aa r X i v : . [ m a t h . AG ] A ug A lifting functor for toric sheaves
Markus Perling ∗ August 2012
Abstract
For a variety X which admits a Cox ring, we introduce a functor from the categoryof quasi-coherent sheaves on X to the category of graded modules over the homogeneouscoordinate ring of X . We show that this functor is right adjoint to the sheafification functorand therefore left-exact. Moreover, we show that this functor preserves torsion-freeness andreflexivity. For the case of toric sheaves, we give a combinatorial characterization of its rightderived functors in terms of certain right derived limit functors. Consider an affine normal variety W = Spec( S ) over an algebraically closed field K , G adiagonalizable group scheme which acts on W , and H ⊆ G a closed subgroup scheme. We denote T the quotient of diagonalizable groups schemes G/H . Moreover, we assume the following. • There exists a Zariski-open G -invariant subset ˆ X of W such that a good quotient X =ˆ X//H exists. We denote π : ˆ X → X the corresponding projection. • X admits an affine T -invariant open covering (this is automatic if T is a torus, see[Sum74]). • The complement Z = W \ ˆ X has codimension at least 2.The actions of G and H on W induce gradings on S by the character groups X ( G ) and X ( H ),respectively, which are compatible via the surjection X ( G ) ։ X ( H ). With this, we requiremoreover the following. • X ( H ) ∼ = A d − ( X ), the divisor class group, and for suitable representatives D χ ∈ A d − there exists an isomorphism of S -modules S ∼ = L χ ∈ X ( H ) Γ (cid:0) O ( D χ ) (cid:1) which is compatiblewith the X ( H )-grading of S . In particular, the latter carries an induced ring structure.These conditions essentially imply that X is a variety which admits a Cox ring, where we admitpossibly some further action by a diagonalizable group scheme. In particular, this class ofvarieties includes the Mori dream spaces. The main application we have in mind is the casewhere T is a torus and X a toric variety such that S is the associated homogeneous coordinatering as defined in [Cox95]. It was shown in [PT10, §
6] that by taking local invariants we obtainan exact and essentially surjective functor which maps an X ( G )-graded S -module E to a T -equivariant quasi-coherent sheaf e E on X , the so-called sheafification functor . Conversely, thereis a functor from the category of T -equivariant sheaves on X to the category of X ( G )-graded S -modules, mapping a quasi-coherent sheaf E to a X ( G )-graded S -module Γ ∗ E := Γ( ˆ X, π ∗ E ).This functor is right inverse to the sheafification functor, i.e., we have g Γ ∗ E ∼ = E for any E . ∗ Fakult¨at f¨ur Mathematik Ruhr-Universit¨at Bochum Universit¨atsstraße 150 44780 Bochum, Germany,
[email protected] ∗ in many cases is not very well behaved. So it usually does not preserveproperties such as torsion-freeness and reflexivity. Also, by being the composition of the right-exact functor π ∗ with the left-exact global section functor, Γ ∗ does not have any exactnessproperties. In general, Γ ∗ is right-exact if ˆ X = W (and thus X is affine) and it is left-exact if π is a flat morphism.The aim of this note is to construct an alternative functor to Γ ∗ , which we are going tocall the lifting functor , which maps a quasi-coherent T -equivariant sheaf E to an X ( G )-graded S -module b E . We will show that the lifting functor has the following two general properties:1. The lifting functor is right adjoint the sheafification functor and therefore left-exact (The-orem 3.8).2. Lifting preserves torsion-freeness and reflexivity. For torsion free sheaves it preservescoherence (Theorem 4.4).The lifting functor is an offspring of recent work on toric sheaves [Per11]. Assume that X is a toric variety and ˆ X the standard quotient presentation as in [Cox95]. By results ofKlyachko [Kly90], [Kly91], any coherent reflexive T -equivariant sheaf E can be described by afinite-dimensional vector space together with a family of filtrations parameterized by the raysof the fan associated to X . In order to represent E by an appropriate Z n -graded module overthe homogeneous coordinate ring, it is a rather straightforward observation that, rather thantaking Γ ∗ E , we can choose a reflexive sheaf which is associated to precisely the same filtrationsas E (this is possible because there is a one-to-one correspondence among the rays of the fansassociated to X and ˆ X , respectively). Our results show that this ad-hoc observation indeed hasa functorial interpretation. In Section 5 we will see that the lifting functor has moreover a verynice interpretation in the combinatorial setting of [Per11]. Let A be any abelian group, S a A -graded K -algebra, and E an A -graded S -module. Then E ∼ = L α ∈ A E α and for any β ∈ A we denote E ( β ) = L α ∈ A E α + β the degree shift of E by β . For any two S -modules E and F , The tensor product E ⊗ R F can be A -graded as follows.Consider first the K -vector space E ⊗ K F and set ( E ⊗ K F ) α = L β ∈ A ( E β ⊗ K F α − β ). Then for α ∈ A we form ( E ⊗ R F ) α as the quotient of ( E ⊗ K F ) α by the subvector space generated by re ⊗ f − e ⊗ rf for e ∈ E, f ∈ F, r ∈ R . Note that E ( α ) ⊗ R F ∼ = E ⊗ R F ( α ) ∼ = ( E ⊗ R F )( α ). For any A -graded S -modules E , F , the graded version of Hom S ( E, F ) by definition is givenby HOM AS ( E, F ) := M α ∈ A Hom AS ( E, F ( α )) , where Hom AS ( E, F ( α )) = { f ∈ Hom S ( E, F ) | f ( E β ) ⊆ F β + α for every β ∈ A } . We can considerin a natural way HOM AS ( E, F ) as a subset of Hom R ( E, F ). Moreover, within the graded setting,HOM AS satisfies the same general functorial properties as the standard Hom (see [Nv04, § A -graded modules, then the set of morphismsbetween modules E and F is given by Hom AS ( E, F ) and not by HOM AS ( E, F ). We will deal with three gradings, given by the character groups X ( T ), X ( G ), and X ( H ),respectively. Any given X ( G )-graded ring S carries an X ( H )-grading as well via the surjec-tion X ( G ) ։ X ( H ). To distinguish between these two gradings, we write the homogeneouscomponents S ( α ) and S χ for the X ( H )- and the X ( G )-grading, respectively, where α ∈ X ( H )and χ ∈ X ( G ). For χ ∈ X ( G ) we may also write S ( χ ) for the X ( H )-homogeneous componentdetermined by the image of χ in X ( H ). Then S ( χ ) has a natural X ( T )-grading which is given2y S ( χ ) ∼ = L η ∈ X ( T ) ( S ( χ ) ) η with ( S ( χ ) ) η = S χ + η . We use the same conventions for X ( G )- and X ( H )-graded S -modules. For any X ( G )-graded S -modules E, F , we have the two graded modules HOM X ( G ) S ( F, E )and HOM X ( H ) S ( F, E ), together with the natural sequence of inclusionsHOM X ( G ) S ( F, E ) ⊆ HOM X ( H ) S ( F, E ) ⊆ HOM S ( F, E )(which even satisfy certain topological properties, [Nv04, § The X ( H )-invariant subring R = S (0) is automatically X ( T )-graded. It is also X ( G )-graded by trivial extension, i.e., we set R χ = 0 for every χ ∈ X ( G ) \ X ( T ). Likewise, every X ( T )-graded R -module can be given an X ( G )-grading. With the notation as in 2.3, note that we have HOM AS ( F, E ) = L α ∈ A Hom AS ( F, E ( α )) = L α ∈ A Hom AS ( F ( − α ) , E )). However, in order to avoid some cumbersome signs, we will usu-ally write expressions like b E = L α ∈ A Hom AS ( S ( α ) , E ), where it is understood that the propergrading is given by ( b E ) α = Hom AS ( S ( − α ) , E ). The sheafification functor as defined in [PT10] maps an X ( H )-graded (respectively X ( G )-graded) S -module E to a quasi-coherent sheaf e E over X as follows. Let open affine covers { U i = Spec( R i ) } i ∈ I and { ˆ U i = π − ( U i ) = Spec( S i ) } i ∈ I on X and ˆ X , respectively, be given,such that U i = ˆ U i //H (both covers can be chosen T and G -invariant, respectively). Then R i = S Hi = ( S i ) (0) for every i ∈ I and we can associate to every U i the R i -module Γ( ˆ U i , E ) (0) ,where by abuse of notation we identify E with its associated quasi-coherent sheaf over W . Theseglue naturally to give a quasi-coherent sheaf of O X -modules. Moreover, if the U i are choosen T -invariant, then the R i are X ( T )-graded, and both the R i and S i are X ( G )-graded by 2.6. Inthis case, E has also an induced T -equivariant structure. For a given morphism of schemes f : U → V and a quasi-coherent sheaf F on V , it is stan-dard to define the pullback f ∗ F as f − F ⊗ f − O V O U . This defines a right-exact functor fromthe category of (quasi-)coherent O V -modules to the category of (quasi-)coherent O U -modules.However, this is not the only conceivable way to define a pull-back functor; instead, one couldconsider the sheaf f ˆ F := H om f − O V ( O U , f − F ) . Clearly, f ˆ is a left-exact functor from the category of quasi-coherent O V -modules to the categoryof quasi-coherent O U -modules. In the affine case, i.e., U = Spec( A ), V = Spec( B ) for somecommutative rings A , B , and F the sheaf corresponding to an B -module F , f ˆ F correspondsto the module Hom B ( A, F ), where the A -module structure is given by ( rg )( r ′ ) = g ( rr ′ ) for r, r ′ ∈ R and g ∈ Hom B ( A, F ).However, the following example shows that f ˆ in general will behave quite pathological. Example 3.1.
Assume R = F = K and T = K [ x ]. Then we have isomorphisms of K -vectorspaces Hom K ( K [ x ] , K ) ∼ = Hom K ( M i ≥ K, K ) ∼ = Y i ≥ Hom K ( K, K ) ∼ = Y i ≥ K. That is, from a one-dimensional K -vector space we have created a K [ x ]-module which hastorsion elements and no countable generating set.3e will see that one can define a better behaved graded version of this pull-back. Underour general assumptions on X and ˆ X , let { U i } i ∈ I and { ˆ U i = π − ( U i ) } i ∈ I be affine T - and G -invariant covers, respectively, as in 2.8. By the general properties of good quotients, the ˆ U i forman affine open covering of ˆ X such that U i = ˆ U i //H for every i ∈ I . We denote U i = Spec( R i )and ˆ U i = Spec( S i ); then R i = S Hi for every i ∈ I . Moreover, both the R i and S i are X ( G )-graded by 2.6. For any character (and thus divisor class of X ) α ∈ X ( H ), there is naturallyassociated the module O ( α ) ∼ = ] S ( α ), which is reflexive and of rank one. This module is adistinguished representative for the isomorphism class of such sheaves associated to the class α .Similarly, if we choose some χ ∈ X ( G ) which maps to α via the surjection X ( G ) ։ X ( H ), weobtain an induced T -equivariant structure on O ( α ), which we denote by O ( χ ) ∼ = ] S ( χ ). Definition 3.2.
