( M, N ) . By Proposition 6.11 we can pass in thecolimit to C -torsion-free representatives e e ′ ∈ Ext c A ( f M ′ , N ) and e e ′′ ∈ Ext c A ( g M ′′ , N ) , which areexact A -subcomplexes of e ′ and e ′′ , respectively. Furthermore, e e ′ is an A -subcomplex of ι ( b e ) ,as it is C -torsion-free and the kernel of e ′ −→ ι ( b e ) is H C ( e ′ ) ; the same holds for e e ′′ . Taking theintersection of e e ′ and e e ′′ as subcomplexes of ι ( b e ) we obtain an A -subcomplex ˘ e of ι ( b e ) , which isnot necessarily exact. Lemma 6.8 yields an exact A -subcomplex e ⊥ ≤ ˘ e which still representsthe same cocycle as e e ′ and e e ′′ and hence e ′ and e ′′ in the colimit, and thus all these cocycles areequal in the colimit. (cid:3) A PPENDIX
A. S
KETCH OF THE PROPER CONSTRUCTIVE SETUP
We now roughly describe the constructive context of this paper. A detailed description wouldrequire a more elaborate preparation and would distract from the main result of this paper, whichin this form should already be self-contained. The standard way to express mathematical no-tions constructively is to provide algorithms for all disjunctions and all existential quantifiersappearing in the defining axioms of a mathematical structure. In the case of Abelian cate-gories this led us to the notion of a computable Abelian or constructively Abelian category [BLH11]. Given that, all constructions which only depend on a category being Abelian be-come computable . The computability of A implies, in particular, that we can compute in its Hom -groups only locally , i.e., we can decide element membership in the
Hom -sets, whethermorphisms are zero, add and subtract morphisms, and hence decide the equality of two mor-phisms. This does not imply that we can “oversee” a
Hom -group in any way, not even beingable to decide its triviality (see
Hom -computability below).For an Abelian category A with thick subcategory C ⊂ A we prove in [BLH14c] that A / C is computable once the Abelian category A is computable and the membership in C ⊂ A isconstructively decidable. A constructive treatment of spectral sequences along these lines can be found in [Bar09] with a computerimplementation in [BLH14a].
N THE
Ext -COMPUTABILITY OF SERRE QUOTIENT CATEGORIES 13
We call
C ⊂ A constructively localizing if there exists algorithms to compute the Gabrielmonad S ◦ Q together with its unit. Formula (Hom) in Section 3 proves that if A is Hom -computable and
C ⊂ A is constructively localizing then A / C is Hom -computable, where
Hom -computability means the computability as an enriched category over a computable monoidalcategory .Theorem 1.1 implies that A / C is Ext -computable if A is Ext -computable and
C ⊂ A isalmost split localizing and constructively localizing and the direct limit is constructive. Wewould define
Ext -computability to be the
Hom -computability of the derived category of A .This would lead too far away.Finally, we note that the entire proof of Theorem 1.1 is constructive and suited for computerimplementation. So if we assume that A is computable and C ⊂ A is constructively almost splitlocalizing then the proof of Theorem 1.1 provides an algorithm to compute images and preim-ages of elements represented as Yoneda cocycles under b Q Ext : lim −→ M ′ ≤ M,M/M ′ ∈C Ext c A ( M ′ , N ) → Ext c Sat C ( A ) ( M, N ) . Furthermore, if A is Hom -computable and has constructively enough pro-jectives or injectives then we can decide equality of (Yoneda) cocycles (cf. [BB08, Appen-dix B]). A
CKNOWLEDGMENTS
We thank Markus Perling and Greg Smith for discussions on the range of applicability ofTheorem 1.1 to coherent sheaves. We are indebted to anonymous referee who spotted a seriousissue in our first version of the injectivity proof. Addressing it helped us to correct, streamline,and simplify the whole argument. R
EFERENCES [Bar09] Mohamed Barakat,
Spectral filtrations via generalized morphisms , submitted ( arXiv:0904.0240 )(v2 in preparation), 2009. 9, 12[BB08] Mohamed Barakat and Barbara Bremer,
Higher extension modules and the Yoneda product , submitted( arXiv:0802.3179 ), 2008. 5, 13[Be˘ı78] A. A. Be˘ılinson,
