Perverse coherent sheaves and the geometry of special pieces in the unipotent variety
aa r X i v : . [ m a t h . R T ] O c t PERVERSE COHERENT SHEAVES AND THE GEOMETRY OFSPECIAL PIECES IN THE UNIPOTENT VARIETY
PRAMOD N. ACHAR AND DANIEL S. SAGE
Abstract.
Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U ⊂ X be an open set whose com-plement has codimension at least 2. We extend the Deligne-Bezrukavnikovtheory of perverse coherent sheaves by showing that a coherent intermediateextension (or intersection cohomology) functor from perverse sheaves on U toperverse sheaves on X may be defined for a much broader class of perversitiesthan has previously been known. We also introduce a derived category versionof the coherent intermediate extension functor.Under suitable hypotheses, we introduce a construction (called “ S -exten-sion”) in terms of perverse coherent sheaves of algebras on X that takes a finitemorphism to U and extends it in a canonical way to a finite morphism to X .In particular, this construction gives a canonical “ S -ification” of appropriate X . The construction also has applications to the “Macaulayfication” problem,and it is particularly well-behaved when X is Gorenstein.Our main goal, however, is to address a conjecture of Lusztig on the ge-ometry of special pieces (certain subvarieties of the unipotent variety of areductive algebraic group). The conjecture asserts in part that each specialpiece is the quotient of some variety (previously unknown in the exceptionalgroups and in positive characteristic) by the action of a certain finite group.We use S -extension to give a uniform construction of the desired variety. Introduction
Let X be a scheme of finite type over a Noetherian base scheme S that admitsa dualizing complex, and let U ⊂ X be an open set whose complement has codi-mension at least 2. Let ˜ U be another scheme, equipped with a finite morphism ρ : ˜ U → U . Consider the problem of completing the following diagram in acanonical way: ˜ U (cid:31) (cid:127) / / ρ (cid:15) (cid:15) ˜ X ρ (cid:15) (cid:15) U (cid:31) (cid:127) / / X In other words: “Construct a canonical new scheme ˜ X that contains ˜ U as an opensubscheme, together with a finite morphism ρ : ˜ X → X that extends ρ .” Onemay want to impose additional conditions, such as requiring ˜ X to obey a regularitycondition or requiring the fibers of ρ to have a specified form. Moreover, the pair Mathematics Subject Classification.
Key words and phrases. perverse coherent sheaves, special pieces in the unipotent variety,Macaulayfication.The research of the first author was partially supported by NSF grant DMS-0500873.The research of the second author was partially supported by NSF grant DMS-0606300. ( ˜
X, ρ ) should satisfy an appropriate universal property. If a group G acts on X with U a G -subscheme and ρ G -equivariant, one would like the entire constructeddiagram to be equivariant. The present paper is motivated by a specific instanceof this problem, arising in a conjecture of Lusztig on the geometry of special pieces(see below for the definition) in reductive algebraic groups.In this paper, we give a general construction (called “ S -extension”) of such ascheme ˜ X and morphism ρ : ˜ X → X , using Deligne’s theory of perverse coherentsheaves on X (following Bezrukavnikov’s exposition [7]), assuming that the cat-egory of coherent sheaves on X has enough locally free objects. (This includes,for example, quasiprojective schemes over S .) This theory parallels the theory ofconstructible perverse sheaves with the major exception that the intermediate ex-tension (or intersection cohomology) functor is not always defined. Indeed, in [7],this functor is only defined in an equivariant setting with strong restrictions on thegroup action. In this paper, we first show that the intermediate extension functormay be defined for a much broader class of perversities. In particular, we studytwo dual perversities, called the “ S ” and “Cohen-Macaulay” perversities.Next, we construct ˜ X as the global Spec of a certain intersection cohomologysheaf with respect to the S perversity. It will be defined whenever ρ ∗ O ˜ U satisfiescertain homological conditions that are weaker than satisfying Serre’s condition S . The scheme ˜ X is locally S outside of ˜ U ; moreover, ρ satisfies a universalproperty related to this condition, and in that sense ˜ X and ρ are canonical. In theparticular case of ˜ U = U and ρ the identity, we obtain a canonical “ S -ification”of U . This construction also has applications to the “Macaulayfication” problem.Indeed, we give necessary and sufficient conditions for X to have a universal finiteMacaulayfication (i.e., universal among appropriate finite morphisms from Cohen-Macaulay schemes).Third, we introduce a derived category version of the coherent intermediateextension functor (from a suitable subcategory of the derived category of coherentsheaves on U to the derived category of coherent sheaves on X ), and we showthat this functor induces an equivalence of categories with its essential image. Onecorollary of this theorem is that when X is Gorenstein, the coherent intermediateextension functor restricted to Cohen-Macaulay sheaves on U is independent ofperversity. Using this, we show that with suitable assumptions on ˜ U and X , thescheme ˜ X produced by S -extension is in fact Cohen–Macaulay or Gorenstein.Our main goal, however, is to apply these results to the aforementioned conjec-ture of Lusztig, which we now recall. Let G be a reductive algebraic group overthe algebraically closed field k , and assume that the characteristic of k is good for G . Let C be a special unipotent class of G in the sense of [21]. The special piece containing C is defined by P = [ C where C ranges over unipotent classes such that C ⊂ C but C C ′ for any special C ′ ⊂ C with C ′ = C .Each special piece is a locally closed subvariety of G , and according to a resultof Spaltenstein [34], every unipotent class in G is contained in exactly one specialpiece.In 1981, Lusztig conjectured that every special piece is rationally smooth [22].This conjecture can be verified in any particular group by explicit calculation ofGreen functions, and indeed, the conjecture was quickly verified for all the ex-ceptional groups following work of Shoji [31] and Benyon–Spaltenstein [5]. In the ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 3 classical groups, however, new techniques were required. In 1989, Kraft and Pro-cesi, relying on their own prior work on singularities of closures of unipotent classes,proved a stronger statement: they showed that every special piece in the classicalgroups is a quotient of a certain smooth variety by a certain finite group F [20]. Inparticular, this implies that special pieces are rationally smooth.A natural question, then, is whether this stronger statement holds in general.The work of Kraft–Procesi makes extensive use of the combinatorics available in theclassical groups, so it is not at all obvious how to generalize their construction to allgroups. However, in 1997, Lusztig succeeded in characterizing the finite group F ina type-independent manner [24, Theorem 0.4]; he identified F as a certain subgroupof ¯ A ( C ). (For any unipotent class C , ¯ A ( C ) denotes Lusztig’s “canonical quotient”of the component group G x / ( G x ) ◦ of the G -stabilizer of a point x ∈ C .) In fact, F is naturally a direct factor of ¯ A ( C ) (see [3, § A ( C ) thestructure of a Coxeter group (see [2]). There is a one-to-one correspondence betweenparabolic subgroups of F and unipotent classes in P . Given a parabolic subgroup H ⊂ F , we denote the corresponding class C H . (The trivial subgroup correspondsto the special class, so this notation is consistent with the earlier notation C .) Conjecture 1.1 (Lusztig) . There is a smooth variety ˜ P with an action of F suchthat P ≃ ˜ P /F and such that C H is precisely the image of those points in ˜ P whose F -stabilizer is conjugate to H . Note that it suffices to examine special pieces for the simple root systems becausethe unipotent variety of a reductive group is the product of the unipotent varietiesof its simple factors.Before addressing this conjecture further, we remark that it is quite easy toproduce candidate varieties that ought to be the preimages of the various C H ’s in ˜ P by using the results of [3] (a paper to which the present paper might be regarded as asort of sequel). Fix x ∈ C H . By [3, Theorem 2.1], we have ¯ A ( C H ) ≃ N ¯ A ( C ) ( H ) /H (where N J ( K ) denotes the normalizer of K in J ) and hence a natural map G x → N ¯ A ( C ) ( H ) /H . Let G xF be the kernel of the composed map G x → N ¯ A ( C ) ( H ) /H → N F ( H ) /H , and let ( ˜ C H ) ◦ = G/G xF . Clearly, this is a connected variety with a freeaction of N F ( H ) /H , and the quotient by that action is C H . Finally, let(1) ˜ C H = ( ˜ C H ) ◦ × N F ( H ) F. An element a ∈ F acts on this variety by a · ( y, f ) = ( y, f a − ). The F -stabilizer ofany point is conjugate to H , and the natural surjective map ρ H : ˜ C H → C H is thequotient of ˜ C H by the action of F .Before stating our main result, we observe that, since quotients of smooth vari-eties are normal, inherent in Lusztig’s conjecture is the subconjecture: Conjecture 1.2.