Let E be a T -equivariant quasi-coherent sheaf on X . Then we set E H := M α ∈ X ( H ) H om O X ( O ( α ) , E ) . and E G := M χ ∈ X ( G ) H om T O X ( O ( χ ) , E ) . We first show the following.
Proposition 3.3.
Let E be a T -equivariant quasi-coherent sheaf on X .(i) Both E H and E G are quasi-coherent subsheaves of π ˆ E , and E H ∼ = E G as O ˆ X -modules.(ii) O HX (and therefore O GX ) is isomorphic to O ˆ X .(iii) If Γ( U i , E ) is a first syzygy module for every i , then so is Γ( ˆ U i , E H ) .(iv) If E is coherent and torsion free, then E H and E G are coherent and torsion free as well.Proof. (i) First note that for every χ ∈ X ( G ) which maps to α ∈ X ( H ), we have a natural inclu-sion of sheaves of K -vector spaces φ χ : H om T O X ( O ( χ ) , E ) ֒ → H om O X ( O ( α ) , E ). Summing overall such characters, we get a map of sheaves P η ∈ X ( T ) φ χ + η : L η ∈ X ( T ) H om T O X ( O ( χ + η ) , E ) →H om O X ( O ( α ) , E ). Locally, we denote E i := Γ( U i , E ) for every U i and this map translatesto an isomorphism of R i -modules L η ∈ X ( T ) Hom X ( T ) R i (( S i ) ( χ + η ) , E i ) → HOM X ( T ) R i (( S i ) ( α ) , E i ).Because ( S i ) ( α ) is a finitely generated R i -module by our general assumptions, the latter is iso-morphic to Hom R i (( S i ) α , E i ) (see [Nv04, Cor. 2.4.4]). So, φ α is indeed an isomorphism and bysumming over all χ ∈ X ( G ), we get an isomorphism P χ ∈ X ( G ) φ χ : E G → E H . Now, Γ( ˆ U i , E H ) ∼ = L α ∈ X ( H ) Hom R i (( S i ) ( α ) , E i ) and therefore E H (and thus E G ) is quasi-coherent. Moreover,observe that locally we have Γ( ˆ U i , π ˆ E ) ∼ = Hom R i ( S i , E ) ∼ = Hom R i ( L α ∈ X ( H ) ( S i ) ( α ) , E i ) ⊇ L α ∈ X ( H ) Hom R i (( S i ) ( α ) , E i ), so E H (and thus E G ) indeed is a subsheaf of π ˆ E .(ii) It suffices to show that for any i the module ˆ R i := L α ∈ X ( H ) Hom R i (( S i ) ( α ) , R i ) isnaturally isomorphic to S i . For this, we observe that for every α ∈ X ( H ) holds ( ˆ R i ) ( α ) =Hom R i (( S i ) ( − α ) , R i ) ∼ = ( S i ) ( α ) , as the ( S i ) ( α ) are reflexive modules of rank one by our general as-sumptions. Therefore we have natural isomorphisms ˆ R i ∼ = L α ∈ X ( H ) ( ˆ R i ) ( α ) ∼ = L α ∈ X ( H ) ( S i ) ( α ) ∼ = S i .(iii) By assumption, we can represent E i as a first syzygy 0 → E i → R ⊕ Ii , where I issome index set. Applying L α ∈ X ( H ) Hom R i (( S i ) ( α ) , ) preserves left-exactness and direct sumsin the right argument, and so we obtain an exact sequence 0 → ˆ E i → ˆ R ⊕ Ii ∼ = S ⊕ Ii , whereˆ E i := L α ∈ X ( H ) Hom R i (( S i ) ( α ) , E i ) ∼ = Γ( ˆ U i , E H ), and the latter isomorphism follows from (ii).4iv) It suffices to show that for any i the module ˆ E i := L α ∈ X ( H ) Hom R i (( S i ) ( α ) , E i ) is torsionfree. Because E i is by assumption torsion free and finitely generated, it can be represented asa first syzygy module 0 → E i → R n i i for some integer n i . Applying (iii), we obtain an exactsequence 0 → ˆ E i → S n i i . Hence, ˆ E i is finitely generated and torsion free.We come now to our main definition. Definition 3.4.
Let E be a T -equivariant quasi-coherent sheaf on X . Then we call the X ( G )-graded S -module b E := Γ( ˆ X, E G )the lifting of E . Remark 3.5.
Note that the S -module b E carries both a X ( G )-grading as well as a X ( H )-grading, which are given by b E ∼ = M α ∈ X ( H ) Hom O X ( O ( α ) , E ) ∼ = M χ ∈ X ( G ) Hom T O X ( O ( χ ) , E )(see 2.7 for our convention on the grading). By construction, lifting is functorial and left-exact. Moreover, if X is smooth, then every sheaf of the form O ( α ) is invertible and wehave natural isomorphisms Hom O X ( O ( α ) , E ) ∼ = Γ( X, E ⊗ O X O ( − α )) for every α . (respectivelyHom T O X ( O ( χ ) , E ) ∼ = Γ( X, E ⊗ O X O ( − χ )) T for every χ ). In this case, our lifting functor isnaturally equivalent to the usual lifting functor Γ ∗ . Proposition 3.6.
The sheafification functor is left-inverse to the lifting functor.Proof.
