Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties
aa r X i v : . [ m a t h . R T ] N ov PARITY SHEAVES, MOMENT GRAPHS AND THE p -SMOOTH LOCUS OF SCHUBERT VARIETIES PETER FIEBIG AND GEORDIE WILLIAMSON
Abstract.
We show that, with coefficients in a field or completelocal principal ideal domain k , the Braden-MacPherson algorithmcomputes the stalks of parity sheaves with coefficients in k . Asa consequence we deduce that the Braden-MacPherson algorithmmay be used to calculate the characters of tilting modules for alge-braic groups and show that the p -smooth locus of a (Kac-Moody)Schubert variety coincides with the rationally smooth locus, if theunderlying Bruhat graph satisfies a GKM-condition. Contents
1. Introduction 21.1. Acknowledgements 62. Equivariant sheaves 62.1. The equivariant derived category of a G -space 72.2. The equivariant functor formalism 72.3. Equivariant cohomology 82.4. The Mayer–Vietoris sequence 92.5. The case of a torus 92.6. An attractive proposition 93. The localisation homomorphism 113.1. One-dimensional orbits 123.2. The localisation theorem – part I 124. The image of the localisation homomorphism 164.1. Sheaves on moment graphs 174.2. Sections of sheaves and the structure algebra 184.3. The costalks of a sheaf 184.4. The moment graph associated to a T -variety 184.5. The functor W X = P W k -smooth locus of a moment graph 346.5. The combinatorics of parity sheaves 357. The case of Schubert varieties 358. p -Smoothness 378.1. Smoothness and p -smoothness 388.2. k -smoothness and the intersection cohomology complex 398.3. k -smoothness and stalks 398.4. On the p -smooth locus of T -varieties 418.5. A freeness result 429. Representations of reductive algebraic groups 439.1. Tilting modules 449.2. Simple rational characters 44References 451. Introduction
In Lie theory, one of the most successful methods to calculate rep-resentation theoretic data (such as character formulae, decompositionnumbers or Jordan–H¨older multiplicities) is to find a geometric or topo-logical interpretation of the problem. In many examples one obtainsrepresentation theoretic information from the stalks of intersection co-homology complexes on an associated algebraic variety (for examplethe flag variety, the nilpotent cone, or the group itself).In the most successful applications of this approach (the Kazhdan-Lusztig conjecture, canonical bases, character sheaves . . . ) the rep-resentation theoretic objects under consideration are assumed to bedefined over a field of characteristic 0. In this case the decompositiontheorem often allows one to recursively calculate the stalks of inter-section cohomology complexes, hence solving (or at least providing acombinatorial algorithm to solve) the representation theoretic problem.However, recently a number of authors have pointed out that geom-etry also has something to say about modular representation theory
ARITY SHEAVES AND MOMENT GRAPHS 3 (see [JMW09b] for a survey). In this article we are motivated by thefollowing two examples of this phenomenon: • For a ring k , the geometric Satake equivalence (cf. [MV07])provides an equivalence of tensor categories between equivari-ant perverse k -sheaves on the affine Grassmannian and rationalrepresentations of the Langlands dual group scheme over k . • In [Fie07b] and [Fie07a] a certain category of sheaves of k -vectorspaces on an affine flag variety was related to representations ofthe k -Lie algebra associated to the Langlands dual root system.Here k is assume to be a field whose characteristic is requiredto be at least the associated Coxeter number. The relation isthen used to give a new proof of Lusztig’s conjecture on thesimple rational characters for reductive groups in almost allcharacteristics.In [JMW09a] (motivated by ideas of Soergel [Soe00] together with adesire to better understand such relationships) a new class of sheaves,the “parity sheaves”, was introduced. These are certain constructiblesheaves on a stratified algebraic variety, which satisfy a parity vanish-ing condition with respect to stalks and costalks. It was shown that,under some additional assumptions, the indecomposable parity sheavesare parametrized in the same way as the intersection cohomology com-plexes. If the coefficients are of characteristic 0 the decompositiontheorem often implies that the indecomposable parity sheaves are iso-morphic to intersection cohomology complexes (up to a shift).In positive characteristics this needs no longer be true. However,with coefficients of positive characteristic parity sheaves are often easierto work with than intersection cohomology complexes. Moreover, forsome representation theoretic applications they may even form theirnatural replacement. For example, • the category considered in [Fie07b, Fie07a] turns out to be thecategory of parity sheaves, • under the geometric Satake equivalence (and under some mildand explicit assumptions on the characteristic of k ) the paritysheaves correspond to tilting modules for the Langlands dualgroup (cf. [JMW09b]).In the above results, fundamental representation theoretic data is en-coded in the stalks of the indecomposable parity sheaves. It is thereforean important problem to find an algorithm for their calculation.For an appropriately stratified complex algebraic variety X withtorus action Braden and MacPherson [BM01] describe an algorithm for PETER FIEBIG AND GEORDIE WILLIAMSON calculating the stalks of intersection cohomology complexes with coeffi-cients in a field of characteristic 0 (using localisation techniques in equi-variant cohomology developed by Goresky, Kottwitz and MacPherson[GKM98]). The torus action has, by assumption, only finitely manyfixed points and one-dimensional orbits. The structure of the one-skeleton of the torus action can be encoded in the “moment graph” ofthe variety: • the vertices and edges are given by the torus fixed points andone-dimensional orbits respectively, with a one-dimensional or-bit incident to those fixed points in its closure, • each edge is labelled by a character of the torus determining anisomorphism of the orbit with C ∗ (this is defined only up to asign).Braden and MacPherson then describe an algorithm (using only com-mutative algebra) to produce a “sheaf” on the moment graph, andshow that its stalks agree with those of the equivariant intersection co-homology complex. Thus the (a priori extremely difficult) computationof the stalks of the intersection cohomology complex may (in principle)be carried out in an elementary way.Now, the Braden–MacPherson algorithm makes sense with coeffi-cients in an arbitrary field k , or even in a local ring. However, simpleexamples show that it does not compute the stalks of intersection co-homology complexes when the coefficients are not of characteristic 0.The central result of this paper is the following: Theorem 1.1.
Suppose that the pair ( X, k ) satisfies the GKM-condition(cf. Section 4.7). Then the Braden-MacPherson algorithm computesthe stalks of indecomposable parity sheaves. In the theorem, k denotes a complete local principal ideal domain.If k is a field, then the GKM-condition may be stated simply: onerequires that, for all pairs of one-dimensional orbits having a commontorus fixed point in their closure, the corresponding characters do notbecome linearly dependent modulo k . This condition can easily be readoff the associated moment graph.In the course of the proof of the above result we provide a version oflocalisation theorem of [GKM98] with coefficients in a ring, i.e. we showthat the hypercohomologies of certain equivariant sheaves on X aregiven by the global sections of associated moment graph sheaves (seeTheorem 4.4). For complete local rings we then show that the Braden–MacPherson algorithm yields the objects associated to parity sheaves(see Theorem 6.10). In contrast to the proof of Braden–MacPherson, ARITY SHEAVES AND MOMENT GRAPHS 5 our arguments are more elementary, as we do not need the theoryof mixed Hodge modules. As in characteristic 0 the decompositiontheorem implies that the parity sheaves are intersection cohomologycomplexes up to a shift, we obtain a new proof of their result.Applying the above theorem to the affine Grassmannian and usingthe Satake equivalence, we obtain:
Theorem 1.2.
Suppose that p > h + 1 , where h denotes the Coxeternumber of our datum. On the moment graph of the affine Grassman-nian and with coefficients in the ring of p -adic integers, the Braden–MacPherson algorithm calculates the characters of tilting modules ofthe Langlands dual group over F p . The moment graph of the affine Grassmannian is GKM for a field k if and only if k is of characteristic 0. We avoid this complication byusing the p -adic integers in the above theorem.We apply the multiplicity one result of [Fie10] to obtain a descrip-tion of the p -smooth locus of Schubert varieties. Recall that an n -dimensional algebraic variety X is p -smooth if for all x ∈ X one hasan isomorphism of graded vector spaces H • ( X, X \ { x } , F p ) ∼ = H • ( C n , C n \ { } , F p ) . The p -smooth locus of X is the largest open p -smooth subvariety. Onesimilarly defines rationally smooth, and the rationally smooth locusby replacing F p by Q above. If X is rationally (resp. p -) smooth itsatisfies Poincar´e duality with rational (resp. F p -) coefficients. Here isour result: Theorem 1.3.
Let G be the moment graph of a (Kac-Moody) Schubertvariety X and suppose that ( G , F p ) is a GKM-pair. Then the p -smoothlocus of X coincides with its rationally smooth locus. In the finite dimensional case, the GKM-condition is always satisfiedif p = 2 and if, in addition, p = 3 in G . This answers a (strongerversion of) a question of Soergel (cf. [Soe00]). In fact, we prove theabove theorem for a larger class of varieties with an appropriate torusaction for fields k that satisfy the GKM-condition.The smooth and rationally smooth locus of Schubert varieties hasbeen the subject of much investigation by a number of authors. See forexample [Car94], [Kum96], [Dye93], [Dye05] and [Ara98]. It a resultknown as Peterson’s theorem that the smooth and rationally smoothlocus agree in simply-laced type, which immediately implies the abovetheorem. However, there are examples in non-simply-laced types of PETER FIEBIG AND GEORDIE WILLIAMSON small rank where the 2-smooth and 3-smooth locus do not agree withthe rationally smooth locus.Lastly let us remark that results of this paper (in particular Section8.5) are used in an essential way in the paper [JW], which shows thatthat Kumar’s criterion for the rational smoothness of Schubert varietiescan be extended to provide a criterion for p -smoothness. In particular,the main result of [JW] provides a means to determine the p -smoothlocus when the underlying moment graph is not GKM, complementingTheorem 1.3. On may also use Theorem 1.3 together with the resultsof [JW] to obtain a novel proof of Peterson’s theorem.1.1. Acknowledgements.
We would like to thank Daniel Juteau andOlaf Schn¨urer for useful conversations and Michel Brion for useful cor-respondence. P.F. gratefully acknowledges the support of the Landess-tiftung Baden–W¨urttemberg as well as the DFG-Schwerpunkt 1388‘Representation Theory”. Both authors gratefully acknowledge thesupport of the Newton Institute in Cambridge, where parts of thispaper were written. 2.
Equivariant sheaves
In this section we recall the construction of the bounded equivariantderived category D bG ( X, k ) that is associated to a topological group G , a ring of coefficients k and a G -space X . To a suitable continuous G -equivariant map f : X → Y one associates the push-forward functors f ∗ , f ! : D bG ( X, k ) → D bG ( Y, k )and the pull-back functors f ∗ , f ! : D bG ( Y, k ) → D bG ( X, k )satisfying a Grothendieck formalism. We then recall the equivariantcohomology H • G ( F ) of X with coefficients in F ∈ D bG ( X, k ) and, finally,the Mayer–Vietoris sequence associated to an open G -stable covering X = U ∪ V .We will be mainly concerned with the following situation: G willeither be a complex algebraic torus, i.e. G ∼ = ( C × ) r for some r > S ) r . Thespace X will be a complex algebraic variety with an algebraic G -action,and endowed with its metric topology. The main reference for thefollowing is [BL94]. ARITY SHEAVES AND MOMENT GRAPHS 7
The equivariant derived category of a G -space. We fix atopological group G . A G -space is a topological space endowed with acontinuous G -action. There always exists a contractible G -space witha topologically free G -action. We fix one of those and call it EG . Forany G -space X we can now define the quotient X G := X × G EG of X × EG by the diagonal G -action. Then we have two maps X × EG p z z uuuuuuuuuu q % % JJJJJJJJJ
X X G . The map q on the right is the canonical quotient map and p is theprojection onto the first factor.Now we fix a ring of coefficients k . For any topological space Y we denote by D ( Y, k ) the derived category of sheaves of k -moduleson Y . By D b ( Y, k ) we denote the full subcategory of objects withbounded cohomology. For a continuous map f : Y → Y ′ we then havethe push-forward functor f ∗ : D ( Y, k ) → D ( Y ′ , k ) and the pull-backfunctor f ∗ : D ( Y ′ , k ) → D ( Y, k ) (see [Spa88]).
Definition 2.1.