Every special piece P is normal. In characteristic zero, we show how this conjecture can be obtained from knownresults on unipotent conjugacy classes in the classical types, G , F , and E . For E and E , there is a conjectural list of all non-normal unipotent conjugacy classclosures due to Broer, Panyushev, and Sommers [11]. Assuming this is true, thenthere would remain 5 special pieces (1 in E and 4 in E ) for which normality isnot known. In positive characteristic, much less is known.In this paper, we will actually construct a variety ˜ P whose algebraic quotientby F is the normalization ¯ P of P . However, we will also show that special pieces PRAMOD N. ACHAR AND DANIEL S. SAGE are unibranch, i.e., the normalization map ν : ¯ P → P is a bijection and in fact ahomeomorphism. This means that P is the topological quotient of ˜ P . In particular,setting ¯ C H = ν − ( C H ), we see that ¯ C H ≃ C H and that ¯ P is again stratified by theunipotent orbits corresponding to parabolic subgroups of F .The main result of the paper is the following. Theorem 1.3. (1)
There is a canonical normal irreducible G -scheme ˜ P to-gether with a finite equivariant morphism ρ : ˜ P → P which extends ρ :˜ C → C ; the pair ( ˜ P , ρ ) is universal with respect to finite morphisms f : Y → P that are S relative to C and whose restriction f | f − ( C ) fac-tors through ρ . (2) The variety ˜ P is rationally smooth. Moreover, if char k = 0 , then ˜ P isGorenstein. (3) The variety ˜ P is endowed with a natural F -action commuting with the G -action. The map ρ is the topological quotient by this action while ¯ ρ : ˜ P → ¯ P is the algebraic quotient. (4) For each class C H ⊂ P , the preimage ρ − ( C H ) = ¯ ρ − ( ¯ C H ) is isomorphic to ˜ C H and contains exactly those closed points whose F -stabilizer is conjugateto H . The first part of this theorem is simply an invocation of the S -extension con-struction. The proof of the Gorenstein property is established by using a theoremof Hinich and Panyushev [19, 26] and the aforementioned results on the derivedintermediate extension functor. We remark that the formalism of the S -extensionconstruction does not yield a concrete description of the resulting scheme in general,but in our setting, the results of [3] (as noted above) allow us to find an explicitstratification (1) for ˜ P .Although we do not prove that ˜ P is smooth, we show that if ˆ P is a smoothvariety containing a dense open set isomorphic to ˜ C and ˆ ρ : ˆ P → P is a finitemorphism extending ρ , then ˆ P is isomorphic to ˜ P . Thus, if Lusztig’s conjectureis true, then our ˜ P is the desired smooth variety. In particular, for the classicalgroups, the ˜ P constructed here coincides with the Kraft–Procesi variety of [20].2. Perverse Coherent Sheaves
The theory of perverse coherent sheaves, following Deligne and Bezrukavnikov [7],closely parallels the much better-known theory of constructible perverse sheaves,but one striking difference is that in the coherent setting, the intermediate exten-sion functor does not always exist. Indeed, in loc.cit., it was only constructed in anequivariant setting with strong assumptions on the group action.In this section, we review the Deligne–Bezrukavnikov theory, and we prove ageneralization of [7, Theorem 2] that allows us to use the intermediate extensionfunctor in a much broader class of examples, including many nonequivariant cases.We begin with the same setting and assumptions as [7]. Let X be a schemeof finite type over a Noetherian base scheme S admitting a dualizing complex,and let G be an affine group scheme acting on X that is flat, of finite type, andGorenstein over S . (For example, the base scheme could be S = Spec k with k afield.) By [17, Corollary V.7.2], a scheme X satisfying these assumptions necessarilyhas finite Krull dimension. Let Coh ( X ) be the category of G -equivariant coherentsheaves on X , and let D ( X ) be the bounded derived category of Coh ( X ). We ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 5 further assume that
Coh ( X ) has enough locally free objects. Let X G -gen be thetopological space consisting of generic points of G -invariant subschemes of X , withthe subspace topology induced by the underlying topological space of X . We adoptthe convention that for any (not necessarily irreducible) G -invariant locally closedsubscheme Y ⊂ X , the codimension of Y is given bycodim Y = min y ∈ Y G -gen dim O y,X . For any point x ∈ X , let i x : { x } → X denote the inclusion map. (This is merely atopological map, not a morphism of schemes.) For brevity, we will write ¯ x for theclosed subspace { x } of X . By [7, Proposition 1], X admits an equivariant dualizingcomplex ω X . By shifting if necessary, we may assume, as in [7, § x ∈ X , i ! x ω X is concentrated in degree codim ¯ x .Although we will always work in this equivariant setting, one can of course obtainnonequivariant versions of our results simply by taking G = 1. Occasionally, wewill explicitly pass from an equivariant category to a nonequivariant one, and makeuse of the fact that all the usual functors on sheaves commute with this forgetfulfunctor. Notational Convention.
Throughout this paper, unless otherwise specified, allgeometric objects will belong to the appropriate category for the equivariant settingwithout further mention. Thus, schemes will be G -schemes, morphisms will be G -morphisms, and sheaves will be G -equivariant. Definition 2.1. A perversity is a function p : X G -gen → Z satisfying(2) p ( y ) ≥ p ( x ) andcodim ¯ y − p ( y ) ≥ codim ¯ x − p ( x ) whenever codim ¯ y ≥ codim ¯ x .(In particular, p ( x ) depends only on codim ¯ x .) For any perversity p , the function¯ p : X G -gen → Z defined by ¯ p ( x ) = codim ¯ x − p ( x ) is also a perversity, called the dual perversity to p .A slightly more general theory could be obtained by imposing the inequalities (2)only when y ∈ ¯ x , as is done in [7] (see also Remark 2.8). For the purposes of thispaper, however, there would be no practical benefit to defining perversities in thisway, and various technical details would become rather more complicated, so wewill confine ourselves to perversities as defined above.Given a perversity p , we define two full subcategories of D ( X ) as follows: p D ( X ) ≤ = { F ∈ D ( X ) | for all x ∈ X G -gen , H k ( i ∗ x F ) = 0 for all k > p ( x ) } p D ( X ) ≥ = { F ∈ D ( X ) | for all x ∈ X G -gen , H k ( i ! x F ) = 0 for all k < p ( x ) } By [7, Theorem 1], ( p D ( X ) ≤ , p D ( X ) ≥ ) is a t -structure on D ( X ). Definition 2.2.
The above t -structure is called the perverse t -structure (with re-spect to the perversity p ) on D ( X ). Its heart, denoted M p ( X ) or simply M ( X ), isthe category of ( G -equivariant) perverse coherent sheaves on X with respect to p .The truncation functors for this t -structure will be denoted τ p ≤ : D ( X ) → p D ( X ) ≤ and τ p ≥ : D ( X ) → p D ( X ) ≥ .We denote the standard t -structure on D ( X ) by ( std D ( X ) ≤ , std D ( X ) ≥ ), andthe associated truncation functors by τ std ≤ and τ std ≥ . The perverse t -structure asso-ciated to the constant perversity p = 0 coincides with the standard t -structure. PRAMOD N. ACHAR AND DANIEL S. SAGE
Now, let U be a locally closed G -invariant subscheme of X , and let Z = U r U .Let U G -gen and Z G -gen be the corresponding subspaces of X G -gen . Given a perversecoherent sheaf on U , we wish to find a canonical way to associate to it a perversecoherent sheaf on U , analogous to the intermediate extension operation on ordinary(constructible) perverse sheaves. This is not always possible, but [7, Theorem 2]gives one set of conditions under which it can be done. In fact, the conditions ofthat theorem can be weakened significantly, at the expense of having intermediateextension defined only on some subcategory of M ( U ) (see Remark 2.7).The following proposition provides a general framework for defining intermediateextension on a subcategory of M ( U ). Later, we will determine the largest possiblesubcategory to which the proposition can be applied.Define a partial order on perversities by pointwise comparison: we say that p ≤ q if p ( x ) ≤ q ( x ) for all x ∈ X G -gen . Proposition 2.3.