We show that ( b E )˜ ∼ = E for any T -equivariant quasi-coherent sheaf on X . The corre-sponding statement about morphisms then will be evident. With notation as in the proof ofProposition 3.3, we have for every i ∈ I Γ( U i , ( b E )˜) = HOM X ( G ) R i ( S i , E i ) (0) = Hom X ( G ) R i (( S i ) (0) , E i ) = Hom X ( T ) R i ( R i , E i ) ∼ = E i . By naturality, the E i glue to yield E .Before we can prove our main result, we need to clarify how homomorphism spaces arerelated under going back and forth under lifting and sheafification. Lemma 3.7. (i) For any X ( G ) -graded S -module E there exists a natural homomorphism of X ( G ) -graded S -modules E → b ˜ E .(ii) Let E , F be T -equivariant quasi-coherent sheaves on X . Then sheafification induces asurjective homomorphism of K -vector spaces Hom X ( G ) S ( b E , b F ) ։ Hom T O X ( E , F ) . (iii) Let E be a X ( G ) -graded S -module and F be a quasi-coherent sheaf on X . Then sheafifi-cation induces a surjective homomorphism of K -vector spaces Hom X ( G ) S ( E, b F ) Hom T O X ( ˜ E, F ) . Proof. (i) Degree-wise we define a map φ χ : E ( χ ) → Hom X ( G ) S ( S ( − χ ) , E (0) ) for χ ∈ X ( G ) bysetting ( φ χ ( e ))( s ) := s · e for every s ∈ S ( − χ ) . We leave it to the reader to check that this indeedyields a X ( G )-homogeneous homomorphism of S -modules.5ii) By revisiting the constructions of the proof of Proposition 3.6, we conclude that thefunctorially induced compositionHom T O X ( E , F ) −→ Hom X ( G ) S ( b E , b F ) −→ Hom T O X ( E , F )is a natural isomorphism. In particular, the second homomorphism is surjective.(iii) By (i) we obtain a homomorphism of S -modules Hom X ( G ) S ( b ˜ E, b F ) → Hom X ( G ) S ( E, b F )which naturally commutes with the maps Hom X ( G ) S ( b ˜ E, b F ) → Hom T O X ( ˜ E, F ) and Hom X ( G ) S ( E, b F ) → Hom T O X ( ˜ E, F ), respectively, which are induced by sheafication. By (ii), the first map issurjective, hence the second must be surjective, too.The can now show our main results, which in particular implies that lifting is left-exact. Theorem 3.8.
The lifting functor from the category of T -equivariant quasi-coherent sheaveson X to the category of X ( G ) -graded S -modules is right adjoint to the sheafification functor.Proof. We first consider the affine situation and assume that ˆ X = Spec( S ) and X = Spec( R ) =Spec( S (0) ). Denote E an X ( G )-graded S -module and F an X ( T )-graded (and therefore X ( G )-graded, see 2.6) R -module. For simplicity, we write b F for the lifting of F . Then we have theisomorphisms of X ( G )-graded R -modulesHOM X ( G ) S ( E, b F ) = HOM X ( G ) S (cid:0) E, HOM X ( G ) R ( S, F ) (cid:1) ∼ = HOM X ( G ) R ( E ⊗ S S, F ) ∼ = HOM X ( G ) R ( E, F ) . Taking invariants with respect to the X ( G )-grading, we getHOM X ( G ) S ( E, b F ) (0) = HOM X ( G ) R ( E, F ) (0) = Hom X ( G ) R ( E , F ) = Hom X ( T ) R ( E , F ) , where the second equality follows form the fact that F is concentrated in X ( H )-degree zero.For the general case, consider a T -equivariant sheaf F on X and an X ( G )-graded S -module E whose restriction to ˆ X corresponds to a G -equivariant quasi-coherent sheaf E . Asabove, denote { U i } i ∈ I , { ˆ U i } i ∈ I a T -invariant (resp. G -invariant) affine cover of X (resp. ˆ X ).The affine case considered before corresponds to isomorphisms Γ (cid:0) ˆ U i , H om G O ˆ Ui ( E| ˆ U i , F G | ˆ U i ) (cid:1) → Γ (cid:0) U i , H om T O Ui ( e E | U i , F | U i ) (cid:1) for every i ∈ I . These isomorphisms commute naturally with therestrictionsΓ( ˆ U i , H om G O ˆ Ui ( E| ˆ U i , F G | ˆ U i )) → Γ( ˆ U i ∩ ˆ U j , H om G O ˆ Ui ∩ ˆ Uj ( E| ˆ U i ∩ ˆ U j , F G | ˆ U i ∩ ˆ U j ))and Γ (cid:0) U i , H om T O Ui ( e E | U i , F | U i ) (cid:1) → Γ (cid:0) U i ∩ U j , H om T O Ui ∩ Uj ( e E | U i ∩ U j , F | U i ∩ U j ) (cid:1) , respectively for i, j ∈ I . Therefore we obtain an induced homomorphismHom G O ˆ X ( E , F G ) = Γ (cid:0) ˆ X, H om G O ˆ X ( E , F G ) (cid:1) → Γ (cid:0) X, H om T O X ( e E, F ) (cid:1) = Hom T O X ( e E, F ) . By the naturality of the local isomorphisms and the property that H om G O ˆ X ( E , F G ) is a sheaf itfollows that this homomorphism is an isomorphism. It remains to show that Hom X ( G ) S ( E, b F ) =Hom G O W ( E, b F ) equals Hom G O ˆ X ( E , F G ). For this, consider the commutative diagramHom G O W ( E, b F ) φ (cid:15) (cid:15) ψ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ Hom G O ˆ X ( E , F G ) ∼ = / / Hom T O X ( e E, F ) , φ is the restriction map and ψ the map induced by the sheafification functor. φ is injectivebecause b F is an extension of F G from ˆ X to W and therefore does not have torsion with supporton Z . Now, ψ is surjective by Lemma 3.7 (iii), hence both φ and ψ are isomorphisms. Remark 3.9.