The equivariant derived category of sheaves on X with coefficients in k is the full subcategory D G ( X, k ) of D ( X G , k ) thatcontains all sheaves F for which there is a sheaf F X ∈ D ( X, k ) suchthat q ∗ F ∼ = p ∗ F X . We denote by D bG ( X, k ) ⊂ D G ( X, k ) the full subcategory of ob-jects with bounded cohomology, i.e. of objects that are contained in D b ( X G , k ).It turns out that the categories D G ( X, k ) and D bG ( X, k ) are indepen-dent of the choice of EG . Since p is a trivial fibration with contractiblefibre EG , the functor p ∗ : D ( X, k ) → D ( X × EG, k ) is a full embedding.We deduce that for
F ∈ D G ( X, k ) the sheaf F X ∈ D ( X, k ) appearingin the definition above is unique up to unique isomorphism, so the map
F 7→ F X even extends to a functor For : D G ( X, k ) → D ( X, k ).2.2.
The equivariant functor formalism.
In order to ensure thatall the functors that we introduce in the following exist we assume that X is a complex algebraic variety endowed with its metric topology, andthat G is a Lie group acting continuously on X .If f : X → Y is a continuous G -equivariant map then we get aninduced map f G := f × G id : X G → Y G and corresponding functors f ∗ G , f G ∗ , f ! G and f G ! between the categories D b ( X G , k ) and D b ( Y G , k ).(Some care is needed in the definition of f ! G and f G ! because X G and Y G are not locally compact in general. In [BL94] this problem is overcome PETER FIEBIG AND GEORDIE WILLIAMSON by considering X G as a direct limit of locally compact spaces. It isalso possible to prove the existence and basic properties of f ! G in arelative setting, see [SHS69].) It turns out that all four functors inducefunctors between the subcategories D bG ( X, k ) and D bG ( Y, k ). By abuseof notation we denote these functors by the symbols f ∗ , f ∗ , f ! and f ! .For a G -stable subvariety i : Y ֒ → X and a sheaf F ∈ D bG ( X, k ) wedefine F Y := i ∗ F . So F Y is an object in D bG ( Y, k ).2.3.
Equivariant cohomology.
The equivariant cohomology H • G ( X, k ) of X with coefficients in k is the (ordinary) cohomology of the space X G , i.e. H • G ( X, k ) := H • ( X G , k ) . In particular, the equivariant cohomology of a point is the cohomologyof the classifying space BG := pt G = EG × G pt = EG/G of G .Now let F ∈ D bG ( X, k ). The equivariant cohomology H • G ( F ) of X with coefficients in F is defined as follows. We denote by π : X → pt themap to a point. Then we have the object π ∗ F ∈ D bG ( pt, k ) ⊂ D b ( BG, k )and we define H • G ( F ) := H • ( π ∗ F ) , where on the right we have the ordinary cohomology of BG with co-efficients in the sheaf π ∗ F . This is naturally a graded module over H • G ( pt, k ) = H • ( BG, k ), so equivariant cohomology is a functor H • G : D bG ( X, k ) → H • G ( pt, k )-mod Z . Here and in the following we denote by A -mod Z the category of Z -graded modules over a Z -graded ring A . For a graded A -module M = L n ∈ Z M n and l ∈ Z we denote by M [ l ] the graded module obtained bya shift such that M [ l ] n = M l + n for all n ∈ Z .Let i : Y ֒ → X be a locally closed G -stable subvariety and F ∈ D bG ( X, k ).The adjunction morphism id → i ∗ i ∗ yields a morphism F → i ∗ i ∗ F = i ∗ F Y . After applying equivariant cohomology this yields a homomor-phism H • G ( F ) → H • G ( i ∗ F Y ) = H • G ( F Y )of H • G ( pt, k )-modules. We call such a homomorphism a restriction ho-momorphism . ARITY SHEAVES AND MOMENT GRAPHS 9
The Mayer–Vietoris sequence.
We will often make use of theequivariant Mayer–Vietoris sequence. Note that the equivariant state-ment is a straightforward consequence of the non-equivariant one (see,for example, [KS94, 2.6.28]).
Proposition 2.2.
Let X = U ∪ V where U, V ⊂ X are open and G -stable. Then, given any F ∈ D bG ( X, k ) , we have a long exact sequenceof equivariant cohomology · · · → H j − G ( F U ∩ V ) → H jG ( F ) → H jG ( F U ) ⊕ H jG ( F V ) →→ H jG ( F U ∩ V ) → H j +1 G ( F ) → . . . . The case of a torus.
Let us suppose now that G = T is acomplex torus, i.e. a topological group isomorphic to ( C × ) r for some r >
0, endowed with the metric topology.For n ≥ C n \ { } ) r together with the T -action given by( t , . . . , t r ) · ( x , . . . , x r ) = ( t · x , . . . , t r · x r ) . The embeddings C n \ { } → C n +1 \ { } that map ( z , . . . , z n ) to( z , . . . , z n ,
0) define a direct system · · · → ( C n \ { } ) r → ( C n +1 \ { } ) r → . . . of T -spaces. The direct limit ( C ∞ \ { } ) r := lim( C n \ { } ) r is a con-tractible space together with a topologically free T -action, hence canbe chosen as a model for ET .We denote by X ∗ ( T ) the character lattice Hom( T, C × ) of T . Let S k := S ( X ∗ ( T ) ⊗ Z k )be the symmetric algebra over the free k -module X ∗ ( T ) ⊗ Z k , gradedin such a way that X ∗ ( T ) ⊗ Z k ⊂ S k is the homogeneous component ofdegree 2. Then the Borel homomorphism (cf. [Bri98], [Jan09]) gives acanonical identification S k ∼ → H • ( BT, k ) = H • T ( pt, k ) . An attractive proposition.
Now let X be a complex T -variety.Recall that a T -fixed point x ∈ X is called attractive if all weights of T on the tangent space to X at x lie in an open half space of X ∗ ( T ) ⊗ Z R . Ifthis is the case then one can find a one parameter subgroup α : C × → T and an open neighbourhood U of x such that(1) lim z → α ( z ) · y = x for all y ∈ U . If, in addition, X is connected and affine, then x is the unique T -fixedpoint of X and (1) holds for all y ∈ X . In particular, the smallest T -stable open neighbourhood of x ∈ X is X itself.Suppose for the remainder of this section that X is connected andaffine, and that x ∈ X is an attractive fixed point. We denote by i : { x } → X the inclusion and by π : X → { x } the projection. If weapply the functor π ∗ to the natural transformation id → i ∗ i ∗ we get anatural transformation π ∗ → π ∗ i ∗ i ∗ . Since π ◦ i is the identity, we geta natural morphism π ∗ → i ∗ of functors from D bT ( X, k ) to D bT ( { x } , k ).The goal of the rest of this section is to prove the following (for similarstatements in the non-equivariant or “weakly equivariant” setting see[Spr84] and [Bra03]): Proposition 2.3.
Suppose that X is connected and affine and that x ∈ X is an attractive fixed point. Then the morphism of functors π ∗ → i ∗ is an isomorphism. We begin with some lemmata. Suppose we have a pair of Cartesiansquares e F e i / / q (cid:15) (cid:15) e X / / q (cid:15) (cid:15) e π / / e F q (cid:15) (cid:15) F i / / X / / π / / F such that q is smooth and surjective, and π ◦ i = id (and hence e π ◦ e i = id).The adjunctions ( π ∗ , π ∗ ) and ( e π ∗ , e π ∗ ) give morphisms of functors π ∗ → i ∗ and e π ∗ → e i ∗ . Lemma 2.4.
Let
F ∈ D b ( X, k ) . Then π ∗ F → i ∗ F is an isomorphismif and only if e π ∗ q ∗ F → e i ∗ q ∗ F is an isomorphism.Proof. Because q is surjective, π ∗ F → i ∗ F is an isomorphism if andonly if q ∗ π ∗ F → q ∗ i ∗ F is an isomorphism. Now q ∗ i ∗ F ∼ → e i ∗ q ∗ F and q ∗ π ∗ F ∼ → e π ∗ q ∗ F by smooth base change. Via these canonical isomor-phisms we obtain a map e π ∗ q ∗ F → e i ∗ q ∗ F . This is the same map (up to isomorphism) as that coming from themorphism e π ∗ → e i ∗ (cf. [BL94, Theorem 1.8].) (cid:3) ARITY SHEAVES AND MOMENT GRAPHS 11
Now suppose a torus T contracts a variety X onto a fixed locus F ⊂ X . Consider the diagram X π (cid:15) (cid:15) X × ET p o o q / / π (cid:15) (cid:15) X × T ET π (cid:15) (cid:15) F i O O F × ET p o o i O O q / / F × T ET i O O . Both p and q are smooth, and so applying the above lemma twice we seethat, given F ∈ D bT ( X, k ), we have that π ∗ F → i ∗ F is an isomorphismin D bT ( F, k ) if and only if π ∗ For( F ) → i ∗ For( F ) is.Given a G -space X , let us call F ∈ D b ( X, k ) naively equivariant ifwe have an isomorphism m ∗ F → p ∗ F where m and p denote the actionand projection maps G × X m / / p / / X. Note that, if G acts freely on X then pullback along X → X/G allowsus to view any
F ∈ D b ( X/G, k ) as a naively equivariant sheaf on X .Note also that if F is naively equivariant for a group G , then it is alsonaively equivariant for any subgroup H ⊂ G . Lemma 2.5.
Suppose that
F ∈ D bG ( X, k ) . Then For( F ) is naivelyequivariant for G .Proof. Consider the quotient map q : X × EG → X × G EG . Then q ∗ F is naively equivariant for G . Then smooth base change applied tothe projection p : X × EG → X yields that For( F ) = p ∗ q ∗ F is naivelyequivariant for G . (cid:3) We can now prove the attractive proposition:
Proof of Proposition 2.3.
The above arguments reduce the proof of theabove to showing that, if
F ∈ D b ( X, k ) is naively equivariant for theaction of a one dimensional torus which contracts X onto x ∈ X , then π ∗ F → i ∗ F is an isomorphism. But this is shown in [Spr82] (see also[Bra03] for another account of this argument). (cid:3) The localisation homomorphism
Throughout this section we assume that k is a unique factorisationdomain and that X is a normal complex algebraic variety (endowedwith its metric topology), acted upon algebraically by a complex torus T ∼ = ( C × ) r . In addition, we assume the following: (A1) The torus acts on X with only finitely many zero- and one-dimensional orbits and the closure of each one-dimensional orbitis smooth.(A2) X admits a covering by open affine connected T -stable subvari-eties, each of which contains an attractive (hence unique) fixedpoint.Note that, by a result of Sumihiro (see [Sum74, KKLV89]), X has acovering by open affine T -stable subvarieties, hence (A2) is automati-cally satisfied if X is proper and each T -fixed point is attractive.Let X T ⊂ X be the subspace of T -fixed points and F ∈ D bT ( X, k ).The restriction homomorphism associated to the inclusion X T ֒ → X , H • T ( F ) → H • T ( F X T ) , is called the localisation homomorphism .As X T is a finite set we have H • T ( F X T ) = L x ∈ X T H • T ( F x ). Followingthe results of [CS74] and [GKM98] we will show that for certain choicesof X , k and F the localisation map is injective and give an explicit de-scription of its image. This is conveniently phrased in terms of momentgraphs (cf. [BM01]), as it turns out that this image is determined bythe restriction of F to the one-dimensional T -orbits in X .3.1. One-dimensional orbits.
Suppose that E ⊂ X is a one-dimen-sional T -orbit. Then E ∼ = T /
Stab T ( x ) for any x ∈ E . Now Stab T ( x )is the kernel of a character α E ∈ X ∗ ( T ) which is well-defined up to asign. From now on we fix a choice of α E for each one-dimensional orbit E in X . Nothing that follows depends on this choice.As before we denote by S k the Z -graded symmetric algebra of the free k -module X ∗ ( T ) ⊗ Z k and identify it with the T -equivariant cohomologyof a point with coefficients in k . Given α ∈ X ∗ ( T ) we often abusenotation and denote by α as well the image of α ⊗ ∈ X ∗ ( T ) ⊗ Z k in S k .Now α E acts as zero on H • T ( E, k ) (see, for example, [Jan03, Section1.9]). As H • T ( F E ) is a H • T ( E, k )-module, we conclude:
Lemma 3.1.
For any one-dimensional T -orbit E in X and any F ∈ D bT ( X, k ) we have α E H • T ( F E ) = 0 . The localisation theorem – part I.