Suppose q , p , and r are perversities with the following properties: q ≤ p ≤ r , r ( x ) − q ( x ) ≤ for all x , and q ( x ) = p ( x ) − and r ( x ) = p ( x ) + 1 for all x ∈ Z G - gen .Define two full subcategories by M q,r ( U ) = q D ( U ) ≤ ∩ r D ( U ) ≥ ⊂ M p ( U ) , M q,r ( U ) = q D ( U ) ≤ ∩ r D ( U ) ≥ ⊂ M p ( U ) , and let j : U ֒ → U be the inclusion map. Then j ∗ : M q,r ( U ) → M q,r ( U ) is anequivalence of categories. Definition 2.4.
The inverse equivalence to that of Proposition 2.3, which is de-noted IC p ( U , · ) : M q,r ( U ) → M q,r ( U ), or simply IC ( U , · ) : M q,r ( U ) → M q,r ( U ), iscalled the intermediate extension functor . Proof.
Our proof is essentially identical to that of [7, Theorem 2]. Let J ! ∗ : D ( U ) → D ( U ) be the functor τ q ≤ ◦ τ r ≥ . We claim that J ! ∗ actually takes values in M q,r ( U ).Given F ∈ D ( U ), let F = τ r ≥ F . Then we have a distinguished triangle( τ q ≥ F )[ − → J ! ∗ ( F ) → F → τ q ≥ F . Note that ( τ q ≥ F )[ − ∈ q D ( U ) ≥ . Now, the condition r ( x ) − q ( x ) ≤ q D ( U ) ≥ ⊂ r D ( U ) ≥ . Clearly, F ∈ r D ( U ) ≥ , so it follows that J ! ∗ F ∈ r D ( U ) ≥ .Since it obviously takes values in q D ( U ) ≤ , J ! ∗ F ∈ M q,r ( U ).Next, note that if F ∈ D ( U ) is such that F | U ∈ M q,r ( U ), then both ( τ r ≥ F ) | U and ( τ q ≤ F ) | U , and hence ( J ! ∗ F ) | U , are isomorphic to F | U . In particular, we cansee now that j ∗ is essentially surjective. Given F ∈ M q,r ( U ), let ˜ F be any objecton D ( U ) such that j ∗ ˜ F ≃ F . (Such an object exists by [7, Corollary 2].) Then F ′ = J ! ∗ ˜ F is an object of M q,r ( U ) such that j ∗ F ′ ≃ F .Now, if φ : F → G is a morphism in M q,r ( U ), then by [7, Corollary 2], we canfind objects F ′ and G ′ in D ( U ) and a morphism φ ′ : F ′ → G ′ such that j ∗ F ′ ≃ F , j ∗ G ′ ≃ G , and j ∗ φ ′ ≃ φ . By applying J ! ∗ , we may assume that F ′ , G ′ , and φ ′ actually belong to M q,r ( U ). This shows that j ∗ is full.To show that j ∗ is faithful, it suffices to show that if φ is an isomorphism, then φ ′ must be as well. Since φ ′ | U is an isomorphism, the kernel and cokernel of φ ′ must be supported on Z . But by [7, Lemma 6], the fact that q ( x ) < p ( x ) < r ( x ) ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 7 for x ∈ Z G -gen implies that F ′ and G ′ have no subobjects or quotients supported on Z . Thus, φ ′ is an isomorphism. Since j ∗ is fully faithful and essentially surjective,it is an equivalence of categories. (cid:3) Remark . It follows from the above proof that for any F ∈ M q,r ( U ), IC ( U , F )is isomorphic to τ q ≤ τ r ≥ ˜ F , where ˜ F is any object of D ( U ) whose restriction to U isisomorphic to F .The above proof could also have been carried out using the functor J ′ ! ∗ = τ r ≥ ◦ τ q ≤ instead of J ! ∗ . From that version of the proof, one sees that IC ( U , F ) is alsoisomorphic to τ r ≥ τ q ≤ ˜ F . Proposition 2.6.
Let p be a perversity, and let z be a generic point of an irre-ducible component of Z of minimal codimension. Among all perversities q which,together with some r , satisfy the assumptions of Proposition 2.3, there is a uniquemaximal one, denoted p − . It is given by p − ( x ) = ( p ( x ) − if p ( x ) ≥ p ( z ) , p ( x ) if p ( x ) < p ( z ) . (3) Similarly, there is a unique minimal perversity among all r of that proposition,denoted p + , and given by p + ( x ) = ( p ( x ) + 1 if codim ¯ x − p ( x ) ≥ codim ¯ z − p ( z ) , p ( x ) if codim ¯ x − p ( x ) < codim ¯ z − p ( z ) . (4) Remark . Although our formulas for p − and p + appear to be different fromthose of [7, Theorem 2], they do in fact coincide under the assumptions of loc. cit.Those assumptions are that U is open and dense in X and that for any x ∈ U G -gen and any z ∈ ¯ x ∩ Z G -gen , we have p ( z ) > p ( x ) and codim ¯ z − p ( z ) > codim ¯ x − p ( x ) . These inequalities cannot hold simultaneously unless codim ¯ x ≤ codim ¯ z −
2. Inparticular, this means U G -gen cannot contain any closed points of U , so one mustnecessarily be in an equivariant setting.Proposition 2.6, on the other hand, applies with no a priori restrictions on X or U . This really does allow us to use the intermediate extension functorin nonequivariant settings, but in practice, it is still necessary to require thatcodim U ≤ codim Z −
2; indeed, if this condition fails, then M p − ,p + ( U ) will bereduced to the zero object. To see this, note that p − ( x ) = p + ( x ) implies thatcodim ¯ x ≤ codim ¯ z −
2, so if codim
U > codim Z −
2, then we have p − ( x ) < p + ( x )for all points x ∈ U . It follows that p − D ( U ) ≤ ⊂ p + D ( U ) ≤− , so any object in M p − ,p + ( U ) will belong to p + D ( U ) ≤− ∩ p + D ( U ) ≥ . The latter category containsonly the zero object. Proof.