From the proofs of 3.6 and 3.8, it follows that the counit of the adjunction isfor every T -equivariant quasicoherent sheaf E given by the natural map ( b E )˜ → E which, usingnotation from the proof of 3.6, is locally given by the natural isomorphisms Hom X ( T ) R i ( R i , E i ) ≡ −→ E i . This is an interesting observation, as it implies that the category of T -equivariant sheaveson X is a reflective localization of the category of X ( G )-graded S -modules by the kernel of thesheafification functor. This was previously only known for the case where X is smooth. As waspointed out to me by M. Barakat and M. Lange-Hegermann, this is relevant for current work[BLH12] related to computational toric geometry. We have seen in Proposition 3.3 that a torsion free coherent sheaf E on X lifts to a torsion freecoherent sheaf E H on ˆ X . In this section we want to give similar and refined criteria for thelifting b E . By Proposition 3.3 (i), properties such as coherence and torsion-freeness do not depend onthe additional T -equivariant structure of E . As our proofs below depend on finding suitable opensubsets on X , which must not necessarily be T -invariant, we will therefore consider without lossof generality only the coarser grading by X ( H ) rather than the X ( G )-grading. Proposition 4.2.
Let D be a Weil divisor on X and denote α ∈ X ( H ) ∼ = A d − ( X ) thecorresponding class. Then \ O ( D ) ∼ = S ( α ) . In particular, b O X ∼ = S .Proof. By the isomorphism O ( D ) ∼ = O ( α ), we have a decomposition as observed in Remark 3.5: \ O ( D ) ∼ = M β ∈ X ( H ) Hom O X ( O ( β ) , O ( α )) ∼ = M β ∈ X ( H ) Γ( ˆ X, O ( α − β )) ∼ = S ( α ) . By the general properties of good quotients, any open subset U of X can be representedas a good quotient ˆ U //H , where ˆ U is the preimage of U in ˆ X under the quotient map. If U = Spec( R ), then from the proof of Propositions 3.3 (ii) and 4.2, we can conclude thatˆ U = Spec( b R ) and R = b R (0) with respect to the natural X ( H )-grading of b R . Theorem 4.4.
Let E be a T -equivariant coherent sheaf on X .(i) If E is torsion free then b E is torsion free and finitely generated.(ii) If E is reflexive then b E is reflexive and finitely generated.Proof. First we note that by the fact that ˆ X has codimension 2 in X , coherence (as well astorsion-freeness and reflexivity, respectively, see [Har80, § E H implies that the S -module b E is finitely generated (and torsion free, respectively reflexive). So, assertion (i) follows fromProposition 3.3 (iv).Now we prove (ii). If E is reflexive, then by [Har80, Proposition 1.1], we can choose for everypoint in X a neighbourhood U = Spec( R ) such that there exists a short exact sequence0 −→ Γ( U, E ) −→ R n −→ F −→ , F is a finitely generated, torsion free R -module. By 4.3, we have U ∼ = ˆ U //H withˆ U = Spec( b R ) and we can lift this sequence to0 −→ Γ( ˆ
U , E H ) −→ b R n −→ G −→ , where G is the homomorphic image of S n in b F and therefore torsion-free by Proposition 3.3 (iv).Applying again [Har80, Proposition 1.1], we conclude that E G locally reflexive and thereforereflexive. Hence, as the complement of ˆ X in W has codimension at least two, the module b E isreflexive by [Har80, Proposition 1.6].We will see in Example 5.6 that in general, coherence is not preserved for sheaves withtorsion. We now assume that X is a d -dimensional toric variety with associated fan ∆ and ˆ X ⊆ Z ∆(1) = W its standard quotient presentation. As a general reference to toric geometry we refer to[Oda88] and [Ful93]; for specifics of our setting see also [Per11]. It is customary to denote M := X ( T ) ∼ = Z d and ˆ T := G , such that X ( G ) ∼ = Z ∆(1) .Moreover, we denote N = M ∗ and ∆ consists of strictly convex polyhedral cones in N ⊗ Z R .We denote { l ρ } ρ ∈ ∆(1) the set of primitive vectors of the rays in ∆, which we interpret as linearforms on M . Elements m ∈ M can be considered as regular functions on T and therefore asrational functions on X . In this case, we write χ ( m ), where χ ( m + m ′ ) = χ ( m ) χ ( m ′ ) for any m, m ′ ∈ M . We have X ( H ) ∼ = A d − ( X ) and the inclusion of M in to Z ∆(1) yields the shortexact sequence 0 −→ M L −−→ Z ∆(1) −→ A d − ( X ) −→ , where L can be represented as a matrix whose rows are formed by the l ρ . For any strictly convexrational polyhedral cone σ ∈ ∆, we get an affine toric variety U σ whose M -graded coordinatering is given by K [ σ M ] with σ M = ˇ σ ∩ M and ˇ σ denotes the dual cone of σ in M ⊗ Z R . Similarly,we get an exact sequence M L σ −−−→ Z σ (1) −→ A d − ( U σ ) −→ , where L σ is the submatrix of L consisting of the rows which correspond to rays in σ (1).We