For any closed connectedsubgroup Γ of T we let X Γ be the subset of Γ-fixed points in X . Letus fix a closed subspace Z ⊂ X which is a discrete union Z = X Γ ⊔ · · · ⊔ X Γ n ARITY SHEAVES AND MOMENT GRAPHS 13 of the fixed points in X of finitely many connected subtori Γ , . . . , Γ n ⊂ T . We set P Z := (cid:26) α E ∈ X ∗ ( T ) (cid:12)(cid:12)(cid:12)(cid:12) E is a one-dimensional T -orbit in X \ Z (cid:27) and define s Z := Y α ∈ P Z α ∈ S k . In addition to (A1) and (A2) we assume from now on:(A3) for each one-dimensional orbit E in X the image of α E ∈ X ∗ ( T )is non-zero in S k .(Of course this condition is vacuous if the characteristic of k is 0.)We now come to the first part of the localisation theorem. In thecharacteristic 0 case it is due to Goresky, Kottwitz and MacPherson(cf. [GKM98]). Theorem 3.2.
Assume that the assumptions (A1), (A2) and (A3) holdand let
F ∈ D bT ( X, k ) . Suppose that H • T ( F ) is a graded free S k -module.Then the restriction homomorphism H ∗ T ( F ) → H ∗ T ( F Z ) is injective and becomes an isomorphism after inverting s Z ∈ S k , i.e. af-ter applying the functor · ⊗ S k S k [1 /s Z ] . The proof of the theorem will take up the rest of this section. Wefollow Brion’s account [Bri98, Section 2] of the characteristic 0 casequite closely, but at points some additional care is needed.Let K ∼ = ( S ) r ⊂ T ∼ = ( C × ) r be the maximal compact subtorus of T . We can regard X as a K -space via restriction of the action. Thisyields a functor res TK : D bT ( X, k ) → D bK ( X, k ) . As T /K is contractible, for any equivariant sheaf
G ∈ D bT ( X, k ) restric-tion gives an isomorphism H • T ( G ) ∼ → H • K (res TK G ) . In particular, we have a canonical isomorphism H • K ( pt, k ) ∼ = S k . Inthe following we write H • K ( G ) for H • K (res TK G ). Hence, for the proof ofTheorem 3.2, it is enough to consider the restriction homomorphism H • K ( F ) → H • K ( F Z )and to show that it is injective and becomes an isomorphism afterinverting s Z . Before we prove this we need a couple of preliminary results. Westate them for the K -equivariant cohomology, however all lemmataexcept Lemma 3.5 are true with T in place of K .First we assume that X = V is a finite dimensional T -module. Let P ⊂ X ∗ ( T ) be the characters occurring in V and s = Q χ ∈ P χ ∈ S k their product. Here is the first step towards the localization theorem. Lemma 3.3. If F ∈ D bK ( V \ { } , k ) then H • K ( F ) is annihilated by apower of s .Proof. Fix an isomorphism(2) V ∼ = C χ ⊕ C χ ⊕ · · · ⊕ C χ m where χ , χ , . . . , χ m ∈ P . (Here, given χ ∈ X ∗ ( T ), C χ denotes the one-dimensional T -module with character χ .) We will use this isomorphismto write elements of V as ( x j ) ≤ j ≤ m . For any 1 ≤ i ≤ m consider thesubset U i = { ( x j ) ∈ V | x i = 0 } . Projection gives us an equivariant map U i → C × χ i . By Lemma 3.1, theequivariant cohomology H • K ( G ) of each G ∈ D bK ( U i , k ) is annihilatedby χ i .However, V \ { } is covered by the sets U i for 1 ≤ i ≤ m and theMayer–Vietoris sequence allows us to conclude that H • K ( F ) is annihi-lated by a power of s . (cid:3) Now let Z ⊂ X be as before. From the above we deduce the secondstep: Lemma 3.4. If F ∈ D bK ( X \ Z, k ) then H • K ( F ) is annihilated by apower of s Z .Proof. First we assume that X is affine and connected and containsan attractive fixed point. In this case Z is necessarily of the form X Γ for a closed subtorus Γ ⊂ T . We recall an argument due to Brion (cf.[Ara98, Proposition 3.2.1-1], or the proof of Theorem 17 in [Bri98])which constructs a finite T -equivariant map π : X → V, where V is a T -module with weights corresponding bijectively to theone-dimensional orbits of T in X . Brion’s construction is as follows:For each one-dimensional orbit E ⊂ X , E is smooth and henceisomorphic, as a T -space, to C α E . For each such orbit we may find aregular function π E : X → C α E such that the restriction of π E to E ARITY SHEAVES AND MOMENT GRAPHS 15 is an equivariant isomorphism of affine spaces. Taking the direct sumover all such π E yields a map X π → V := M E C α E . We claim that π is finite. Because x ∈ X is attractive, we can finda rank one subtorus of T inducing a positive grading on the regu-lar functions on X . By the graded Nakayama lemma π is finite ifand only if π − (0) is finite. If π − (0) is not finite, then it containsa one-dimensional T -orbit (again by the attractiveness of x ), but thiscontradicts the construction.Now let V Γ ⊂ V be the subspace of Γ-fixed points. Because eachfibre of π is finite and π is equivariant it follows that π − ( V Γ ) = X Γ .Choose a decomposition V = V ′ ⊕ V Γ of T -modules and let V → V ′ denote the projection. We get an inducedmap π ′ : X \ X Γ → V ′ \ { } and the result follows from Lemma 3.3 because H • K ( F ) ∼ = H • K ( π ′∗ F ) . Hence we proved the lemma in the case of affine X .By our assumption (A2), the general case follows from the Mayer–Vietoris sequence. (cid:3) Lemma 3.5.
For any equivariant sheaf
F ∈ D bK ( X, k ) we have anisomorphism H • K ( F Z ) ∼ = lim → H • K ( F U ) , where the direct limit takes place over all K -stable open neighbourhoods U of Z .Proof. By assumption X has a covering by open subvarieties, all iso-morphic to closed subvarieties of affine spaces with linear T -actions.Thus we may choose a basis of open neighbourhoods { U i } i ∈ I of Z whichare K -stable. (This is where we need the compactness of K .)Now we may write EK as a countable direct limit of (finite dimen-sional) manifolds with free K -action (for example, by taking EK = ET as in Section 2.5). Hence X K can be written as a countable union ofcompact subsets. Because X K is regular, we conclude that X K is para-compact (cf. [MS74, Section 5.8] and [Dug66, Theorem 6.5]). It is straightforward to see that { ( U i ) K } i ∈ I give a basis of open neighbour-hoods of Z K . It then follows from [KS94, Remark 2.6.9] that we havean isomorphism H • K ( F Z ) = H • ( F Z K ) ∼ = lim → H • ( F ( U i ) K ) = lim → H • K ( F U )as claimed. (cid:3) Now we are ready to prove Theorem 3.2.
Proof.
Let
F ∈ D bT ( X, k ) and assume that H • T ( F ) is free as an S k -module. We have to show that the restriction map H • K ( F ) → H • K ( F Z )is injective, and becomes an isomorphism after inverting s Z .Let U be an open K -stable neighbourhood of Z ⊂ X . We haveinclusions U j ֒ → X i ← ֓ X \ U and hence a distinguished triangle: i ! i ! F → F → j ∗ j ∗ F [1] → . Applying Lemma 3.4 (and remembering that i ∗ ∼ = i ! ) we deduce that H • K ( i ! i ! F ) is annihilated by a power of s Z . As H • K ( F ) is free, therestriction map H • K ( F ) → H • K ( F U ) is injective. It also follows that itbecomes an isomorphism after inverting s Z .To finish the proof, note that, by Lemma 3.5, H • K ( F Z ) ∼ = lim → H • K ( F U ) . Because H • K ( F ) → H • K ( F U ) is injective for all U it follows that H • K ( F ) → H • K ( F Z ) is injective. Lastly, this map becomes an isomorphism after in-verting s Z because the direct limit commutes with tensor products. (cid:3) The image of the localisation homomorphism
We are now going to describe the image of the localisation homomor-phism under a certain further restriction on the ring k which is calledthe GKM-condition . For this it is convenient to use the language ofsheaves on moment graphs. We start by recalling the main definitionsand constructions in the theory of moment graphs. In particular, wedefine the Z -graded category G -mod Z k of k -sheaves on a moment graph G and associate to any such sheaf F its space of global sections Γ( F ).To a T -space X with finitely many zero- and one-dimensional orbitswe associate a moment graph G X and define a functor W : D bT ( X, k ) → G X -mod Z k ARITY SHEAVES AND MOMENT GRAPHS 17 between Z -graded categories. We then show that under some assump-tions on F ∈ D bT ( X, k ), the equivariant cohomology of X with coeffi-cients in F coincides with the space of global sections of W ( F ), i.e. H • T ( F ) = Γ( W ( F )) . Sheaves on moment graphs.
Let Y ∼ = Z r be a lattice of finiterank. Definition 4.1.
An (unordered) moment graph G over Y is given bythe following data: • A graph ( V , E ) with set of vertices V and set of edges E . • A map α : E → Y \ { } . We assume that two vertices of a moment graph are connected by atmost one edge.Let G = ( V , E , α ) be a moment graph. We write E : x —— y for anedge E that connects the vertices x and y . If we also want to denotethe label α = α ( E ) of E , then we write E : x α ——— y . As before wedenote by S k = S ( Y ⊗ Z k ) the symmetric algebra of Y over k , whichwe consider as a graded algebra with Y ⊗ Z k sitting in degree 2. Definition 4.2. A k -sheaf M on a moment graph G is given by thefollowing data: • a graded S k -module M x for any vertex x ∈ V , • a graded S k -module M E with α ( E ) M E = 0 for any E ∈ E , • a homomorphism ρ x,E : M x → M E of graded S k -modules forany vertex x lying on the edge E . For a k -sheaf M on G and l ∈ Z we denote by M [ l ] the shifted k -sheaf with stalks M [ l ] x = ( M x )[ l ], M [ l ] E = ( M E )[ l ] and shifted ρ -homomorphisms. A morphism f : M → N between k -sheaves M and N on G is given by a collection of homomorphisms of graded S k -modules f x : M x → N x and f E : M E → N E for all vertices x and edges E that are compatible with the maps ρ , i.e. such that thediagram M xρ M x,E (cid:15) (cid:15) f x / / N xρ N x,E (cid:15) (cid:15) M E f E / / N E commutes for all vertices x that lie on the edge E . We denote by G -mod Z k the category whose objects are k -sheaves on G and whosemorphisms are the morphisms between k -sheaves. It is Z -graded bythe functor M M [1]. Sections of sheaves and the structure algebra.
The structurealgebra over k of a moment graph G is Z k = ( ( z x ) ∈ Y x ∈V S k (cid:12)(cid:12)(cid:12)(cid:12) z x ≡ z y mod α ( E )for all edges E : x —— y (cid:27) . Coordinatewise addition and multiplication makes Z k into an S k -algebra.It is Z -graded if we consider the product in the definition in the gradedsense.Let M by a k -sheaf on G . For any subset I of V we define the spaceof sections of M over I byΓ( I , M ) := ( m x ) ∈ Y x ∈I M x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ x,E ( m x ) = ρ y,E ( m y )for all edges E : x —— y with x, y ∈ I . Coordinatewise multiplication makes Γ( I , M ) into a Z k -module (as α ( E ) ρ x,E ( m x ) = 0 for any edge E with vertex x ). Again it is agraded module when the product is taken in the category of graded S k -modules.We call the space Γ( M ) := Γ( V , M ) the space of global sections . If I ⊂ J are two subsets of V , then the canonical projection L x ∈J M x → L x ∈I M x induces a restriction map Γ( J , M ) → Γ( I , M ).4.3. The costalks of a sheaf.