Let us first show that p − is a perversity. Suppose codim ¯ x ≥ codim ¯ y , so p ( x ) ≥ p ( y ). If p ( x ) ≥ p ( y ) ≥ p ( z ) or p ( z ) > p ( x ) ≥ p ( y ), then the conditions (2)obviously hold because they hold for p . Now suppose p ( x ) ≥ p ( z ) > p ( y ). Thestrictness of the second inequality implies that codim ¯ x > codim ¯ y . In this situation,we clearly have p − ( x ) = p ( x ) − ≥ p ( y ) = p − ( y ) andcodim ¯ x − p − ( x ) = codim ¯ x − p ( x ) + 1 > codim ¯ y − p ( y ) = − codim ¯ y − p − ( y ) . PRAMOD N. ACHAR AND DANIEL S. SAGE
Thus, p − is a perversity.Let q and r be perversities satisfying the assumptions of Proposition 2.3. Therequirement that q ( x ) = p ( x ) − x ∈ Z G -gen implies that q ( x ) = p ( x ) − p − ( x ) for all x with codim ¯ x ≥ codim ¯ z . For all such points, of course,we have p ( x ) ≥ p ( z ). Now suppose x is such that codim ¯ x < codim ¯ z , so that p ( z ) ≥ p ( x ). If p ( z ) > p ( x ), it is trivial that q ( x ) ≤ p − ( x ), while if p ( z ) = p ( x ),then q ( x ) ≤ q ( z ) = p ( z ) − p ( x ) − p − ( x ). Thus, q ( x ) ≤ p − ( x ) for all x ∈ X G -gen , so q ≤ p − , and p − has the desired maximality property.The proofs of the corresponding statements for p + are similar. (cid:3) Remark . If we were to change the definition of “perversity” by imposing theinequalities (2) only when y ∈ ¯ x , then this result could be improved, i.e., p − couldbe replaced be a larger perversity and p + by a smaller one, resulting in a largerdomain category for IC ( U , · ). Let us call a sequence of points x , y , x , . . . , y k , x k +1 in X G -gen a lower chain (resp. upper chain ) if the following conditions hold:(1) x i , x i +1 ∈ ¯ y i for all i , and x k +1 ∈ ¯ y k ∩ Z G -gen , and(2) p ( x i +1 ) = p ( y i ) and p ( x i ) = p ( y i ) − codim ¯ y i + codim ¯ x i (resp. p ( x i +1 ) = p ( y i ) − codim ¯ y i + codim ¯ x i +1 and p ( x i ) = p ( y i )) for all i .Let S (resp. T ) be the set of all points of X G -gen occurring in some lower (resp. up-per) chain, and define p ⊖ ( x ) = ( p ( x ) − x ∈ S , p ( x ) otherwise, and p ⊕ ( x ) = ( p ( x ) + 1 if x ∈ T , p ( x ) otherwise.It is not difficult to prove an analogue of Proposition 2.6 using these formulasinstead of p − and p + . 3. Notation and Preliminaries
In this section, we introduce some useful notation and terminology, and we provea number of lemmas on perverse coherent sheaves. To simplify the discussion, wehenceforth assume that U is actually an open dense subscheme of X and that Z has codimension at least 2. Let j : U ֒ → X be the inclusion map. For the mostpart, we will consider only “standard” perversities, defined as follows. Definition 3.1.
A perversity p is said to be standard if(5) p ( x ) = p − ( x ) = p + ( x ) = 0 if codim ¯ x = 0.Note that if p is standard, so is its dual ¯ p . Remark . The assumption that codim Z ≥ p , when there is no risk of ambiguity, we write D ( X ) − , ≤ = p − D ( X ) ≤ and D ( X ) + , ≥ = p + D ( X ) ≥ , or even simply D − , ≤ and D + , ≥ . Next, let M p, ± ( U ) = D ( U ) − , ≤ ∩ D ( U ) + , ≥ and M p, ± ( X ) = D ( X ) − , ≤ ∩ D ( X ) + , ≥ . Then we have an intermediate extension functor IC ( X, · ) : M p, ± ( U ) → M p, ± ( X ).These categories will usually be denoted simply M ± ( U ) and M ± ( X ), respectively.Let D denote the coherent (Serre-Grothendieck) duality functor R H om ( · , ω X ).By [7, Lemma 5], D takes M p ( X ) to M ¯ p ( X ) and M p, ± ( U ) to M ¯ p, ± ( U ). ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 9
Two specific standard perversities will be particularly useful in the sequel: s ( x ) = ( c ( x ) = ( codim ¯ x if codim ¯ x < codim Z ,codim ¯ x − x ≥ codim Z .We call s the “ S perversity” and c the “Cohen–Macaulay perversity” for reasonsthat are made clear in Lemma 3.9. These two perversities are dual to one an-other, and they are extremal among all standard perversities (see Lemma 3.3). Forconvenience, we also record the corresponding “ − ” and “+” perversities: s − ( x ) = 0 c + ( x ) = codim ¯ xs + ( x ) = c − ( x ) = codim ¯ x if codim ¯ x < codim Z + 1,codim ¯ x − x = codim Z + 1,codim ¯ x − x ≥ codim Z .It is clear that s − and c + are the smallest and largest possible perversities,respectively, that take the value 0 on generic points of X . Since the “ − ” and “+”operations respect the partial order on perversities, we have the following result. Lemma 3.3.
Every standard perversity p satisfies s ≤ p ≤ c . (cid:3) We will use the following observation repeatedly.
Lemma 3.4.
Let F be a coherent sheaf on U . The complex IC ( X, F ) is definedif and only if F ∈ D ( U ) + , ≥ , or equivalently, if depth O x F x ≥ p + ( x ) for all x ∈ U G - gen .If IC ( X, F ) is defined, then given a coherent extension G of F to X , we have G ≃ IC ( X, F ) if and only if G ∈ D ( X ) + , ≥ , or equivalently, if depth O x G x > p ( x ) for all x ∈ Z G - gen .Proof. The exactness of i ∗ x implies that H k ( i ∗ x F ) = 0 unless k = 0, and since we areassuming that p is a standard perversity, this cohomology vanishes for k > p − ( x ) ≥
0. Thus, F ∈ D ( U ) − , ≤ automatically. The depth-condition characterization of D ( U ) + , ≥ comes from the well-known fact that the lowest degree in which H k ( i ! x F )is nonzero is depth O x F x . The same arguments apply to G as well. (cid:3) Next, recall (see [14, § I.3.3]) that to any quasicoherent sheaf of algebras F on X ,one can canonically associate a new scheme Y and an affine morphism f : Y → X such that f ∗ O Y ≃ F . Moreover, f is finite if and only if F is coherent. Thisprocedure is often called “global Spec”; we will use the notation Y = Spec F .Coherent sheaves of algebras and the global Spec operation play a major role inthe sequel. The following proposition relates these to the IC functor. Proposition 3.5.
Let F be a coherent sheaf of algebras on X . Form Y = Spec F ,and let f : Y → X be the canonical map. Then IC ( Y, O f − ( U ) ) is defined if andonly if IC ( X, F | U ) is. If both are defined, then IC ( Y, O f − ( U ) ) ≃ O Y if and only if IC ( X, F | U ) ≃ F .Remark . Here, the notation IC ( Y, O f − ( U ) ) is to be understood in terms of theintermediate extension functor associated to the open inclusion f − ( U ) ֒ → Y andthe perversity p ′ = p ◦ f : Y G -gen → Z . The fact that f is a finite morphism implies that the complement of f − ( U ) has the same codimension as Z . In ad-dition, codim f − (¯ x ) = codim ¯ x for any x ∈ X G -gen , so p ′ does indeed satisfy theinequalities (2).Moreover, since Y is finite over X , it satisfies our basic hypotheses for definingperverse coherent sheaves—it is of finite type over S , and Coh ( Y ) has enough locallyfree objects. (To see the latter, note that if G is a coherent O Y -module, then there isa locally free O X -module F which surjects to f ∗ G . This gives a surjective morphismof O Y -modules f ∗ F → f ∗ f ∗ G . Composing with the surjection f ∗ f ∗ G → G exhibits G as a quotient of the locally free O Y -module f ∗ F .) Proof.
By Lemma 3.4, it suffices to show O f − ( U ) ∈ D ( f − ( U )) + , ≥ if and only if F | U ∈ D ( U ) + , ≥ , and then that O Y ∈ D ( Y ) + , ≥ if and only if F ∈ D ( X ) + , ≥ . Weprove both assertions simultaneously.Let x ∈ X G -gen , and let Y x = f − ( x ). The latter is a finite set of points, and( f | Y x ) ∗ is clearly an exact functor that kills no nonzero sheaf. Now, we have R ( f | Y x ) ∗ i ! Y x O Y ≃ i ! x f ∗ O Y ≃ i ! x F , so the lowest degree in which H k ( i ! x F ) is nonzero is the same as the lowest degreein which H k ( i ! Y x O Y ) is nonzero. Let i y,Y x be the inclusion of a point y into Y x .Then i ! y,Y x = i ∗ y,Y x is also an exact functor; it kills no nonzero sheaf whose supportcontains y . Since i ! y = i ! y,Y x ◦ i ! Y x , we conclude that the lowest degree in which some H k ( i ! y O Y ) = H k ( i ! y,Y x i ! Y x O Y ) is nonzero is the same as the lowest degree in which H k ( i ! x F ) is nonzero. In particular, considering the degree k = p + ( x ), we see that O Y ∈ D ( Y ) + , ≥ if and only if F ∈ D ( X ) + , ≥ , and likewise for O f − ( U ) and F | U . (cid:3) Note that the proof in fact shows that if f is a finite morphism, then f ∗ is t -exact.In the remainder of the section, we prove a handful of results specific to the S and Cohen–Macaulay perversities. Proposition 3.7.