start by recalling some facts about toric sheaves on affine toric varieties and posetrepresentations from [Per04] and [Per11]. Assume that σ is a cone and S = K [ N σ (1) ] thehomogeneous coordinate ring. For any m, m ′ ∈ M we write m ≤ σ m ′ if and only if m ′ − m ∈ σ M . This way we get a preordered set ( M, ≤ σ ), which is partially ordered if dim σ = d .Equivalently, M becomes a small category, where the morphisms are given by pairs ( m, m ′ )whenever m ≤ σ m ′ . By the preorder ≤ σ , M also becomes a topological space. Its topology isgenerated by open sets U ( m ) = { m ′ ∈ M | m ≤ σ m ′ } for every m ∈ M . Proposition 5.2 ([Per04, Prop. 5.5] & [Per11, Prop. 2.5]) . The following categories areequivalent:(i) Toric sheaves on U σ .(ii) M -graded K [ σ M ] -modules.(iii) Functors from ( M, ≤ σ ) to the category of K -vector spaces.(iv) Sheaves of K -vector spaces on M . E of ( M, ≤ σ ), the associated sheaf assigns to any opensubset U of M the limit lim ←− E m , with m ∈ U (see [Per11, Prop. 2.5]).Similarly, N σ (1) induces a partial order “ ≤ ” on Z σ (1) , which is compatible with ≤ σ n thefollowing way. Lemma 5.3. L σ ( m ) ≤ L σ ( m ′ ) if and only if m ≤ σ m ′ .Proof. We observe L σ ( m ) ≤ L σ ( m ′ ) ⇔ L σ ( m ′ ) − L σ ( m ) ∈ N n ⇔ l ρ ( m ′ − m ) ≥ ρ ∈ σ (1) ⇔ m ′ − m ∈ σ M .So, with respect to a fixed cone σ , it is natural to write m ≤ m ′ instead of L σ ( m ) ≤ L σ ( m ′ ),i.e., m ≤ m ′ if and only if m ≤ σ m ′ . Moreover, for every c ∈ Z n there exists some m ∈ M suchthat c ≤ m . To see this, we observe that we always can choose some m ∈ σ M with l ρ ( m ) > ρ ∈ σ (1) and some integer r > c ≤ r · m . So, for every c ∈ Z σ (1) we obtaina nonempty open subset U c of M which is given as U c = [ c ≤ m U ( m )By Proposition 5.2, every M -graded module E is equivalent to a sheaf of K -vector spaces on M which assigns to every open subset U of M the vector space E ( U ) = lim ←− E m , where the limit istaken over all m ∈ U . We use this to define a representation b E of ( Z σ (1) , ≤ ) by setting E c := E ( U c ) . By the functoriality of sheaves we have restriction maps E c → E c ′ whenever c ≤ c ′ . Hencewe obtain a functor from ( Z σ (1) , ≤ ) to the category of K -vector spaces and thus a Z σ (1) -graded S -module E := L c ∈ Z σ (1) E c by Proposition 5.2. Clearly this construction is functorial. Proposition 5.4.
Denote b E ∼ = L c ∈ Z σ (1) b E c the Z σ (1) -graded lifting of the sheaf over U σ associ-ated to E in the sense of Definition 3.4. Then the modules b E and E are naturally isomorphic.In particular, b E c ∼ = Hom MK [ σ M ] ( S ( c ) , E ) is naturally isomorphic to E c for every c ∈ Z σ (1) .Proof. We write c = (cid:0) c ρ | ρ ∈ σ (1) (cid:1) . We can consider S ( c ) as an M -graded K [ σ M ]-submoduleof the group ring K [ M ] with S ( c ) ∼ = L m Kχ ( m ), where the sum is taken over all m ∈ M with l ρ ( m ) ≥ − c ρ . Choose a minimal set of generators s , . . . , s t of S ( c ) with degrees m , . . . , m t .Then any M -homogeneous homomorphism is determined by the images of the s i in the homoge-neous components E m i . Hence, we can identify Hom MK [ σ M ] ( S ( c ) , E ) in a natural way with a sub-vector space of L ti =1 E m i consisting of tuples ( e , . . . , e t ) such that χ ( m − m i ) e i = χ ( m − m j ) e j whenever m i , m j ≤ σ m . But this vector space has the universal properties of the limit lim ←− E m and thus we can naturally identify it with b E c = lim ←− E m . The isomorphism of the modules b E and E then follows from the naturality of this identification. Remark 5.5.
By Theorem 3.8 the lifting functor is left-exact and to any toric sheaf E we canconsider its right derived modules b E = b E (0) , b E (1) , . . . By Proposition 5.4 we have now a very nice interpretation of these modules, as we can identifythem degree-wise with the right derived functors of the limit functor lim ←− . Right derived limitfunctors lim ←− i have been pioneered by Roos [Roo61] and have since been studied extensively.Roos also gives a combinatorial analog of the Cech complex which allows in simple cases theexplicit computation of the derived functors. We can now understand the left-exactness ofthe lifting functor combinatorially by the fact that the posets { m ∈ M | l ρ ( m ) ≥ − c ρ for all9 ∈ σ (1) } are not filtered, i.e., for any m, m ′ ∈ U c there may not exist any m ′′ ∈ U c with m ′′ ≤ σ m and m ′′ ≤ σ m ′ , which otherwise would imply the exactness of the limit functor (see[Jen72, Cor. 7.2]).The following example shows both that lifting in general does not preserve exactness, andthe existence of nontrivial right derived modules b E ( i ) . Example 5.6.