Let M be a k -sheaf on G and let x be avertex. Then we define the costalk M x of M at x to be the S k -module M x := { m ∈ M x | ρ x,E ( m ) = 0 for all edges E that contain x } . We can identify M x in an obvious way with the kernel of the restrictionhomomorphism Γ( V , M ) → Γ( V \ { x } , M ).4.4. The moment graph associated to a T -variety. To a complex T -variety X satisfying (A1) we associate the following moment graph G X = ( V , E , α ) over the lattice X ∗ ( T ): • We set V := X T . • The vertices x and y , x = y , are connected by an edge if thereis a one-dimensional orbit E such that E = E ∪ { x, y } . Wedenote this edge by E as well. • We let α ( E ) = α E ∈ X ∗ ( T ) be the chosen character.Note that only those one-dimensional orbits E in X give rise to anedge that pick up two distinct fixed points in their closure. ARITY SHEAVES AND MOMENT GRAPHS 19
The functor W . Suppose that E ⊂ X is a one-dimensional T -orbit, and suppose that x ∈ E is a fixed point in its closure. Let F bean object in D bT ( X, k ). Then the restriction homomorphism H • T ( F E ∪{ x } ) → H • T ( F x )is an isomorphism by the attractive Proposition 2.3. Hence we candefine a homomorphism ρ x,E from H • T ( F x ) to H • T ( F E ) by composingthe inverse of the above homomorphism with the restriction homomor-phism H • T ( F E ∪{ x } ) → H • T ( F E ): ρ x,E : H • T ( F x ) ∼ ← H • T ( F E ∪{ x } ) → H • T ( F E ) . Now we can define the functor W . To an equivariant sheaf F ∈ D bT ( X, k ) on X we associate the following k -sheaf W ( F ) on G X : • For a vertex x ∈ V we set W ( F ) x := H • T ( F x ). • For a one-dimensional orbit E we set W ( F ) E := H • T ( F E ) (notethat α E H • T ( F E ) = 0 by Lemma 3.1). • In case that x ∈ E we let ρ x,E : W ( F ) x → W ( F ) E be the mapconstructed above.This construction clearly extends to a functor W : D bT ( X, k ) → G X -mod Z k .4.6. The case X = P . Suppose that T acts linearly on P via anon-trivial character α . In this case the moment graph is0 α —— ∞ . For
F ∈ D bT ( P , k ) the sheaf W ( F ) consists of the stalks H • T ( F ), H • T ( F ∞ ) and the space H • T ( F C × ) together with the maps H • T ( F ) ρ , C × −→ H • T ( F C × ) ρ ∞ , C × ←− H • T ( F ∞ ) . A consequence of the Mayer–Vietoris sequence is the following lemma.
Lemma 4.3.
Let
F ∈ D bT ( P , k ) . Then the image of the restrictionhomomorphism H • T ( F ) → H • T ( F ) ⊕ H • T ( F ∞ ) is { ( z , z ∞ ) | ρ , C × ( z ) = ρ ∞ , C × ( z ∞ ) } . The localisation theorem – part II.
Now we assume that X satisfies the assumptions (A1), (A2) and (A3). Let F ∈ D bT ( X, k ). If H • T ( F ) is a free S k -module, then Theorem 3.2 shows that we can view H • T ( F ) as a submodule of L x ∈ X T H • T ( F x ) = L x ∈ X T W ( F ) x . The spaceof global sections Γ( W ( F )) is a submodule of this direct sum as well.In this section we want to prove that these two submodules coincide.We need some more notation. For α ∈ X ∗ ( T ) let us define X α to bethe subvariety of all T -fixed points in X and all one-dimensional orbits E ⊂ X such that kα ∩ kα E = 0. Then X = X T for all rings k , but ingeneral X α depends on the ring k . We define P α := (cid:26) α E ∈ X ∗ ( T ) (cid:12)(cid:12)(cid:12)(cid:12) E is a one-dimensional T -orbit in X \ X α (cid:27) and s α := Y α E ∈ P α α E ∈ S k . We need some additional assumptions on our data:(A4a) For any α ∈ X ∗ ( T ) the space X α is a disjoint union of pointsand P ’s.(A4b) If E is a one-dimensional T -orbit and n ∈ Z is such that α E isdivisible by n in X ∗ ( T ), then n is invertible in k .Note that (A4a) and (A4b) imply that the greatest common divisorof s α for all α ∈ X ∗ ( T ) is 1. For the proof of the next theorem wewill only need this fact, but we need the stronger statements (A4a)and (A4b) later. Note also that (A4a) guarantees that we can applyTheorem 3.2 with Z = X α and s Z = s α .Let G X be the moment graph associated to X . For α ∈ X ∗ ( T ) wedenote by G αX the moment graph obtained from G X by deleting all edges E with kα E ∩ kα = 0. Then (A4a) is equivalent to:(A4a) ′ The moment graph G αX is a (discrete) union of moment graphswith only one or two vertices.Now we can state the second part of the localisation theorem. Theorem 4.4.
Suppose that (A1), (A2), (A3), (A4a) and (A4b) hold.Let
F ∈ D bT ( X, k ) and suppose that H • T ( F ) and H • T ( F X T ) are free S k -modules. Then H • T ( F ) = Γ( W ( F )) as submodules of L x ∈ X T H • T ( F x ) = L x ∈ X T W ( F ) x . For the proof of the above statement we use similar arguments asthe ones given in [CS74], [GKM98] or [Bri98]. Again we follow [Bri98]closely.
Proof.
As a first step let
F ∈ D bT ( X, k ) be any sheaf and α ∈ X ∗ ( T ).Let Γ α ( W ( F )) be the sections of the sheaf W ( F ) on the moment graph G αX (so we only consider the edges E with kα E ∩ kα = 0). By (A3),Γ( W ( F )) = \ α ∈ X ∗ ( T ) Γ α ( W ( F )) . ARITY SHEAVES AND MOMENT GRAPHS 21
By (A4a), X α is a discrete union of points and P ’s. Hence, if wedenote by r α : H • T ( F X α ) → H • T ( F X T ) the restriction map, then Lemma4.3 yields Γ α ( W ( F )) = r α ( H • T ( F X α )). Hence:Γ( W ( F )) = \ α ∈ X ∗ ( T ) r α ( H • T ( F X α )) . So we have to show that H • T ( F ) = T α ∈ X ∗ ( T ) r α ( H • T ( F X α )).Clearly H • T ( F ) is contained in the intersection T α ∈ X ∗ ( T ) r α ( H • T ( F X α )).Hence it remains to show that if f ∈ H • T ( F X T ) is in r α ( H • T ( F X α )) forall α ∈ X ∗ ( T ), then f is contained in H • T ( F ).By Theorem 3.2 the injection i : H • T ( F ) → H • T ( F X T ) becomesan isomorphism after inverting s . By assumption, H • T ( F ) is a free S k -module. We choose a basis e , . . . , e m for H • T ( F ) and denote by e ∗ , . . . , e ∗ m ∈ Hom( H • T ( F ) , S k ) the dual basis. Because i becomes an iso-morphism after inverting s , we can find ˜ e ∗ , . . . , ˜ e ∗ m ∈ Hom S k ( H • T ( F X T ) , S k [1 /s ])such that e ∗ j = ˜ e ∗ j ◦ i for 1 ≤ j ≤ m . Note that f is in H • T ( F ) if andonly if ˜ e ∗ j ( f ) is contained in S k for 1 ≤ j ≤ m .By Theorem 3.2, the map H • T ( F ) ֒ → H • T ( F X α )becomes an isomorphism after inverting s α . As f is contained in H • T ( F X α ), we conclude ˜ e ∗ j ( f ) ∈ S k [1 /s α ] for any 1 ≤ j ≤ m . Hence,˜ e ∗ j ( f ) ∈ \ α ∈ X ∗ ( T ) S k [1 /s α ] . But T α ∈ X ∗ ( T ) S k [1 /s α ] = S k as the greatest common divisor of all s α is1. Hence ˜ e ∗ j ( f ) ∈ S k , which is what we wanted to show. (cid:3) Equivariant parity sheaves
In the following sections we consider equivariant parity sheaves on astratified variety, which were introduced in [JMW09a]. It turns out thatthe equivariant cohomology of such a sheaf is free over the symmetricalgebra, so by the results in the previous sections it can be calculated bymoment graph techniques. We determine the corresponding sheaves onthe moment graph explicitely: we show that these are the sheaves thatare constructed by the Braden-MacPherson algorithm (cf. [BM01]).For all of the above, we need an additional datum: a stratificationof the variety.
Stratified varieties.
We assume from now that the T -variety X is endowed with a stratification X = G λ ∈ Λ X λ by T -stable subvarieties X λ . We write D bT, Λ ( X, k ) for the full subcat-egory of D bT ( X, k ) consisting of objects which are constructible withrespect to this stratification. In addition to the assumptions (A1) and(A2) we assume:(S1) For each λ ∈ Λ there is a T -equivariant isomorphism X λ ∼ = C d λ ,where C d λ carries a linear T -action.(S2) The category D bT, Λ ( X, k ) is preserved by under Grothendieck-Verdier duality. (This is satisfied, for example, if the stratifica-tion is Whitney.)By (A1) and (A2) each stratum X λ contains a unique fixed point.We denote this fixed point by x λ .The topology of X gives us a partial order on the set Λ: We set λ ≤ µ if and only if X λ ⊂ X µ . We use the following notation for an arbitrarypartially ordered set Λ: For λ ∈ Λ we set {≥ λ } := { ν ∈ Λ | ν ≥ λ } and we define {≤ λ } , { > λ } , etc. in an analogous fashion. Definition 5.1.
Let K be a subset of Λ . • We say that K is open , if for all γ ∈ K , λ ∈ Λ with λ ≥ γ wehave λ ∈ K , i.e. if K = S γ ∈K {≥ γ } . • We say that K is closed if Λ \K is open, i.e. if K = S γ ∈K {≤ γ } . • We say that K is locally closed if it is the intersection of anopen and a closed subset of Λ . For a subset K of Λ the set K + := S γ ∈K {≥ γ } is the smallest opensubset containing K , and K − := S γ ∈K {≤ γ } is the smallest closedsubset containing K . K is locally closed if K = K − ∩ K + .For any subset K of Λ we define X K = G γ ∈K X γ ⊂ X If K is open (closed, locally closed), then X K is an open (closed, lo-cally closed, resp.) subvariety in X . In particular, for any λ ∈ Λ thesubvariety X ≤ λ := X {≤ λ } is closed. For F ∈ D bT, Λ ( X, k ) we define F K := F X K .5.2. Equivariant parity sheaves.
For λ ∈ Λ we denote by i λ : X λ → X the inclusion. We now give the definition of an equivariant paritysheaf on X : ARITY SHEAVES AND MOMENT GRAPHS 23
Definition 5.2.
Let ? either denote the symbol ∗ or the symbol ! , andlet P ∈ D bT ( X, k ) . • P is ?-even (resp. ?-odd ) if for all λ ∈ Λ the sheaf i ? λ P isa direct sum of constant sheaves appearing only in even (resp.odd) degrees. • P is even (resp. odd ) if it is both ∗ -even and ! -even (both ∗ -oddand ! -odd, resp.). • P is parity if it may be written as a sum P = P ⊕ P with P even and P odd. Note that, by assumption (S1), for all λ ∈ Λ, all T -equivariant localsystems on X λ are trivial and we have H • T ( X λ , k ) = H • T ( pt, k ) = S k . Hence, we have the following classification of indecomposable paritysheaves (see [JMW09a, Theorem 2.9]):
Theorem 5.3.
Suppose that k is a complete local ring. For all λ ∈ Λ there exists, up to isomorphism, at most one indecomposable paritysheaf P ( λ ) extending the equivariant constant sheaf k X λ . Moreover, anyindecomposable parity sheaf is isomorphic to P ( λ )[ i ] for some λ ∈ Λ and some integer i . Note that in this paper (in contrast to [JMW09a]) we normaliseindecomposable parity sheaves in such a way that the restriction of P ( λ ) to X λ is the constant sheaf in degree 0. Also, in [JMW09a]parity sheaves a considered with respect to an arbitrary “pariversity” † : Λ → Z / Z . In this paper we only consider parity sheaves withrespect to the constant pariversity, which corresponds to the abovedefinition. Proposition 5.4.
Let λ ∈ Λ and assume that P ( λ ) exists. We have D ( P ( λ )) ∼ = P ( λ )[2 d λ ] where d λ denotes the complex dimension of X λ .Proof. This is a simple consequence of the above theorem, together withthe fact that D preserves parity and the fact that D k X λ ∼ = k X λ [2 d λ ]. (cid:3) Short exact sequences involving parity sheaves.
Let Q bea parity sheaf on X and let J ⊂
Λ be an open subset with closedcomplement I = Λ \ J . Denote by j : X J → X and i : X I → X thecorresponding inclusions. Consider the distinguished triangle(3) i ! i ! Q → Q → j ∗ j ∗ Q [1] → . Lemma 5.5. (1)
The above triangle gives rise to a short exact se-quence → H • T ( i ! Q ) → H • T ( Q ) → H • T ( Q J ) → . (2) Let P be another parity sheaf on X . Then the above trianglegives rise to a short exact sequence → Hom • ( i ∗ P , i ! Q ) → Hom • ( P , Q ) → Hom • ( P J , Q J ) → . Proof.
We may assume without loss of generality that Q is even. Thenthe distinguished triangle in (3) is a distinguished triangle of !-evensheaves. If P (resp. Q ′ ) is ∗ -even (resp. !-even) then an induction (see[JMW09a, Corollary 2.8]) shows that Hom( P , Q ′ [ n ]) = 0 for odd n .Then (2) follows and part (1) is the case P = k X . (cid:3) Further properties of equivariant parity sheaves.