Coherent sheaves on P n and problems in linear algebra , Funktsional. Anal. iPrilozhen. (1978), no. 3, 68–69. MR 509388 (80c:14010b) 4[BLH11] Mohamed Barakat and Markus Lange-Hegermann, An axiomatic setup for algorithmic homologicalalgebra and an alternative approach to localization , J. Algebra Appl. (2011), no. 2, 269–293,( arXiv:1003.1943 ). MR 2795737 (2012f:18022) 12[BLH13] Mohamed Barakat and Markus Lange-Hegermann, On monads of exact reflective localizations ofAbelian categories , Homology Homotopy Appl. (2013), no. 2, 145–151, ( arXiv:1202.3337 ).MR 3138372 3, 13 as we call it in [BLH13]. Enriched categories are usually required to be small. In an algorithmic setting any category is small, as thepossible states of the computer memory is a set. ... of Abelian groups, k -vector spaces, etc., depending on the context. I.e., constructively localizing and that we can algorithmically construct the maximal almost C -complement ofobjects in A [BLH14a] Mohamed Barakat and Markus Lange-Hegermann, The homalg package – A homo-logical algebra
GAP4 meta-package for computable Abelian categories , 2007–2014,( http://homalg.math.rwth-aachen.de/index.php/core-packages/homalg-package ).12[BLH14b] Mohamed Barakat and Markus Lange-Hegermann,
Characterizing Serre quotients with no sectionfunctor and applications to coherent sheaves , Appl. Categ. Structures (2014), no. 3, 457–466,( arXiv:1210.1425 ). MR 3200455 1[BLH14c] Mohamed Barakat and Markus Lange-Hegermann, Gabriel morphisms and the computability of Serrequotients with applications to coherent sheaves , ( arXiv:1409.2028 ), 2014. 12[BS98] M. P. Brodmann and R. Y. Sharp,
Local cohomology: an algebraic introduction with geometric ap-plications , Cambridge Studies in Advanced Mathematics, vol. 60, Cambridge University Press, Cam-bridge, 1998. MR 1613627 (99h:13020) 4[CLS11] David A. Cox, John B. Little, and Henry K. Schenck,
Toric varieties , Graduate Studies in Mathematics,vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322 (2012g:14094) 8[DE02] Wolfram Decker and David Eisenbud,
Sheaf algorithms using the exterior algebra , Computationsin algebraic geometry with Macaulay 2, Algorithms Comput. Math., vol. 8, Springer, Berlin, 2002,pp. 215–249. MR MR1949553 4[EFS03] David Eisenbud, Gunnar Fløystad, and Frank-Olaf Schreyer,
Sheaf cohomology and free resolu-tions over exterior algebras , Trans. Amer. Math. Soc. (2003), no. 11, 4397–4426 (electronic).MR MR1990756 (2004f:14031) 4[Gab62] Pierre Gabriel,
Des catégories abéliennes , Bull. Soc. Math. France (1962), 323–448. MR 0232821(38 A course in homological algebra , second ed., Graduate Texts in Math-ematics, vol. 4, Springer-Verlag, New York, 1997. MR MR1438546 (97k:18001) 5[HSS06] Milena Hering, Hal Schenck, and Gregory G. Smith,
Syzygies, multigraded regularity and toric vari-eties , Compos. Math. (2006), no. 6, 1499–1506. MR 2278757 (2007k:13025) 2[ML95] Saunders Mac Lane,
Homology , Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint ofthe 1975 edition. MR MR1344215 (96d:18001) 4[MS04] Diane Maclagan and Gregory G. Smith,
Multigraded Castelnuovo-Mumford regularity , J. ReineAngew. Math. (2004), 179–212. MR 2070149 (2005g:13027) 2[MS05] Diane Maclagan and Gregory G. Smith,
Uniform bounds on multigraded regularity , J. AlgebraicGeom. (2005), no. 1, 137–164. MR 2092129 (2005g:14098) 2[Oor64] F. Oort, Yoneda extensions in abelian categories , Math. Ann. (1964), 227–235. MR 0162836 (29
A lifting functor for toric sheaves , Tohoku Math. J. (2) (2014), no. 1, 77–92,( arXiv:1110.0323 ). MR 3189480 4[Smi00] Gregory G. Smith, Computing global extension modules , J. Symbolic Comput. (2000), no. 4-5, 729–746, Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). MR 1769664(2001h:14013) 2[Smi13] Gregory G. Smith, Computing global extension modules , Mini-Workshop: Constructive homologicalalgebra with applications to coherent sheaves and control theory, Oberwolfach Reports, no. 25, MFO,Oberwolfach, 2013, ( ). 2D
EPARTMENT OF MATHEMATICS , U
NIVERSITY OF K AISERSLAUTERN , 67653 K
AISERSLAUTERN , G ER - MANY
E-mail address : [email protected] L EHRSTUHL B FÜR M ATHEMATIK , RWTH A
ACHEN U NIVERSITY , 52062 A
ACHEN , G
ERMANY
E-mail address ::