Let σ ⊂ N R ∼ = R the the cone generated by the primitive vectors l = (1 , , l = (0 , , l = ( − , , l = (0 , , m ⊂ K [ σ M ] the maximal homogeneous idealand consider K = K [ σ M ] / m as a simple module in degree 0. Now for a given c = ( c , c , c , c ) ∈ Z , it is straightforward to see that 0 ∈ M is a minimal element in { m ∈ M | l i ( m ) ≥ − c i } ifand only if c , c ≤ , c = c = 0 or c = c = 0 , c , c ≤ . If c satisfies one of these conditions, then b K c ∼ = K and b K c = 0 otherwise. So there is no lowerbound for the c i such that b K c vanishes and so b K cannot be finitely generated. We observe that b K is Artinian and is supported precisely on those torus orbits which get contracted to the fixedpoint under the quotient map A K → U σ .Moreover, by Lemma 4.2, we have \ K [ σ M ] ∼ = S and the long exact derived sequence of0 → m → K [ σ M ] → K → −→ b m −→ S −→ b K −→ b m (1) . By degree-wise inspection one can see that b m = ( x , x , x , x ), and therefore b m (1) cannot befinitely generated as well.By adjointness, the lifting functor transports injective M -graded K [ σ M ]-modules to injective Z σ (1) -graded S -modules. In [Per11], codivisorial modules have been considered. For given c ∈ Z σ (1) , such a module can be defined as K [ − M c,I ] = L m ∈− M c,I Kχ ( m ), where I is anysubset of σ (1) and M c,I = { m ∈ M | l ρ ( m ) ≥ c ρ for ρ ∈ I } . If c = L σ ( m ) for some m ∈ M , then K [ − M c,I ] is an injective object in M - K [ σ M ]-Mod. However, if K [ − M c,I ] is not injective, thefollowing example shows that lifting can exhibit a more bizarre behavior than in the previousexample. Example 5.7.
Let K [ σ M ] be as in Example 5.6 and consider the module K [ − M c,I ] with c = 0 and I = { , } . A similar computation as in Example 5.6 shows that \ K [ − M c,I ] ( c , ,c , ∼ = K − c − c whenever c + c ≤
0. So this module exhibits an infinite family of graded componentsof any finite dimension. This shows that the lifting functor does not respect combinatorialfiniteness in the sense of [Per11].
Rather than limits, we can also consider colimits associated to representations of ( M, ≤ σ ).That is, for any M -graded K [ σ M ]-module E , there is its colimit lim −→ E m . As the preorderedset ( M, ≤ σ ) is filtered, forming the colimit is exact. Given an M -graded K [ σ M ]-module E ∼ = L m ∈ M E m , we can associate to it the colimit E := lim −→ E m . Similarly, for the lifted S -module b E we have the colimit b E := lim −→ b E c , which is formed over the poset ( Z σ (1) , ≤ ). Proposition 5.9.
In the above situation we have E = b E .Proof. It is easy to see that for every c ∈ Z σ (1) we can find some m ∈ M such that c ≤ m .Conversely, for every m ∈ M we can find some c ∈ Z σ (1) such that m ≤ c . It follows that lim −→ E m and lim −→ b E c are cofinal.If dim σ < d , then we have m ≤ σ m ′ and m ′ ≤ σ m whenever m ′ − m ∈ σ ⊥ M . In particular,such a pair ( m, m ′ ) is an isomorphism in the category M . The following proposition states that,10p to natural equivalence, we do not loose anything essential if we pass from the preordered set( M, ≤ σ ) to M/σ ⊥ M with the induced partial order: Proposition 5.10 ([Per11, Prop. 2.8]) . Let Λ ⊆ σ M be a subgroup. Then there is an equiva-lence of categories between the category of M -graded K [ σ M ] -modules and the category of M/ Λ -graded K [ σ M / Λ] -modules. Note that we state Proposition 5.10 in slightly greater generality than [Per11].