The fol-lowing properties of the equivariant cohomology of parity sheaves willbe useful when we come to relate parity sheaves and Braden-MacPhersonsheaves in the next section.
Proposition 5.6.
Suppose that P is an equivariant parity sheaf on X .Then the following holds: (1) For any open subset J of Λ the equivariant cohomology H • T ( P J ) is a free S k -module. (2) For any open subset J of Λ the restriction homomorphism H • T ( P ) → H • T ( P J ) is surjective. (3) Assume that (A4b) holds and suppose that E ⊂ X λ is a one-dimensional T -orbit. Then the restriction map ρ λ,E : H • T ( P x λ ) → H • T ( P E ) is surjective with kernel α E H • T ( P x λ ) .Proof. Note that (2) has already been shown in the previous lemma.For (1), first note that if we choose an open subset
J ⊂
Λ then P J isa parity sheaf on X J . Hence it is enough to show that H • T ( P ) is a free S k -module. Choose x ∈ Λ minimal, let I = { x } and J = Λ \ { x } . Wehave an exact sequence0 → H • T ( i ! P ) → H • T ( P ) → H • T ( P J ) → . As P is a parity sheaf, i ! P is a direct sum of constant sheaves andso H • T ( i ! P ) is a free S k -module. Using induction we can assume that H • T ( P J ) is a free S k -module. Hence H • T ( P ) is free.Let us prove (3). Since E ∪ { x λ } is contained in X λ , the restrictionof P to E ∪ { x λ } is isomorphic to a sum of shifted constant sheaves.Hence it is enough to show that if T acts on C via the character α = 0 ARITY SHEAVES AND MOMENT GRAPHS 25 such that n is invertible in k if α is divisible by n in X ∗ ( T ), then themap ρ , C × : H • T ( { } , k ) → H • T ( C × , k )identifies with the canonical quotient map S k → S k /αS k . With char-acteristic 0 coefficients this is proved in [Jan09, Section 1.10]. Thedivisibility assumption guarantees that the argument given there alsoworks with coefficients in k . (cid:3) Obtaining parity sheaves via resolutions.
Up until now wehave only discussed various properties of parity sheaves, without dis-cussing their existence. We now show that the existence of certainproper morphisms to the varieties X λ guarantees the existence of par-ity sheaves.Assume that, for all λ ∈ Λ, there exists a T -variety f X λ and a propersurjective morphism π λ : f X λ → X λ such that:(R1) each f X λ is smooth and admits a T -equivariant closed embedding f X λ ֒ → P ( V ) for some T -module V ,(R2) the fixed point set f X λT is finite,(R3) π λ ∗ k f X λ is constructible with respect to the stratification Λ (thatis, π λ ∗ k f X λ ∈ D bT, Λ ( X, k )).Note that we do not assume that the morphisms π λ are birational. Theorem 5.7.
Assume that k is a complete local principal ideal do-main. With the above assumptions we have: (1) For all λ ∈ Λ there exists an indecomposable parity sheaf P ( λ ) ∈ D bT ( X, k ) with support equal to X λ and such that P ( λ ) X λ ∼ = k X λ . (2) For all µ ≤ λ the restriction homomorphism H • T ( P ( λ )) → H • T ( P ( λ ) x µ ) is surjective. (3) The cohomology H • T ( P ( λ )) is self-dual of degree X λ . Thatis, Hom • S k ( H • T ( P ( λ )) , S k ) ∼ = H • T ( P ( λ ))[2 dim X λ ] . Before proving the theorem we state and prove two propositions. Forthis we need some more notation. Given a T -variety Z we write ω Z for the T -equivariant dualising sheaf in D bT ( Z, k ). Note that, up toreindexing, H • T ( ω Z ) is the T -equivariant Borel-Moore homology of Z .Let us fix µ ≤ λ and set F := π − λ ( x µ ). We have: Proposition 5.8. (1) H • T ( ω F ) and H • T ( ω f X λ ) are free S k -modulesconcentrated in even degrees. (2) The canonical map H • T ( ω F ) → H • T ( ω f X λ ) is a split injection of S k -modules.Proof. As x µ is attractive, there exists a one parameter subgroup γ : C × → T which contracts an open neighbourhood of x µ in X onto x µ as z ∈ C × goes to 0. Moreover, we can choose γ such that f X λ C × = f X λT .Now consider the Bialynicki-Birula’s minus decomposition of f X λT with respect to γ . That is, for each x ∈ f X λT set Y − x := { y ∈ f X λ | lim z →∞ γ ( z ) · y = x } . Then a theorem of Bialynicki-Birula ([BB81]) asserts that each Y − x is alocally closed T -stable subvariety of f X λ isomorphic to an affine space.Our choice of γ implies that F = π − λ ( x µ ) = G x ∈ F T Y x . Moreover, by assumption we can find a T -equivariant embedding f X λ ֒ → P ( V )and we may decompose V ∼ = L V i where V i denotes the i th weightspace of the C ∗ -action on V given by γ . If we set V ≤ i = L j ≤ i V i thenit is straightforward to check that the filtration ∅ ⊂ · · · ⊂ P ( V ≤ i ) ⊂ P ( V ≤ i +1 ) ⊂ · · · ⊂ P ( V )induces filtrations of f X λ and F by closed subvarieties such that eachsuccessive difference is a disjoint union of Bialynicki-Birula cells. A sim-ple induction (see, for example, [Ful97] for the non-equivariant case)shows that both H • T ( ω f X λ ) and H • T ( ω F ) are free S k -modules with ba-sis given by the equivariant fundamental classes of closures of theBialynicki-Birula cells. The two statements of the lemma then fol-low. (cid:3) Proposition 5.9.
With notation as above we have: (1) π λ ∗ k f X λ is parity and its support is equal to X λ . (2) For all µ ≤ λ the restriction homomorphism H • T ( π λ ∗ k f X λ ) → H • T (( π λ ∗ k f X λ ) x µ ) is surjective. ARITY SHEAVES AND MOMENT GRAPHS 27
Proof.
The support claim follows from the surjectivity of π λ . We nowexplain why π λ ∗ k f X λ is parity. As π λ is proper and f X λ is smooth, π λ ∗ k f X λ is self-dual up to a shift and so it is enough to show that π λ ∗ k f X λ is !-even. As π λ ∗ k f X λ is constructible with respect to the Λ-stratification, itis enough to show that, for all µ , i ! x µ π λ ∗ k f X λ is a direct sum of constantsheaves concentrated in even degrees, where i x µ denotes the inclusion { x µ } ֒ → X . A devissage argument shows that this is the case if andonly if H • T ( i ! x µ π λ ∗ k f X λ ) is a free S k -module.By proper base change i ! x µ π λ ∗ k f X λ is isomorphic (up to a shift) to π λ ∗ ω F . Hence it is enough to show that H • T ( ω F ) is a free S k -moduleconcentrated in even degrees. This is the case by Proposition 5.8(1)above.For the second statement of the proposition note that the restrictionhomomorphism H • T ( π λ ∗ k f X λ ) → H • T (( π λ ∗ k f X λ ) x µ ) is dual (again, up toa shift) to the canonical map H • T ( ω F ) → H • T ( ω f X λ ) which is a splitinjection by Proposition 5.8(2). (cid:3) Proof of Theorem 5.7.
By Proposition 5.9, π λ ∗ k f X λ ∈ D bT ( X, k ) is par-ity. If we let Q denote an indecomposable summand of π λ ∗ k f X λ con-taining X λ in its support then Q is also parity and, by Theorem 5.3,we have Q X λ ∼ = k X λ [ i ] for some integer i . It follows that we can take P ( λ ) := Q [ − i ].Another consequence of Theorem 5.3 is that any indecomposableparity sheaf P ( λ ) occurs as a direct summand of π λ ∗ k f X λ [ i ] ∈ D bT ( X, k )for some i . Hence, to show Part (2) of the theorem it is enough tocheck that the map H • T ( π λ ∗ k f X λ ) → H • T (( π λ ∗ k f X λ ) x µ )is surjective. This is the case by Proposition 5.9.By Proposition 5.4 we have D P ( λ ) ∼ = P ( λ )[2 dim X λ ]. We also knowthat H • T ( P ( λ )) is a free S k -module by Proposition 5.8 (recall that ω f X λ ∼ = k f X λ because f X λ is smooth). HenceHom • S k ( H • T ( P ( λ )) , S k ) ∼ = H • T ( P ( λ ))[2 dim X λ ] . as X λ is proper. (cid:3) Parity sheaves and the functor W . In this section we begindiscussing the relationship between parity sheaves and the localisationfunctor W . In particular, we show that W is fully faithful on morphismsof all degrees between parity sheaves.For the rest of this section we assume (A1)-(A4a/b) and (S1), (S2). Proposition 5.10.
Let P ( λ ) be a parity sheaf. Then the localisationhomomorphism H • T ( P ( λ )) → H • T ( P ( λ ) X T ) identifies H • T ( P ( λ )) with theglobal sections of W ( P ( λ )) .Proof. In order to apply Theorem 4.4 we only need to check that H • T ( P ( λ )) and H • T ( P ( λ ) X T ) are free S k -modules. This is the case for H • T ( P ( λ )) by Proposition 5.6. For H • T ( P ( λ ) X T ) note that because P ( λ )is parity, the restriction of P ( λ ) to any T -fixed point is a direct sum ofequivariant constant sheaves. (cid:3) Theorem 5.11.
The functor W is fully faithful when restricted to par-ity sheaves, i.e. if P and P ′ are parity sheaves on X , then Hom • D bT ( X,k ) ( P , P ′ ) → Hom •G -mod Z k ( W ( P ) , W ( P ′ )) is an isomorphism.Proof. Without loss of generality we can assume that both P and P ′ are even. Let λ ∈ Λ be a minimal element and set J = Λ \ { λ } .Denote by j : X J = X \ X λ → X the corresponding open inclusion andby i : X λ → X the complementary closed inclusion. Then we have adistinguished triangle i ! i ! P ′ → P ′ → j ∗ j ∗ P ′ [1] → which gives rise, by Lemma 5.5, to a short exact sequence0 → Hom • ( i ∗ P , i ! P ′ ) → Hom • ( P , P ′ ) → Hom • ( P J , P ′J ) → • ( P , P ′ ) → Hom • ( P J , P ′J ) is inducedby the restriction to an open subspace, hence we can fit the above shortexact sequence into a commutative diagram0 / / Hom( i ∗ P , i ! P ′ ) / / (cid:15) (cid:15) Hom( P , P ′ ) / / (cid:15) (cid:15) Hom( P J , P ′J ) (cid:15) (cid:15) / / / / K / / Hom( W ( P ) , W ( P ′ )) / / Hom • ( W ( P J ) , W ( P ′J )) / / . As P J and P ′J are parity sheaves on X J we can, by induction onthe number of strata, assume that the vertical map on the right isan isomorphism. Hence we can finish the proof by showing that thevertical map on the left is an isomorphism as well.Now K is the space of all morphisms f : W ( P ) → W ( P ′ ) with f µ = 0and f E = 0 for vertices µ and edges E of J . By Proposition 5.6, (3),it identifies with the set of homomorphisms from the stalk W ( P ) λ intothe costalk W ( P ′ ) λ . By definition we have W ( P ) λ = H • T ( i ∗ P ). Nowlet us look at the short exact sequence0 → H • T ( i ! P ′ ) → H • T ( P ′ ) → H • T ( P ′J ) → ARITY SHEAVES AND MOMENT GRAPHS 29 given by Lemma 5.5. By Proposition 5.10, H • T ( P ′ ) and H • T ( P ′J ) canbe identified with the sections of W ( P ′ ) over Λ and J , respectively.Hence we may identify H • T ( i ! P ′ ) = W ( P ′ ) λ . As i ∗ P and i ! P ′ are free sheaves on X λ , we deduce from the above thatthe homomorphism Hom( i ∗ P , i ! P ′ ) → K in the commutative diagramabove is an isomorphism as well. (cid:3) Braden-MacPherson sheaves on a moment graph
We return now to the theory of sheaves on a moment graph. We firstmotivate the definition of the Braden-MacPherson sheaves by consid-ering the problem of extending local sections. Then we prove one ofour main results, namely that the functor W sends parity sheaves toBraden-MacPherson sheaves.6.1. Extending local sections.