For any coherent sheaf E on U such that IC s ( X, E ) is defined,there is a canonical isomorphism IC s ( X, E ) ≃ j ∗ E . In particular, j ∗ E is coherent.Proof. We begin by observing that IC s ( X, E ) is actually a sheaf ( i.e., that it isconcentrated in degree 0). Indeed, IC s ( X, E ) is perverse with respect to s − . Thisperversity is constant with value 0, so the resulting t -structure is just the standard t -structure.Next, we show that j ∗ E is coherent. Note that the smallest value taken by s + on Z G -gen is 2. Now, E is, by assumption, a perverse coherent sheaf on U withrespect to s + . According to [7, Corollary 3], the complex τ std ≤ ( Rj ∗ E ) has coherentcohomology. But that object is simply j ∗ E , the nonderived push-forward of E .Since j ∗ E is concentrated in degree 0, it obviously lies in D − , ≤ ( X ), so by Re-mark 2.5, IC s ( X, E ) can be calculated as τ + ≥ ( j ∗ E ). Thus, the truncation functorgives us a canonical morphism j ∗ E → IC s ( X, E ). On the other hand, we havethe usual adjunction morphism IC s ( X, E ) → j ∗ j ∗ IC s ( X, E ) ≃ j ∗ E . Both thesemorphisms have the property that their restrictions to U are simply the identitymorphism of E . The compositions IC s ( X, E ) → j ∗ E → IC s ( X, E ) and j ∗ E → IC s ( X, E ) → j ∗ E are then both identity morphisms of the appropriate objects, because their restric-tions to U are the identity morphism of E , and the functors IC s ( X, · ) and j ∗ areboth fully faithful. Thus, IC s ( X, E ) ≃ j ∗ E . (cid:3) ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 11
We remark that the notation “ IC s ( X, E )” is still useful, in spite of the aboveproposition, because IC s ( X, E ) is not always defined, whereas j ∗ E is. Most of thestatements in Section 4 become false if we drop the assumption that IC ( X, E ) bedefined and replace that object by j ∗ E , which is not coherent in general. Moreover,most of the proofs rely, at least implicitly, on the fact that the S -perversity givesrise to a nontrivial t -structure on D ( X ). Definition 3.8.
A scheme X is said to be locally S at x ∈ X if depth O x ≥ min { , dim O x } . X is S if it is locally S at every point. Lemma 3.9. IC s ( X, O U ) is defined if and only if U is locally S at all points x ∈ U G - gen such that codim ¯ x ≥ codim Z . In that case, the following conditions areequivalent: (1) IC s ( X, O U ) ≃ O X . (2) X is locally S at all points of Z G - gen .Similarly, IC c ( X, O U ) is defined if and only if U is locally Cohen–Macaulay at allpoints of U G - gen . In that case, the following conditions are equivalent: (1) IC c ( X, O U ) ≃ O X . (2) X is locally Cohen–Macaulay at all points of Z G - gen .Proof. The proofs of the two parts of this lemma are essentially identical; wewill treat only the S case. By Lemma 3.4, IC s ( X, O U ) is defined if and only ifdepth O x ≥ s + ( x ) for all x ∈ U G -gen , i.e., ifdepth O x ≥ x < codim Z − x = codim Z − x ≥ codim Z .The first two cases above hold trivially. (In the case codim ¯ x = codim Z −
1, we havedim O x ≥
1, and any local ring of positive dimension has positive depth.) Since Z has codimension at least 2, the last case holds only if O x is S . Thus, IC s ( X, O U ) isdefined if and only if U is locally S at points x ∈ X G -gen with codim ¯ x ≥ codim Z .The same argument applied to O X shows the equivalence of conditions (1) and (2)above. (cid:3) A similar proof gives the following result:
Lemma 3.10. IC s ( X, F ) is defined if and only if depth F x ≥ min { , dim F x } at allpoints x ∈ U G - gen such that codim ¯ x ≥ codim Z . Finally, for the last two lemmas of this section, we assume that G is a linearalgebraic group over S = Spec k for some algebraically closed field k , and that X is a variety over k . In this case, we can extract a bit more geometric informationfrom the preceding results. Recall that in this setting, the notion of “orbit” is well-behaved: X is a union of orbits, each of which is a smooth locally closed subvariety,isomorphic to a homogeneous space for G . The following lemma deals with localcohomology on an orbit. Lemma 3.11.
Let F ∈ D ( X ) , and let C be a G -orbit in X . Suppose that there issome p ∈ Z such that for any generic point x of C , we have H k ( i ! x F ) = 0 for all k < p . Then, for any y ∈ C , we have H k ( i ! y F ) = 0 for all k < p + depth O y,C ,where O y,C is the local ring at y of the reduced induced scheme structure on C . Proof.
We begin by noting that any equivariant coherent sheaf on C is locally free.Indeed, given a coherent sheaf E on C , consider the function φ : C → Z definedby φ ( y ) = dim k ( y ) k ( y ) ⊗ O y,C E y , where k ( y ) is the residue field of the local ring O y,C . This function is constant on closed points of C (because they form a single G -orbit), and hence, by the semicontinuity theorem, on all of C . By, for instance, [18,Ex. II.5.8], since φ is constant and C is reduced, E is locally free.Now, let i y,C : { y } ֒ → C and j C : C ֒ → X be the inclusion maps. (The formeris merely a topological map; the latter is a morphism of schemes.) Recall that i ! y,C ( j ! C F ) ≃ R H om ( O y,C , i ! y F ), where j ! C is the right adjoint to Rj C ∗ in the settingof Grothendieck duality for coherent sheaves (as constructed in, say, [17]), but i ! y,C and i ! y are the Verdier-duality right adjoints to ( i y,C ) ! and ( i y ) ! , respectively.By the argument given in [7, Lemma 2(b)], the vanishing assumptions on H k ( i ! x F )for x a generic point of C imply that H k ( j ! C F ) vanishes for all k < p ; further-more, the lowest nonzero cohomologies of i ! y,C j ! C F and of i ! y F occur in the samedegree. Now, j ! C F is a bounded complex of locally free sheaves on C , so there issome open subscheme C ⊂ C containing y such that j ! C F | C is in fact a complexof free sheaves. Recall, as in the proof of Lemma 3.4, that H k ( i ! y,C O y,C ) van-ishes in degrees k < depth O y,C = depth O y,C . It follows that the cohomology of i ! y,C j ! C F = i ! y,C ( j ! C F | C ) vanishes in degrees k < p + depth O y,C . (cid:3) We conclude with the following refinement of Lemma 3.9.
Lemma 3.12.
Assume that G acts on X with finitely many orbits. If IC s ( X, O U ) is defined, then the following conditions are equivalent: (1) IC s ( X, O U ) ≃ O X . (2) X is locally S at all points of Z .Similarly, if IC c ( X, O U ) is defined, the following conditions are equivalent: (1) IC c ( X, O U ) ≃ O X . (2) X is locally Cohen–Macaulay at all points of Z .Proof. As in the proof of Lemma 3.9, we treat only the S case. Since G acts withfinitely many orbits, every closed G -invariant subvariety contains an open orbit,so every point of X G -gen is a generic point of some G -orbit. It suffices to showthat part (2) of Lemma 3.9 is equivalent to part (2) of the present lemma. Thatassertion follows from Lemma 3.11: we see that for any x ∈ Z G -gen and any y in the G -orbit C containing x , we have depth O y ≥ depth O x , since depth O x,C ≥ (cid:3) S -Extension Our goal in this section is to use coherent intermediate extension with respectto the S -perversity to construct new schemes and then to use powerful generalproperties of the intermediate extension functor to deduce various properties ofthose schemes. Throughout this section, all IC ’s will be with respect to the S -perversity unless otherwise specified.The construction involves the global Spec operation (see Proposition 3.5 andthe comments preceding it) on coherent sheaves of commutative algebras. Hence-forth, all sheaves of algebras that we consider will be assumed to be coherent andcommutative. We reemphasize the fact that we are working in the equivariant set-ting, so that schemes are G -schemes, morphisms are G -morphisms, and sheaves are G -equivariant. ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 13
Proposition 4.1.