Now, we are ready to consider the non-affine case. Denote { U σ } σ ∈ ∆ the standard coveringof X and { ˆ U σ = Spec( S σ ) } σ ∈ ∆ the corresponding cover of ˆ X given by the preimages of the U σ .If we take a T -equivariant, i.e., toric sheaf E on X , we see by Proposition 5.10 that the S σ -modules Γ( ˆ U σ , E ˆ T ) are naturally equivalent to the lifts of Γ( U σ , E ) to K [ N σ (1) ]. In particular,it is straightforward to check that coherence, torsion-freeness, and reflexivity are preserved bypassing back and forth between K [ N σ (1) ] and S σ . Given a quasi-coherent sheaf E on X , we obtain a family of colimits E σ := lim −→ Γ( U σ , E ) m for σ ∈ ∆. For every pair of faces τ, σ such that τ is a face of σ , the restriction Γ( U σ , E ) → Γ( U τ , E ) induces a map of directed families over ( M, ≤ σ ) and ( m, ≤ τ ), respectively, and by theuniversal property of colimits we obtain an induced K -linear isomorphism E σ → E τ (see [Per04, § E σ → E to identify the E σ with E =: E . For the case that E is coherent, it hasbeen shown in [Per04, § E equals the rank of E . We can do the same constructionfor b E and obtain a colimit b E , which, using Proposition 5.9, we can in a natural way identifywith E . This construction becomes most interesting for the case that E (and thus b E by Theorem4.4) is finitely generated and torsion-free. Then the maps Γ( U σ , E ) M · χ ( m ′ ) −−−−→ Γ( U σ , E ) m + m ′ are injective for every σ ∈ ∆, m ∈ M , and m ′ ∈ σ M . It follows that the induced mapsΓ( U σ , E ) m → E are injective as well for every σ ∈ ∆ and m ∈ M , and analogously so for theinduced maps b E c → E for c ∈ Z ∆(1) . This allows a greatly condensed representation of torsionfree toric sheaves in terms of families of subvector spaces of a fixed vector space E which areparameterized by the family of posets { ( M, ≤ σ ) } σ ∈ ∆ (see [Per04, Theorem 5.18]).For the case of reflexive sheaves we have the following structural theorem due to Klyachko. Theorem 5.14 ([Kly90], [Kly91], see also [Per04]) . The category of coherent reflexive toricsheaves on a toric variety X is equivalent to the category of vector spaces E endowed withfiltrations ⊆ · · · ⊆ E ρ ( i ) ⊆ E ρ ( i + 1) ⊆ · · · ⊆ E for ρ ∈ ∆(1) which are full in the sense that E ρ ( i ) = 0 for i << and E ρ ( i ) = E for i >> . Over U σ , we observe that for a torsion free K [ σ M ]-module E we have lim ←− E m equalsthe intersection T m ≤ σ m ′ E m ′ in E . Therefore, given E and E ρ ( i ) for ρ ∈ σ (1) as in Theorem5.14, one one constructs a reflexive module E from this data by setting E = L m ∈ M E m and E m = T ρ ∈ σ (1) E ρ (cid:0) l ρ ( m ) (cid:1) ⊆ E .By Theorem 4.4 we know that for a reflexive toric sheaf E on X , its lifting b E is reflexiveas well. The fan b ∆ associated to ˆ X in general contains more cones than ∆, but we have aone-to-one correspondence between ∆(1) and ˆ∆(1) given by, say, ρ ˆ ρ . So we know a priorithat E and b E are described by the same number of filtrations. The following result shows thatthese filtrations (in an almost tautological sense) indeed coincide and, moreover, that lifting isindeed “the” correct functor to translate reflexive toric sheaves into Z ∆(1) -graded S -modules. Theorem 5.16.
A toric sheaf E is coherent and reflexive if and only if b E is coherent and eflexive. Moreover, if E and b E are reflexive, then they are canonically described by the samedata, i.e., b E = E and b E ˆ ρ ( i ) = E ρ ( i ) for any ρ ∈ ∆(1) . In particular, lifting induces anequivalence of categories between the category of reflexive toric sheaves on X , the category ofreflexive toric sheaves on ˆ X , and the category of reflexive Z ∆(1) -graded S -modules.Proof. The statements on coherence and reflexivity follow from Theorem 4.4. It suffices toconsider the case that X is affine, i.e., X = U σ . So, assume that E is a reflexive M -graded K [ σ M ]-module, given by filtrations E ρ ( i ) of the vector space E . From this data we can constructa reflexive Z σ (1) -graded S -module F by setting F = E and F ˆ ρ ( i ) = E ρ ( i ). Similarly, if we startwith the reflexive S -module F , we get a reflexive K [ σ M ]-module E ′ by simply identifying thefiltrations. We show that F ∼ = b E and E ′ = F (0) = E .The equality E ′ = F (0) = E follows from the fact that E m = T ρ ∈ σ (1) E ρ (cid:0) l ρ ( m ) (cid:1) = T ρ ∈ σ (1) F ˆ ρ (cid:0) l ρ ( m ) (cid:1) = F m (see 5.15), where in the latter equation we identify m with its image L σ ( m ) ∈ Z σ (1) . Now consider b E c for some c ∈ Z n . By 5.15 we have b E c = lim ← E m = T c ≤ m E m = T c ≤ m T ρ ∈ ∆(1) E ρ (cid:0) l ρ ( m ) (cid:1) ⊆ E . Now by the fact that the l ρ are primitive elements in N , wecan always choose for any ρ ∈ ∆(1) some m ∈ M such that l ρ ( m ) = c ρ . It follows that b E c = T ρ ∈ ∆(1) E ρ ( c ρ ) = F c .For the equivalence of categories, it suffices to remark that for any two reflexive toric sheaves E , F , there is a natural bijection Hom( E , F ) → Hom( b E , b F ), as any homomorphism of vectorspaces E → F which respects the filtrations also respects any of their intersections. Remark 5.17.
For E reflexive, one can easily show that the S -module b E is isomorphic to(Γ ∗ E )ˇˇ, the reflexive hull of Γ ∗ E . Note that more generally, if E is torsion free, then b E does notnecessarily coincide with Γ ∗ E modulo torsion. Remark 5.18.
In [Per11], reflexive M -graded K [ σ M ]-modules have been investigated in termsof the vector space arrangements underlying the associated filtrations. Given such a module E , it is not difficult to see that in general not all possible intersections are realized as thegraded components Γ( U σ , E ) m = T ρ ∈ σ (1) E ρ (cid:0) l ρ ( m ) (cid:1) . However, for the vector space arrangementunderlying the filtrations associated to b E , all possible intersections indeed are realized this way.In this sense, on can consider vector space arrangement in E underlying the filtrations associatedto b E as the intersection completion of the vector space arrangement underlying the filtrationsassociated to E . References [BLH12] M. Barakat and M. Lange-Hegermann. Characterizing Serre quotients with no sectionfunctor and applications to coherent sheaves. In preparation, 2012.[Cox95] D. A. Cox. The Homogeneous Coordinate Ring of a Toric Variety.
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