Let G = ( V , E , α ) be a moment graph.Suppose that each edge is given a direction. Then, for x, y ∈ V , we set x y if and only if there is a directed path from x to y . We assumethat this determines a partial order on V , i.e. we assume that thereare no non-trivial closed directed paths. We call this datum a directedmoment graph .Recall that we call a subset J of V open if it contains with anyelement all elements that are larger in the partial order, i.e. all elementsthat can be reached by a directed path. For a sheaf M and an opensubset J of V we call an element in Γ( J , M ) a local section .Now we want to find some conditions on M that ensure that eachlocal section can be extended to a global section, i.e. which ensurethat the restriction Γ( M ) → Γ( J , M ) from global to local sections issurjective for any open set J . For this we need the following definitions.For a vertex x of G we define V δx := { y ∈ V | there is an edge E : x → y } . So V δx is the subset of vertices y that are bigger than x in the partialorder and that are connected to x by an edge. We denote by E δx := { E ∈ E | E : x → y } the set of the corresponding edges. Then there is an obvious correspon-dence E δx ∼ → V δx (as we assume that two vertices are connected by atmost one edge). For a sheaf M and a vertex x we define the map u x : Γ( { > x } , M ) → M E ∈E δx M E as the compositionΓ( { > x } , M ) ⊂ M y>x M y p → M y ∈V δx M y ρ → M E ∈E δx M E , where p is the projection along the decomposition and ρ = L y ∈V δx ρ y,E .We let M δx := u x (Γ( { > x } , M )) ⊂ M E ∈E δx M E be the image of this map. Moreover, we define the map d x := ( ρ x,E ) TE ∈E δx : M x → M E ∈E δx M E . The connection of the above definitions with the problem of extendinglocal sections is the following. Suppose that m ′ ∈ Γ( { > x } , M ) is asection and that m x ∈ M x . Then the concatenated element ( m x , m ′ ) ∈ L y ≥ x M y is a section over {≥ x } if and only if u x ( m ′ ) = d x ( m x ). Lemma 6.1.
For a sheaf M on the moment graph G the following areequivalent: (1) For any open subsets J ′ ⊂ J of V the restriction map Γ( J , M ) → Γ( J ′ , M ) is surjective. (2) For any vertex x ∈ V , the restriction map Γ( {≥ x } , M ) → Γ( { > x } , M ) is surjective. (3) For any x ∈ V , the map d x : M x → L E ∈E δx M E contains M δ x in its image.Proof. Clearly property (2) is a special case of property (1). Let usprove the converse, so let us assume that (2) holds. It is enough toprove property (1) in the special case that J = J ′ ∪ { x } for a singleelement x , since we get the general case from this by induction. So let m = ( m y ) be a section in Γ( J \{ x } , M ). Since { > x } ⊂ J \{ x } we canrestrict m and get a section m ′ in Γ( { > x } , M ). By assumption thereis an element m x ∈ M x such that ( m x , m ′ ) is a section over {≥ x } . As x is not connected to any vertex in J \ {≥ x } it follows that ( m x , m )is a section over J . Hence (2) implies (1).Let us show that (2) is equivalent to (3). Now (2) means that forany section m over { > x } we can find m x ∈ M x such that ( m x , m ) isa section over {≥ x } . But ( m x , m ) is a section over {≥ x } if and onlyif d x ( m x ) = u x ( m ). Hence, a section m over { > x } can be extended tothe vertex x if and only of u x ( m ) is contained in the image of d x . (cid:3) For later use we prove the following statement.
ARITY SHEAVES AND MOMENT GRAPHS 31
Lemma 6.2.
Let x be a vertex of G and M a sheaf on G . Then thefollowing are equivalent: (1) The map Γ( {≥ x } , M ) → M x is surjective. (2) The image of the map d x : M x → L E ∈E δx M E is contained in M δx .Proof. Suppose that (1) holds and let s ∈ M x . Then there is a section m of M over {≥ x } with m x = s . If we denote the restriction of m to { > x } by m ′ , then this means that d x ( s ) = u x ( m ′ ). So d x ( s ) iscontained in the image of u x , which is M δx .Conversely, suppose that (2) holds and let s ∈ M x . Then thereis a section m ′ of M over { > x } such that d x ( s ) = u x ( m ′ ). Hence( s, m ′ ) is a section over {≥ x } , hence s is contained in the image ofΓ( {≥ x } , M ) → M x . (cid:3) Braden–MacPherson sheaves.
The most important class ofsheaves on a moment graph is the following.
Definition 6.3.
A sheaf B on the moment graph G is called a Braden–MacPherson sheaf if it satisfies the following properties: (1)
For any x ∈ V , the stalk B x is a graded free S k -module of finiterank and only finitely many B x are non-zero. (2) For a directed edge E : x → y the map ρ y,E : B y → B E issurjective with kernel α ( E ) B y , (3) For any open subset J of V the map Γ( B ) → Γ( J , B ) is sur-jective. (4) The map Γ( B ) → B x is surjective for any x ∈ V . Here is a classification theorem.
Theorem 6.4.
Assume that k is a local ring and suppose that themoment graph is such that for any vertex w the set {≤ w } is finite.Then the following holds. (1) For any vertex w there is an up to isomorphism unique Braden–MacPherson sheaf B ( w ) on G with the following properties: • We have B ( w ) w ∼ = S k and B ( w ) x = 0 unless x ≤ w . • B ( w ) is indecomposable in G -mod Z k . (2) Let B be a Braden–MacPherson sheaf. Then there are w , . . . , w n ∈V and l , . . . , l n ∈ Z such that B ∼ = B ( w )[ l ] ⊕ · · · ⊕ B ( w n )[ l n ] . The multiset ( w , l ) ,. . . , ( w n , l n ) is uniquely determined by B .Remark . We need the locality assumption on k in order to ensurethat projective covers exist in the category of graded S k -modules. Proof.
We first prove the existence part of statement (1). For w ∈ V we define a sheaf B ( w ) by the following inductive construction:(1) We start with setting B ( w ) w = S k and B ( w ) x = 0 if x w .(2) If we have already defined B ( w ) y , then we set, for each edge E : x → y , B ( w ) E := B ( w ) y /α ( E ) B ( w ) y and we let ρ y,E : B ( w ) y → B ( w ) E be the canonical map.(3) Suppose that we have already defined B ( w ) y for all y in anopen subset J and suppose that x ∈ V is such that J ∪ { x } is open as well. By step (2) we have also defined the spaces B ( w ) E for each edge E : x → y originating at x , as well as themaps ρ y,E : B ( w ) y → B ( w ) E . We now define B ( w ) x and themaps ρ x,E for those edges E . We can already calculate the sec-tions of B ( w ) over { > x } , as well as the S k -modules B ( w ) δx ⊂ L E ∈E δx B ( w ) E . Now we define d x : B ( w ) x → B ( w ) δx as aprojective cover in the category of graded S k -modules. Thecomponents of d x (with respect to the inclusion B ( w ) δx ⊂ L E ∈E δx B ( w ) E ) give us the maps ρ x,E .Let us check that B ( w ) is indeed a Braden–MacPherson sheaf. Since B ( w ) x is projective for all x ∈ V it is a graded free S k -module andthe finiteness assumptions hold as well, so B ( w ) fulfills property (1).Property (2) is assured by step (2) in the inductive construction of B ( w ). Property (3) is, by Lemma 6.1, equivalent to the fact that forall x ∈ V the map d x : B ( w ) x → L E ∈E δx B ( w ) E contains B ( w ) δ x inits image. This is clear by step (3). In addition, step (3) also yieldsthat the image of d x is contained in B ( w ) δx . By Lemma 6.2 this isequivalent to the surjectivity of Γ( {≥ x } , B ( w )) → B ( w ) x . We havealready seen that the restriction map Γ( B ( w )) → Γ( {≥ x } , B ( w )) issurjective. Hence also Γ( B ( w )) → B ( w ) x is surjective, hence B ( w )also satisfies property (4) of a Braden–MacPherson sheaf.Now we prove statement (2) of the above theorem using the aboveexplicitly defined objects B ( w ). Note that this also gives the unique-ness part of statement (1), which we have not yet proven. So let B bean arbitrary Braden–MacPherson sheaf. We prove by induction on theset of open subsets J of V that there are ( w , l ),. . . ,( w n , l n ) such that B J ∼ = B ( w ) J [ l ] ⊕ · · · ⊕ B ( w n ) J [ l n ] . (Here and in the following we denote by F I the obvious restriction ofa sheaf F to the subgraph corresponding to the vertices in a subset I of V .) ARITY SHEAVES AND MOMENT GRAPHS 33
So suppose that J is open, that x ∈ J is minimal and we have adecomposition as above for J ′ = J \ { x } . We get, in particular,Γ( { > x } , B ) ∼ = Γ( { > x } , B ( w )[ l ]) ⊕ · · · ⊕ Γ( { > x } , B ( w n )[ l n ])and B δx ∼ = B ( w ) δx [ l ] ⊕ · · · ⊕ B ( w n ) δx [ l n ] . Now d x : B x → B δx is surjective, by property (3) of a Braden–MacPhersonsheaf and Lemma 6.1, and L B ( w i ) x [ l i ] → L B ( w i ) δx [ l i ] is a projec-tive cover by construction. Hence we have a decomposition B x = L B ( w i ) x [ l i ] ⊕ R for some graded free S k -module R which lies in thekernel of d x . Each isomorphism R ∼ = S k [ m ] ⊕ · · · ⊕ S k [ m r ] then yieldsan isomorphism B J ∼ = B ( w ) J [ l ] ⊕ · · · ⊕ B ( w n ) J [ l n ] ⊕ B ( x ) J [ m ] ⊕ · · · ⊕ B ( x ) J [ m r ] , which is our claim for J . The above construction also yields the unique-ness of the multiset ( w , l ),. . . ,( w n , l n ). (cid:3) Directed moment graphs from stratified varieties.
Supposethat X is a complex T -variety satisfying (A1). In Section 4.4 we con-structed an (undirected) moment graph G X from this datum. Supposenow that, in addition, we are given a stratification X = F λ ∈ Λ X λ sat-isfying (S1) and (S2). Recall that for each λ ∈ Λ we denote by x λ theunique fixed point in X λ . Hence we now have identifications betweenthe set of fixed points in X , the set Λ of strata and the set of verticesof G X .From this we obtain a direction of each edge as follows. Suppose thatthe one-dimensional orbit E contains x λ and x µ in its closure. Theneither X λ ⊂ X µ or X µ ⊂ X λ . We direct the corresponding edge of G X towards µ in the first case, and towards λ in the second case. Wedenote by ≤ the partial order on the vertices of G X generated by therelation λ ≤ µ if there is an edge E : λ → µ . The following propositionshows that this is the same order as the one induced by the closurerelations on the strata: Proposition 6.5.
We have λ ≤ µ if and only if X λ ⊂ X µ .Proof. Clearly, if λ ≤ µ then X λ ⊂ X µ . For the converse we show:(4) If X λ ⊂ X µ then there exists an edge E : λ → ν such that X ν ⊂ X µ .Let us assume that (4) holds. Then, if X λ ⊂ X µ , we can find a chain λ → ν → · · · → ν m → µ and so λ ≤ µ . It remains to show (4). Let U be an affine T -stable neighbourhood of x λ in X µ and let N λ ⊂ U be a T -invariant normal slice to the stratum X λ at x λ . Because x λ ∈ N λ is attractive, we can find a cocharacter γ : C × → T such that lim z → γ ( z ) · x = x λ for all x ∈ N λ . It followsthat Y := ( N λ \ { x λ } ) / C × is a projective variety. By Borel’s fixedpoint theorem, T has a fixed point on Y and hence a one-dimensionalorbit on N λ . This one-dimensional orbit is contained in some X ν , henceconnects x λ with x ν . By construction we have X ν ⊂ X µ . (cid:3) The k -smooth locus of a moment graph. In this subsectionwe assume that k is a field and that the (directed) moment graph G contains a largest vertex w . This moment graph carries the indecom-posable Braden–MacPherson sheaf B := B ( w ) over k . Definition 6.6.
The k -smooth locus Ω k ( G ) of G is the set of all vertices y of G such that B y is a free S k -module of (ungraded) rank . In [Fie10] the k -smooth locus is determined for a large class of pairs( G , k ). In order to formulate the result, let B := Γ( B ) be the space ofglobal sections of B . We consider this as a graded Z k -module. Definition 6.7.
We say that B is self-dual of degree l ∈ Z if there isan isomorphism Hom • S k ( B , S k ) ∼ = B [ l ] of graded Z k -modules. The following is an analogue of the assumption (A4a) for momentgraphs.
Definition 6.8.