Let E be a sheaf of O U -algebras on U . Then IC ( X, E ) can bemade into a sheaf of O X -algebras in a unique way that is compatible with the algebrastructure on E .Proof. This is immediate from Proposition 3.7 and the fact that the algebra struc-ture on E determines a unique algebra structure on j ∗ E . (cid:3) Definition 4.2.
Let U ⊂ X be an open subscheme whose complement has codi-mension at least 2. A morphism of schemes f : Y → X is said to be S relative to U if for all x ∈ X G -gen such that codim ¯ x ≥ codim Z , we have H k ( i ! x f ∗ O Y ) = 0 if k < Remark . Note that if f is finite and S relative to U , then the image under f of any generic point of an irreducible component of Y must lie in U . Indeed,if y is such a generic point and x = f ( y ), then H ( i ! y O Y ) = 0, which impliesthat H ( i ! x f ∗ O Y ) = 0 by the argument given in the proof of Proposition 3.5. Inparticular, f − ( U ) cannot be empty; in fact, it is open dense. Remark . If f is finite, the definition of “ S relative to U ” is equivalent torequiring that f ∗ O Y ≃ IC ( X, f ∗ O f − ( U ) ), and hence, according to Proposition 3.5,to requiring that IC ( Y, O f − ( U ) ) ≃ O Y . (Note that the proposition applies since f − ( U ) is open dense by the previous remark.)In particular, by Lemma 3.9, id : X → X is S relative to U if and only if IC ( X, O U ) is defined and X is locally S outside U . Moreover, if f : Y → X is afinite morphism with Y S and f − ( U ) dense, then f is S relative to U . Theorem 4.5.
Let ρ : ˜ U → U be a finite morphism such that IC ( X, ρ ∗ O ˜ U ) isdefined, and let ˜ X denote the scheme Spec IC ( X, ρ ∗ O ˜ U ) . The natural morphism ρ : ˜ X → X is universal with respect to finite morphisms f : Y → X which are S relative to U and whose restriction f | f − ( U ) factors through ρ . In other words, if f : Y → X is any finite morphism that is S relative to U and such that f | f − ( U ) factors through ρ , then f factors through ρ in a unique way.In addition, ρ is a finite morphism, ρ − ( U ) ≃ ˜ U , and ρ | ˜ U = ρ . Moreover, ˜ U isa dense open subscheme of ˜ X , and id : ˜ X → ˜ X is S relative to ˜ U . Here is a diagram: f − ( U ) (cid:31) (cid:127) / / g & & NNNNNNN f | f − U ) (cid:30) (cid:30) >>>>>>>>>>>>> Y g f (cid:21) (cid:21) ˜ U (cid:31) (cid:127) ˜ / / ρ (cid:15) (cid:15) ˜ X ρ (cid:15) (cid:15) U (cid:31) (cid:127) j / / X As usual, the universal property enjoyed by ˜ X and ρ characterizes them uniquelyup to unique isomorphism.Note that by Lemma 3.10, the condition that IC ( X, ρ ∗ O ˜ U ) be defined is equiv-alent to requiring that depth( ρ ∗ O ˜ U ) x ≥ min { , dim( ρ ∗ O ˜ U ) x } at all points x ∈ U G -gen such that codim ¯ x ≥ codim Z . Moreover, since IC ( X, ρ ∗ O ˜ U ) ∈ M ± ( X ), thissheaf satisfies the analogous depth conditions for all x ∈ X G -gen with codim ¯ x ≥ codim Z . By Proposition 3.7, this is equivalent to saying that j ◦ ρ : ˜ U → X is S relative to U . Definition 4.6.
The scheme ˜ X constructed in Theorem 4.5 is called the S -extension of ρ : ˜ U → U .It should be noted that in general, an S -extension may be locally S only at thepoints in X G -gen . It is in fact an S scheme when G is trivial (so that X G -gen = X )or when Lemma 3.12 can be invoked. Moreover, if IC ( X, ρ ∗ O ˜ U ) is defined in thenonequivariant case, i.e., the depth condition on ( ρ ∗ O ˜ U ) x described above holdsfor x ∈ U and not just x ∈ U G -gen , then both equivariant and nonequivariant S -extensions of ρ are defined. Since the nonequivariant universal mapping property isstronger than the equivariant universal property, we obtain the following corollary. Corollary 4.7. If IC ( X, ρ ∗ O ˜ U ) is defined in the nonequivariant case, then thenonequivariant and equivariant S -extensions of ρ are canonically isomorphic. Before proving the theorem, we consider a few examples in which S -extensionhas an elementary description. Example . If the map j ◦ ρ : ˜ U → X is already finite (for example, if ˜ U is asingle point), then ˜ X = ˜ U . Example . Note that the complement of ˜ U in ˜ X must have codimension at least2, since Z has codimension at least 2 in X and ρ is finite. Recall that according toSerre’s criterion, a scheme is normal if and only if it is S and regular in codimension1. Thus, if ˜ U is normal, the fact that ˜ X is locally S outside ˜ U implies that ˜ X isalso normal.In particular, suppose that U is a normal subscheme of the integral scheme X and that ρ : ˜ U → U is an isomorphism. Then ρ : ˜ X → X is simply the usualnormalization of X . In view of Proposition 3.7, we see that the normalization of X has a remarkably simple description as Spec ( j ∗ O U ). Example . As a slight generalization of the previous example, let us now supposeonly that ˜ U is normal and that X is integral. An elementary construction of ˜ X is given as follows. Given an affine open subscheme V = Spec A of X , let K bethe fraction field of ρ − ( V ), and let B be the integral closure of the image of thenatural map A → K induced by ρ . Let ˜ V = Spec B . The various ˜ V ’s obtainedin this way as V ranges over affine open subschemes of X can be glued together toform a scheme ˜ X ′ . This scheme enjoys a universal property similar to that of thenormalization of a scheme. By comparing with the universal property of ˜ X , it iseasy to verify that ˜ X and ˜ X ′ are in fact canonically isomorphic. We thus obtainan alternative elementary description of IC ( X, ρ ∗ O ˜ U ); it is the sheaf V B . Proof of Theorem 4.5.
We begin by establishing various properties of ρ and ˜ X .Since IC ( X, ρ ∗ O ˜ U ) is coherent, ρ is finite. From the definition of Spec , we knowthat ρ − ( U ) ≃ Spec IC ( X, ρ ∗ O ˜ U ) | U ≃ Spec ρ ∗ O ˜ U . Now, ρ is finite, and there-fore affine, so Spec ρ ∗ O ˜ U is canonically isomorphic to ˜ U . Identifying these twoschemes, we also see that ρ | ˜ U = ρ . Moreover, Proposition 3.5 tells us that IC ( ˜ X, O ˜ U ) ≃ O ˜ X , and then by Lemma 3.9 and Remark 4.4, we see that id : ˜ X → ˜ X is S relative to ˜ U .As we have previously observed, ρ finite implies that ρ ∗ is exact and t -exact.Thus, ρ ∗ O ˜ X ≃ IC ( X, ρ ∗ O ˜ U ), and ρ is S relative to U . We have already seenthat ρ | ˜ U factors through ρ ; indeed, with the obvious identifications, it equals ρ . ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 15
Finally, Remark 4.3 tells us that all generic points of irreducible components of ˜ X lie in ˜ U , and hence that ˜ U is dense in ˜ X .It remains to show that ˜ X and ρ are universal with respect to these properties.Let f : Y → X be a finite morphism that is S relative to U , and assume that f | f − ( U ) factors through ρ . Let V = f − ( U ), and let g : V → ˜ U be the morphismsuch that f | V = ρ ◦ g . Then g gives rise to a morphism of sheaves ρ ∗ O ˜ U → f ∗ O V on U and therefore to a morphism of perverse coherent sheaves ρ ∗ O ˜ X ≃ IC ( X, ρ ∗ O ˜ U ) → IC ( X, f ∗ O V ) ≃ f ∗ O Y . Applying the global Spec functor to this morphism of sheaves ρ ∗ O ˜ X → f ∗ O Y , weobtain a morphism of schemes g : Y → ˜ X ; this is the desired morphism suchthat f = ρ ◦ g . The uniqueness of g follows from the fact that there is a uniquemorphism IC ( X, ρ ∗ O ˜ U ) → IC ( X, f ∗ O V ) whose restriction to U is the morphism ρ ∗ O ˜ U → f ∗ O V induced by g . (cid:3) As usual, any object characterized by a universal property comes with a unique-ness theorem, but for S -extension, there is an even stronger uniqueness property. Proposition 4.11.
Let ˆ X be a scheme containing ˜ U as a dense open set, and let ˆ ρ : ˆ X → X be a finite morphism extending ρ : ˜ U → U . If ˆ X is locally S outsideof ˜ U , then ˆ X is isomorphic to ˜ X .Proof. Since ˆ X is locally S outside of ˜ U , its structure sheaf is an IC sheaf: O ˆ X ≃ IC ( ˆ X, O ˜ U ). The functor ˆ ρ ∗ is exact and t -exact, so ˆ ρ ∗ O ˆ X ≃ IC ( X, ˆ ρ ∗ O ˜ U ). But nowˆ X ≃ Spec ρ ∗ O ˆ X ≃ ˜ X . (cid:3) Remark . The developments of this section are closely related to the ideas inSection 5.10 of EGA4, Part II [16], which (translated into our notation) deals withthe S condition for sheaves of the form j ∗ F and schemes of the form Spec j ∗ O U .The assumptions in loc. cit. are a bit different (e.g., the last part of our Proposi-tion 3.7 must be imposed as a hypothesis), and the specific setting of Theorem 4.5is not treated there. Nevertheless, it is easy to imagine adapting the methods usedthere to prove that ρ : ˜ X → X is S relative to U . However, the universal propertyof ρ and the uniqueness statement in Proposition 4.11 are consequences of the factthat the IC functor is an equivalence of categories with its essential image. Prov-ing those statements in the language of [16, Section 5.10] would likely amount tounwinding the proof of Proposition 2.3 and the construction of the perverse coher-ent t -structure in [7]. The conciseness and clarity of the uniqueness arguments areperhaps the main benefit of using perverse coherent sheaves here. Remark . Our main goal in this paper is to apply the S -extension constructionin the setting of special pieces, where the hypotheses of Example 4.10 hold. Indeed, U will be normal, and ρ : ˜ U → U will be a surjective ´etale morphism; see Section 7.Since an elementary construction of the S -extension is available in that setting(as was known to C. Procesi [28]), one could in principle forego developing themachinery of the functor Spec IC ( X, · ). However, we will also require the resultsof Section 5, and those seem to be much easier to state and prove in the context ofperverse coherent sheaves than in a purely ring-theoretic setting.We conclude this section with a remark on the “Macaulayfication” problem:given a scheme, find a Cohen–Macaulay scheme that is birationally equivalent to it. (For varieties over a field of characteristic 0, this problem is solved by Hiron-aka’s Theorem.) Kawasaki, extending early work of Faltings [15], has shown howto construct a Macaulayfication of any Noetherian scheme over a ring (of arbitrarycharacteristic) with a dualizing complex [25], but the resulting scheme is not canon-ical. (It does not have the obvious universal property.) Indeed, just by consideringvarieties that fail to be Cohen–Macaulay at a single closed point, Brodmann hasexhibited a family of examples which do not have a universal Macaulayfication [8].There may, however, be a finite Macaulayfication that is universal among appro-priate finite morphisms from Cohen-Macaulay schemes. The following theoremaddresses the problem in a way that is reminiscent of Example 4.9. Theorem 4.14 (Macaulayfication) . Let X be an scheme of finite type over a Noe-therian base scheme S admitting a dualizing complex, and suppose Coh ( X ) hasenough locally free sheaves. Let U be an open Cohen–Macaulay subscheme whosecomplement has codimension at least . Then X has a finite Macaulayfication if andonly if IC c ( X, O U ) is a sheaf (where c is the Cohen–Macaulay perversity). In thiscase, the unique finite Macaulayfication is ˜ X c def = Spec IC c ( X, O U ) and coincideswith the S -extension of id : U → U . The scheme ˜ X c is universal with respect to fi-nite morphisms to X which are S relative to U . In particular, any finite morphism f : Y → X with Y Cohen-Macaulay and f − ( U ) dense factors uniquely though ˜ X c . This theorem is stated without a group action for convenience. An equivariantversion akin to Theorem 4.5 can be proved by a similar argument.
Proof.
Let f : Y → X be a finite morphism such that f − ( U ) is Cohen–Macaulayand dense in Y . The last condition allows us to apply Lemma 3.9: Y is Cohen–Macaulay if and only if IC c ( Y, O f − ( U ) ) ≃ O Y . Next, by Proposition 3.5, we seethat Y is Cohen–Macaulay if and only if we have IC c ( X, f ∗ O f − ( U ) ) ≃ f ∗ O Y . Inparticular, if f is an isomorphism over U , then Y is a Macaulayfication of X ifand only if IC c ( X, O U ) is a sheaf. However, since c ≥ s , we know that whenever IC c ( X, O U ) is a sheaf, it must coincide with IC s ( X, O U ). We now see that ˜ X c isjust the “ S -ification” of X , and the universal property follows from Theorem 4.5.Finally, a finite morphism f : Y → X with Y Cohen–Macaulay and f − ( U ) denseis S relative to U , so the universal property applies in this situation. (cid:3) Note that this construction does not coincide with Kawasaki’s Macaulayfica-tion; the latter involves blow-ups and accordingly is never finite. Instead, thistheorem generalizes a result of Schenzel [30] relating finite Macaulayfications and“ S -ifications” for a certain class of local rings. (Schenzel’s result is essentiallyTheorem 4.14 in the special case X = Spec A , where A is a local domain that isa quotient of a Gorenstein ring.) However, the above scheme-theoretic statementappears not to have been previously known.Finally, we remark that Theorem 4.5 can be generalized to define a Cohen–Macaulay extension (or a p -extension for any perversity p with s ≤ p ≤ c ) ofappropriate ρ : ˜ U → U . Let ρ be a finite morphism such that IC p ( X, ρ ∗ O ˜ U ) isdefined and a sheaf, and let ˜ X p denote the scheme Spec IC p ( X, ρ ∗ O ˜ U ). A sim-ilar argument to that given in the proof of Theorem 4.5 shows that the naturalmorphism ρ : ˜ X p → X is universal with respect to finite morphisms f : Y → X which are “ p relative to U ” (defined in the obvious way) and whose restriction ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 17 f | f − ( U ) factors through ρ . However, since IC p ( X, ρ ∗ O ˜ U ) and IC s ( X, ρ ∗ O ˜ U ) co-incide when the former is a sheaf, we see that ˜ X p is just the S -extension, and so ρ is in fact universal with respect to finite morphisms S relative to U for which U has dense preimage.5. Middle Extension in the Derived Category