We say that the pair ( G , k ) is a GKM-pair if α E isnon-zero in Y ⊗ Z k for any edge E and if for any distinct edges E and E ′ containing a common vertex we have kα E ∩ kα E ′ = 0 . Note that this can be considered, for given G , as a requirement onthe characteristic of k . The main result of [Fie10] is the following: Theorem 6.9 ([Fie10, Theorem 5.1]) . Suppose that ( G , k ) is a GKM-pair and that B is self-dual of degree l . Then we have Ω k ( G ) = (cid:26) x ∈ V (cid:12)(cid:12)(cid:12)(cid:12) for all y ≥ x the numberof edges containing y is l (cid:27) . We are going to apply this statement later in order to study the k -smooth locus of T -varieties. ARITY SHEAVES AND MOMENT GRAPHS 35
The combinatorics of parity sheaves.
Let X be a complex T -variety, k a complete local principal ideal domain. Assume that thesedata satisfy the assumptions (A1)–(A4a/b), (S1), (S2) and (R1)–(R3).We now come to the principal result of this paper. Theorem 6.10.
Suppose that
P ∈ D bT ( X, k ) is a parity sheaf. Then W ( P ) is a Braden–MacPherson sheaf. More precisely, W ( P ( λ )) ∼ = B ( λ ) .Proof. We have to show that W ( P ) satisfies the four properties listedin Definition 6.3. If we translate this definition into our situation wesee that we have to check the following:(1) For each x ∈ X T the cohomology H • T ( P x ) is a graded free mod-ule over S k .(2) For each one-dimensional orbit E that is contained in the stra-tum associated to the fixed point x , the map H • T ( P x ) → H • T ( P E )is surjective with kernel α ( E ) H • T ( P x ).(3) For each open union X J ⊂ X of strata the restriction homo-morphism H • T ( P ) → H • T ( P J )is surjective.(4) For each x ∈ X T the homomorphism H • T ( P ) → H • T ( P x ) is sur-jective.Part (1) follows directly from the definition of a parity sheaf, the parts(2) and (3) are stated in Proposition 5.6. Part (4) follows from Theorem5.7 and the fact P is the direct sum of shifted copies of P ( λ )’s. The laststatement follows, as P ( λ ) is indecomposable if and only if W ( B ( λ ))is, by Theorem 5.11. (cid:3) The case of Schubert varieties
We now discuss a special and important case of the general theorydeveloped in the previous section, namely the case of Schubert varietiesin (Kac-Moody) flag varieties. For a detailed construction of thesevarieties in the Kac-Moody setting the reader is referred to [Kum02].We fix some notation. Let A be a generalised Cartan matrix of size l and let g = g ( A ) = n − ⊕ h ⊕ n + denote the corresponding Kac-MoodyLie algebra with Weyl group W , Bruhat order ≤ , length function ℓ andsimple reflections S = { s i } i =1 ,...,l . To A one may also associate a Kac-Moody group G with Borel subgroup B and connected algebraic torus T ⊂ B . Given any subset I ⊂ { , . . . , l } one has a standard parabolic subgroup P I containing B and standard parabolic subgroup W I ⊂ W .The set G/P I may be given the structure of an ind- T -variety and iscalled a Kac-Moody flag variety.For each w ∈ W one may consider the Schubert cell X Iw := BwP I /P I ⊂ G/P I and its closure, the Schubert variety , X Iw = G v ≤ w BvP I . Each Schubert cell is isomorphic to a (finite dimensional) affine spaceand each Schubert variety is a (finite dimensional) projective algebraicvariety. The partition of
G/P I into Schubert cells gives a stratificationof G/P I .The following proposition shows that the results of this article maybe applied to any closed union of finitely many B -orbits in G/P I : Proposition 7.1.
Let X ⊂ G/P I be a closed subset which is the unionof finitely many Schubert cells. Then X together with its stratificationinto Schubert cells satisfies our assumptions (A1), (A2), (S1), (S2),(R1), (R2) and (R3).Proof. The assumptions (A1), (A2), (S1) are standard properties ofKac-Moody Schubert varieties (see [Kum02, Chapter 7]) and (S2) fol-lows because we have an equivalence D bT, Λ ( X, k ) ∼ = D bB ( X, k ). Givena Schubert variety X Iw ⊂ X , let π : e X → X Iw denote a Bott-Samelsonresolution (see [Kum02, 7.1.3]). Then e X is a smooth T -variety withfinitely many T -fixed points which admits a T -equivariant closed lin-ear embedding into a projective space. Lastly, the variety e X is even a B -variety, and the map π is B -equivariant. So properties (R1), (R2)and (R3) hold as well. (cid:3) We now describe the moment graph of
G/P I . The identification of h with the Lie algebra of T allows us to identify the lattice of characters X ∗ ( T ) with a lattice in h ∗ . Moreover, under this identification, allthe roots of g ( A ) lie in X ∗ ( T ). Let R ⊂ X ∗ ( T ) denote the subset ofreal roots, and R + the subset of real positive roots. Then we have abijection R + ∼ → { reflections in W } α s α . The following proposition follows from [Kum02, Chapter 7]:
Proposition 7.2.
We have:
ARITY SHEAVES AND MOMENT GRAPHS 37 • The T -fixed points are in bijection with the set W/W I : W/W I ∼ → ( G/P I ) T wW I wP I . • There is a one-dimensional T -orbit with xW I and yW I in itsclosure if and only if s α xW I = yW I for some reflection s α ∈ W in which case T acts on this orbit with character ± α . We complete this section by discussing what the arithmetic assump-tions (A3), (A4a) and (A4b) mean in the case of Kac-Moody flag va-rieties. First note that the lattice Z R ⊂ X ∗ ( T ) spanned by the realroots determines a surjection of algebraic tori s : T ։ T ′ so that X ∗ ( T ′ ) = Z R . The action of T on G/P I is trivial on thekernel of s and we obtain an action of T ′ on G/P I . (In the case ofa finite dimensional Schubert variety this corresponds to the fact thatone may always choose the adjoint form of a reductive group in orderto construct the flag variety.) Because real roots are never divisible in Z R = X ∗ ( T ′ ) it follows that (A4b) (and hence (A3)) is always satisfiedfor Kac-Moody Schubert varieties viewed as T ′ -varieties.The condition (A3) is more subtle. If we fix a field k then condition(A3) is satisfied if and only if no two distinct roots in R + become lin-early dependent modulo k . One may check that in the finite cases wehave to exclude characteristic 2 in non-simply laced types and charac-teristic 3 in type G .In the affine case the situation is radically different. Suppose that b G is the affine Kac-Moody group associated to a semi-simple group G .Recall that there exists a an element δ ∈ b h ∗ such that the set of realroots of b G is equal to { α + nδ } where α ∈ h ∗ is a root of G , and n ∈ Z .It follows that condition (A3) is satisfied for b G/ b P and any parabolicsubgroup b P = b G if and only if k is of characteristic 0.However, if one restricts oneself to a fixed a Schubert variety X ⊂ b G/ b P the GKM-condition for X may yield interesting restrictions onthe characteristic of k (see [Fie07a]).8. p -Smoothness In this section we recall the definition and basic properties of the p -smooth locus of a complex algebraic variety X . Our main goal isTheorem 8.8 for which we need Proposition 8.6, where we show that an(a priori weaker) condition on the stalks of the intersection cohomologycomplex is enough to conclude p -smoothness. Throughout this section all varieties will be irreducible and k denotesa ring (assumed to be a field from Sections 8.2 to 8.4). Dimension willalways refer to the complex dimension. Given a point y in a variety Y we denote by i y : { y } ֒ → Y its inclusion.8.1. Smoothness and p -smoothness. If x is a smooth point of avariety X of dimension n a simple calculation (using the long exactsequence of cohomology, excision and the cohomology of a 2 n − H d ( X, X \ { x } , k ) = ( k, if d = 2 dim X ,0 , otherwise.The isomorphism(5) H d ( i ! x k X ) ∼ = H d ( X, X \ { x } , k )motivates the following. Definition 8.1.
A variety X is k -smooth if, for all x ∈ X , one hasan isomorphism H d ( i ! x k X ) ∼ = ( k, if d = 2 dim X, , otherwise.The k -smooth locus of X is the largest open k -smooth subvariety of X . We define p -smooth (respectively the p -smooth locus ) to mean F p -smooth (respectively the F p -smooth locus ). Proposition 8.2.
We have inclusions Q -smoothlocus ⊃ p -smoothlocus ⊃ Z -smoothlocus ⊃ smoothlocus . Proof.
The fact that the Z -smooth locus contains the smooth locus fol-lows from the above discussion. For all rings k one has an isomorphism(6) i ! x k X ∼ = i ! x Z X L ⊗ Z k. As the category of Z -modules is hereditary, every object in D b ( { x } , Z )is isomorphic to its cohomology. It then follows from (6) that, for afield k , the condition of k -smoothness is satisfied at x if and only if:(1) H d ( i ! x Z X ) is torsion except for d = 2 n , where the free part is ofrank 1,(2) all torsion in H • ( i ! x Z X ) is prime to the characteristic of k .The claimed inclusions now follow easily. (cid:3) Remark . The above proof shows that, if k is a field, then the k -smoothlocus only depends on the characteristic of k . ARITY SHEAVES AND MOMENT GRAPHS 39 k -smoothness and the intersection cohomology complex. Until Section 8.4 we assume that k is a field. Let us denote by ( D ≤ ( X, k ) , D ≥ ( X, k ))the standard t -structure on D ( X, k ) with heart Sh ( X, k ), the abeliancategory of sheaves of k -vector spaces on X . We denote the corre-sponding truncation and cohomology functors by τ ≤ , τ > and H d .Let us a fix a Whitney stratification X = F λ ∈ Λ X λ and denote forall λ ∈ Λ by i λ : X λ ֒ → X the inclusion. Recall that the intersectioncohomology complex of X , IC ( X, k ) ∈ D ( X, k ), is uniquely determinedby the properties:(1) i ∗ λ IC ( X, k ) ∼ = k X λ for the open stratum X λ ⊂ X ;and, for all strata X λ of strictly positive codimension,(2) H d ( i ∗ λ IC ( X, k )) = 0 for d ≥ codim X X λ ,(3) H d ( i ! λ IC ( X, k )) = 0 for d ≤ codim X X λ . Note that under this normalisation IC ( X, k ) is not Verdier self-dual.Rather D IC ( X, k ) ∼ = IC ( X, k )[2 dim X ]. Conditions (2) and (3) areequivalent to the conditions(2*) H d ( i ∗ x IC ( X, k )) = 0 for d ≥ codim X X λ ,(3*) H d ( i ! x IC ( X, k )) = 0 for d ≤ dim X + dim X λ for all x ∈ X λ and strata X λ of strictly positive codimension. (Thisfollows from the fact if y ∈ Y is a smooth point, then one has anisomorphism i ! y k X λ ∼ = i ∗ y k X λ [ − Y ].) Proposition 8.3.
A variety X is k -smooth if and only if IC ( X, k ) ∼ = k X .Proof. If X is k -smooth then the constant sheaf k X satisfies (1), (2*)and (3*) above and hence k X ∼ = IC ( X, k ). On the other hand, if k X ∼ = IC ( X, k ) then D k X ∼ = k X [2 dim X ] and for all x ∈ X we have i ! x k X ∼ = i ! x ( D k X )[ − X ] ∼ = D ( i ∗ x k X [2 dim X ])and hence H d ( i ! x k X ) = H − d ( i ∗ x k X [2 dim X ]) = ( k, if d = 2 dim X ,0 , otherwise,and so x is k -smooth. (cid:3) k -smoothness and stalks. Given a morphism f : X → Y ofcomplex algebraic varieties we write f ∗ for the non-derived direct im-age functor. The functor f ∗ is left t -exact with respect to the standard t -structure. Given F ∈ Sh ( X, k ) we have f ∗ F ∼ = τ ≤ f ∗ F canonically. Lemma 8.4.
Given
F ∈ D ≥ ( X, k ) and a morphism f : X → Y wehave a natural isomorphism τ ≤ f ∗ F ∼ = f ∗ τ ≤ F . Proof.
Applying f to the distinguished triangle τ ≤ F → F → τ > F → yields a distinguished triangle f ∗ τ ≤ F → f ∗ F → f ∗ τ > F → . Now f ∗ is left t -exact for the t -structure ( D ≤ ( X, k ) , D ≥ ( X, k )) andso τ ≤ f ∗ τ > F = 0. Hence if we apply τ ≤ to the above distinguishedtriangle we obtain the required isomorphism f ∗ τ ≤ F = τ ≤ f ∗ τ ≤ F ∼ → τ ≤ f ∗ F . (cid:3) Lemma 8.5.
We have an isomorphism τ ≤ IC ( X, k ) ∼ = j ∗ k U , where j : U ֒ → X denotes the inclusion of a smooth, open, dense subvarietyof X .Proof. Choose a stratification of X which has U as the only stratumof dimension n , and write X i for the union of all strata of codimension ≤ i (so that X = U and X n = X ). We have a chain of inclusions X j ֒ → X j ֒ → X j ֒ → · · · j n − ֒ → X n − j n − ֒ → X n . The Deligne construction (see [BBD82, Proposition 2.1.11]) gives anisomorphism IC ( X, k ) ∼ = ( τ ≤ n − ◦ j n − ∗ ) ◦ ( τ ≤ n − ◦ j n − ∗ ) ◦ . . . ( τ ≤ ◦ j ∗ k U )Repeatedly applying the above lemma yields τ ≤ IC ( X, k ) ∼ = j n − ∗ ◦ · · · ◦ j ∗ ◦ j ∗ k U ∼ = j ∗ k U , as claimed. (cid:3) Proposition 8.6.
A variety X is k -smooth if and only if IC ( X, k ) x ∼ = k for all x ∈ X .Proof. If X is k -smooth then IC ( X, k ) ∼ = k X by Proposition 8.3 and so IC ( X, k ) x ∼ = k for all x ∈ X . It remains to show the converse. Choosean open, dense, smooth subvariety U of X and let j : U ֒ → X denoteits inclusion. The adjunction morphism k X → j ∗ j ∗ k X is an injection on stalks, as may easily be checked from the definition of j ∗ . (It is an isomorphism if and only if X is unibranched, however wewill not need this fact.) It follows from our assumptions that IC ( X, k )lies in D ≤ ( X, k ) and so we have an isomorphism τ ≤ IC ( X, k ) ∼ → IC ( X, k ) . ARITY SHEAVES AND MOMENT GRAPHS 41
By the above lemma we also have an isomorphism j ∗ k U ∼ = τ ≤ IC ( X, k ).It follows that all stalks of j ∗ k U are one-dimensional and that we havean isomorphism k X ∼ → j ∗ k U ∼ → IC ( X, k ) . Our claim now follows from Proposition 8.3. (cid:3)
On the p -smooth locus of T -varieties. Now let X = F λ ∈ Λ X λ be an irreducible, complex, stratified T -variety, and let k be a field.Assume that these data satisfy the assumptions (A1)–(A4a/b), (S1),(S2) and (R1)–(R3) and let Ω k ( G ) denote the k -smooth locus of themoment graph G of X . The following proposition shows that the k -smooth locus of X and of its moment graph agree. Proposition 8.7.
All points of a stratum X λ belong to the k -smoothlocus of X if and only if λ ∈ Ω k ( G ) .Proof. Let U denote the p -smooth locus of X . It is a union of strata byour assumption (S2). Because we have assumed that X is irreduciblethere exists a unique open dense stratum X λ ⊂ X . Let P be the corre-sponding indecomposable parity sheaf normalised so that its restrictionto X λ is k X λ .In the following it will be useful to work with non-equivariant sheaves.Note that the non-equivariant analogue of Theorem 5.3 is valid (see[JMW09a, Theorem 2.12]) and P := For( P ) is the indecomposablenon-equivariant parity sheaf with support X .Let U ′ denote the largest open union of strata X λ for which P X λ ∼ = k X λ . We claim U = U ′ .Indeed, if U ′ denotes this set then P U ′ satisfies the properties (1) and(2) of the IC-complexes and hence also satisfies (3) because D ( P U ′ ) ∼ = P U ′ [2 dim X ]. Hence P U ′ ∼ = IC ( U, k ) and so U ′ ⊂ U by Propositions 8.3and 8.6. On the other hand, IC ( U, k ) ∼ = k U is certainly indecomposableand ∗ -even. It is even !-even because D IC ( U, k ) ∼ = IC ( U, k )[2 dim X ].Hence P U ∼ = k U by the classification of parity sheaves, together withthe fact that the restriction of an indecomposable parity sheaf to anopen union of strata is either zero or indecomposable (see [JMW09a,Proposition 2.11]).Now, by Theorem 6.10, W ( P ) ∼ = B ( λ ) and hence P X µ ∼ = k X λ if andonly if B ( λ ) µ ∼ = S k . The proposition then follows by definition of the k -smooth locus of the moment graph of X . (cid:3) Combining this result with Theorem 6.9 yields:
Theorem 8.8. A T -fixed point x µ ∈ X µ belongs to the p -smooth locusof X if and only if for all λ ≥ µ the number of one-dimensional T -orbits having x λ in their closure is equal to the complex dimension of X . Moreover, X µ belongs to the k -smooth locus if and only if its T -fixedpoint x µ does. A freeness result.
In this subsection k denotes a complete localring and p denotes the characteristic of the residue field of k . Let X = F λ ∈ Λ X λ be an irreducible, complex, stratified T -variety. Assumethat these data satisfy the assumptions (A1)–(A4a/b), (S1), (S2). Wefurther assume that there exists an indecomposable parity sheaf P corresponding to the unique open stratum X µ ⊂ X . (For example, X could be open in a stratified variety satisfying (R1), (R2) and (R3).)For any λ ∈ Λ let X >λ = G γ>λ X γ and X ≥ λ = G γ ≥ λ X γ For any λ ∈ Λ we can find a T -stable affine neighbourhood U of x λ and a T -invariant affine normal slice N ⊂ U to the stratum X λ . Theaim of this section is to show the following result: Proposition 8.9. H • T ( P N \{ x λ } ) is torsion free over k . Of course this result has no content if k is a field. However it seems tobe quite useful if k is, for example, the p -adic integers. Before turningto the proof of this result we state a corollary, which is of centralimportance to [JW]: Corollary 8.10. If X >λ is p -smooth then H • T ( N \ { x λ } , k ) is a free k -module.Proof. If X >λ is p -smooth then the constant sheaf with coefficients in k is self-dual and hence parity. Hence the restriction of P to X >λ isisomorphic to the constant sheaf (cf. [JMW09a, Proposition 2.11]).The result then follows from Proposition 8.9. (cid:3) Proof of Proposition 8.9.
Consider the Cartesian diagram: N / / U { x λ } i O O / / X λ ˜ i O O Without loss of generality we may assume that X λ is a closed stratum in X . In this case we have seen in the course of the proof of Theorem 5.11 ARITY SHEAVES AND MOMENT GRAPHS 43 that we have an isomorphism H • T (˜ i ! P ) ∼ = W ( P ) λ . Moreover, because N ֒ → U is a normally non-singular inclusion we have H • T ( i ! P N ) = H • T (˜ i ! P ) ∼ = W ( P ) λ . One the other hand, by the attractive proposition, we have H • T ( P N ) = H • T ( P x λ ) = W ( P ) λ .Now consider the open-closed decomposition: { x λ } i ֒ → N j ← ֓ N \ { x λ } . This leads to a distinguished triangle i ! i ! P → P U → j ∗ j ∗ P [1] → . Taking hypercohomology and using the above observations we concludethat we have an exact sequence0 → W ( P ) λ → W ( P ) λ → H • T ( P N \{ x λ } ) → H • T ( P N \{ x λ } ) ֒ → M E : λ → γ W ( P ) E Now each W ( P ) E is isomorphic to a direct sum of shifts of S/ ( α E ).By assumption (A4b) no character α E is p -divisible in X ( T ) and henceeach S/ ( α E ) is torsion free over k . It follows that H • T ( P N \{ x λ } ) is torsionfree over k . (cid:3) Representations of reductive algebraic groups
Let G be a simple reductive algebraic group over F p and let Rep G denote the category of rational representations of G . It is a fundamentalproblem in representation theory to determine the characters of thesimple and tilting modules in Rep G . For simple modules there existsa conjecture, due to Lusztig, in the case that the characteristic p islarger than the Coxeter number h associated to G . For tilting modulesthere is no general conjecture. Schur-Weyl duality can be used to showthat knowledge of the characters of tilting modules for G = GL n ( F p )implies dimension formula for the simple modules for S m for m ≤ n incharacteristic p .We want to explain how the above results allow one to reinterpretthese two basic problems using the geometry of certain Schubert vari-eties in the (complex) affine Grassmannian associated with the Lang-lands dual group. To this end let T ⊂ B ⊂ G denote a maximal torus and Borel subgroup of G respectively. Let X ∗ ( T ) denote the charac-ter lattice and X + ( T ) denote the subset of dominant weights. Then X + ( T ) parametrises both the simple and tilting modules in Rep G .9.1. Tilting modules.
Let G ∨ C denote the complex Langlands dualgroup of G , G ∨ C (( t )) its loop group, c T ∨ C = T ∨ C × C × the extended torusand G r ∨ := G ∨ C (( t )) /G ∨ C [[ t ]] the corresponding affine Grassmannian.Then X + ( T ) also parametrises the G ∨ C [[ t ]]-orbits on G r ∨ and G r ∨ sat-isfies our assumptions when viewed with the action of c T ∨ C (indeed, theclosures of G ∨ C [[ t ]]-orbits are examples of Kac-Moody Schubert vari-eties). Recall that the geometric Satake equivalence [MV07] establishesa tensor equivalence between the abelian category of rational represen-tations of G and the tensor category of G ∨ C [[ t ]]-equivariant perversesheaves on G r ∨ .Recall the following two results which are Theorem 5.1 and Corollary5.8 of [JMW09b]:(1) If p > h + 1, then parity sheaves correspond under the geomet-ric Satake isomorphism to tilting modules. More precisely, theindecomposable parity sheaf P ( λ ) corresponds to the indecom-posable tilting module T ( λ ).(2) The rank of H • T ( P ( λ ) µ ) is equal to the dimension of the µ -weightspace of the tilting module T ( λ ).With the above results in mind, it seems natural to expect that theBraden-MacPherson algorithm can be used to calculate the charactersof tilting modules. There is a problem, however: the moment graph ofthe affine Grassmannian satisfies the GKM-condition if and only if k is of characteristic 0.To get around this problem we take k to be the ring Z p of p -adicnumbers. For this the GKM-condition is satisfied. Moreover it is shown[JMW09a, Proposition 2.41] that the graded ranks of the stalks ofparity sheaves depend only on the characterstic of the residue field.The following theorem then follows from the above discussion and ourmain theorem: Theorem 9.1.
Suppose that p > h + 1 (see [JMW09b] for betterbounds). When conducted with coefficients in the ring k = Z p of p -adicnumbers, the Braden-MacPhersons algorithm computes the charactersof tilting modules. More precisely, for any character µ ∈ X ∗ ( T ) , thedimension of the µ -weight space of T ( λ ) is equal to the rank of B ( λ ) µ . Simple rational characters.
We now turn to the applicationof the above results to Lusztig’s conjecture. Let I ∨ ⊂ G ∨ C (( t )) be the ARITY SHEAVES AND MOMENT GRAPHS 45
Iwahori subgroup containing B ∨ and let F l ∨ := G ∨ C (( t )) /I ∨ denote theaffine flag variety with its c T ∨ C -action.In [Fie07b] a certain subcategory I ⊂ D b b T ( F l ∨ , k ) of special equi-variant sheaves was considered and a functor Φ : I → R was defined,where R is a category of projective objects in a category C naturallyassociated to the Lie algebra of G . For the application to Lusztig’sconjecture one needs to consider only objects in I that are supportedon a certain Schubert variety X res ⊂ F l ∨ .An intermediate step in the construction of Φ was a functor from I to the category of Braden–MacPherson sheaves on the moment graphassociated to F l ∨ . It turns out that I is the category of parity sheaveson F l ∨ (with respect to the stratification by Schubert cells). Indeed,the category I is generated from the skyscraper sheaf on the pointstratum on F l ∨ by repeatedly applying the functors π ∗ s π s ∗ for simplereflections s , where π s : F l ∨ → F l ∨ s is the projection onto the partialaffine flag variety associated to s . Now parity sheaves are preservedby these functors (cf. [JMW09a, Proposition 4.9]). From the resultsin [Fie07b] we can hence deduce that the ranks of the stalks of paritysheaves determine baby Verma multiplicities for projective objects in C .These multiplicities in turn determine the characters of simple rationalrepresentations of G . Using the results of this paper we can rephrasethe above as follows. Given λ ∈ Λ let IC ( X λ , Z ) denote the intersectioncohomology complex of X λ with integral coefficients (cf. [Jut09]). Theorem 9.2.
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