Recall that the intermediate extension functor IC ( X, · ) : M ± ( U ) → M ± ( X ) isan equivalence of categories. (We make no assumptions about the perversity in thissection.) In particular, we haveHom D ( X ) ( IC ( X, E ) , IC ( X, F )) ≃ Hom M ( U ) ( E , F ) , for E , F in M ± ( U ). The goal of this section is introduce a derived version of thisfunctor. We will construct a functor of triangulated categories from a suitable sub-category of D ( U ) to D ( X ) that “extends” IC ( X, · ), and then prove a generalizationof Proposition 2.3 in this setting.For most of this section, we make the following assumption:(Q) There is a class of projective objects Q in M ± ( U ) such that(i) every object of M ± ( U ) is a quotient of some object in Q , and(ii) for every object A in Q , IC ( X, A ) is a projective sheaf on X .For example, condition (Q) holds if X is a quasiaffine scheme that is locally S outside U , and p is the S perversity. In that case, IC ( X, O U ) = O X . Every objectin M ± ( U ) is in fact a sheaf. Moreover, since U is quasiaffine, every coherent sheafon U is a quotient of a free sheaf, so we can take Q to be the class of free sheaveson U .Let M ± ( U ) be the abelian category M ± ( U ) ∩ Coh ( U ). (In some cases, such aswith the S perversity, it happens that M ± ( U ) = M ± ( U ).) Let M ± ( X ) be thesubcategory of M ± ( X ) consisting of objects F such that j ∗ F ∈ M ± ( U ). Clearly, IC ( X, · ) and j ∗ restrict to give equivalences of categories between M ± ( U ) and M ± ( X ).Now, M ± ( U ) is a full subcategory of Coh ( U ) with enough projective objects thatare also projective in Coh ( U ) (namely, the objects of Q ). It follows that the boundedderived category DM ± ( U ) can be identified with a full triangulated subcategory of D ( U ). For brevity, we henceforth write D ± ( U ) for DM ± ( U ).Let D ± ( X ) be the full subcategory of D ( X ) consisting of those objects A forwhich p H n ( A ) ∈ M ± ( X ) for all n . This is a triangulated subcategory of D ( X ) (be-cause the subcategory M ± ( X ) of M ( X ) is stable under extensions). The perverse t -structure on D ± ( X ) (that is, the t -structure induced by the perverse t -structureon D ( X )) has heart M ± ( X ).Following [4, § t -structure on a full triangulated subcategory ofthe derived category of an abelian category, there is a realization functor from thederived category of the heart of the t -structure to the original derived category. Inour situation, we obtain a functor real : DM ± ( X ) → D ( X ), where DM ± ( X ) is thebounded derived category of the abelian category M ± ( X ). We now briefly review itsconstruction. This requires the machinery of filtered derived categories; we refer thereader to [4, § DF ( X ) be the boundedfiltered derived category of coherent sheaves on X , and let DF bˆete ( X ) be the fullsubcategory of DF ( X ) consisting of objects whose filtration is stupid (“bˆete”) withrespect to the perverse t -structure on D ± ( X ). Forgetting the filtration gives us a functor ω : DF bˆete ( X ) → D ( X ); on the other hand, by [4, Proposition 3.1.8], thereis an equivalence of categories G : DF bˆete ( X ) → CM ± ( X ), where CM ± ( X ) is thecategory of complexes of objects of M ± ( X ). We first define real : CM ± ( X ) → D ( X )by real = ω ◦ G − . By [4, Proposition 3.1.10], real factors through DM ± ( X ) andthus gives rise to a functor real : DM ± ( X ) → D ( X ). This functor is compatiblewith cohomology in the following sense: for all n , there is an isomorphism of functors H n ≃ p H n ◦ real : DM ± ( X ) → M ± ( X ).We define D IC ( X, · ) : D ± ( U ) → D ( X ) by D IC ( X, · ) = real ◦ IC ( X, · ). Lemma 5.1.
The functor D IC ( X, · ) : D ± ( U ) → D ( X ) takes values in D ± ( X ) , andthere are isomorphisms of functors p H n ( D IC ( X, · )) ≃ IC ( X, p H n ( · )) for all n .Proof. Since IC ( X, · ) : M ± ( U ) → M ( X ) is exact, the functor on derived categories IC ( X, · ) : D ± ( U ) → DM ( X ) respects cohomology: IC ( X, p H n ( · )) (or, equivalently, IC ( X, std H n ( · ))) is isomorphic to H n ( IC ( X, · )). Next, H n ≃ p H n ◦ real, so IC ( X, p H n ( · )) ≃ H n ( IC ( X, · )) ≃ p H n (real( IC ( X, · ))) ≃ p H n ( D IC ( X, · )) . Since IC ( X, · ) takes values in M ± ( X ), it is obvious that D IC ( X, · ) takes values in D ± ( X ). (cid:3) Lemma 5.2.
There is an isomorphism of functors j ∗ D IC ( X, · ) ≃ id : D ± ( U ) → D ± ( U ) .Proof. Since IC ( X, · ) : M ± ( U ) → M ± ( X ) and j ∗ : M ! ∗ ( U ) → M ± ( U ) are inverseequivalences of categories, they give rise to equivalences of the corresponding cat-egories of complexes, as well as of the the corresponding derived categories. Theseequivalences are such that the square in the center of the diagram below commutes. DF bˆete ( X ) j ∗ ≀ (cid:15) (cid:15) G ∼ / / ω * * CM ± ( X ) j ∗ ≀ (cid:15) (cid:15) / / DM ± ( X ) j ∗ ≀ (cid:15) (cid:15) real / / D ( X ) j ∗ l l DF bˆete ( U ) G ∼ / / ω CM ± ( U ) IC ( X, · ) O O / / D ± ( U ) j ∗ O O D IC ( X, · ) In the setting of filtered derived categories, the restriction functor j ∗ : DF ( X ) → DF ( U ) respects stupidity of the filtration (because j ∗ takes M ± ( X ) to M ± ( U ))and so gives rise to a functor j ∗ : DF bˆete ( X ) → DF bˆete ( U ) that makes the leftmostsquare in the diagram above commute. Here DF bˆete ( U ) is defined with respectto the perverse t -structure on D ± ( U ) (which is simply a shift of the standard t -structure). It is clear that restriction commutes with forgetting the filtration, so j ∗ ◦ ω ≃ ω ◦ j ∗ . Together, these statements imply that j ∗ ◦ real : DM ± ( X ) → D ± ( U ) is isomorphic to j ∗ : DM ± ( X ) → D ± ( U ). Composing with IC ( X, · ) : D ± ( U ) → DM ± ( X ), we find that j ∗ ◦ real ◦ IC ( X, · ) ≃ j ∗ ◦ IC ( X, · ), or in otherwords, j ∗ ◦ D IC ( X, · ) ≃ id. (cid:3) Definition 5.3.
An object F of M ± ( U ) is said to be short if IC ( X, F ) ∈ std D ( X ) ≤ .For example, if p is the S perversity, all objects in M ( U ) are short. Indeed, they,as well as all other short objects we will actually encounter, satisfy the strongercondition that their images under IC ( X, · ) belong to std D ( X ) ≤ , but the weakercondition above suffices for the statements we wish to prove. ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 19
Proposition 5.4. If F ∈ M ± ( U ) is short, there are natural isomorphisms Hom D ( X ) ( IC ( X, E ) , IC ( X, F )[ n ]) ≃ Hom D ( U ) ( E , F [ n ])(6) R H om D ( X ) ( IC ( X, E ) , IC ( X, F )) ≃ Rj ∗ R H om D ( U ) ( E , F )(7) for all n ∈ Z and all E ∈ M ± ( U ) .Remark . According to [4, Remarque 3.1.17(ii)], the isomorphism (6) alwaysexists for n ≤
1, without assuming condition (Q) or the shortness of F . Proof.
As we have just remarked, the natural morphism(8) Hom D ( U ) ( E , F [ n ]) → Hom D ( X ) ( IC ( X, E ) , IC ( X, F )[ n ]) . induced by IC ( X, · ) is an isomorphism for n ≤
1. Lemma 5.2 implies that thismorphism is always injective (in other words, that IC ( X, · ) is faithful), so it simplyremains to show that it is surjective for n >
1. We proceed by induction.Given f : D IC ( X, E ) → D IC ( X, F )[ n ], choose a surjective map g : A → E with A ∈ Q . By assumption, IC ( X, A ) is a projective sheaf, so Hom( IC ( X, A ) , G ) = 0 forall G ∈ std D ( X ) ≤− . Since F is short, we have IC ( X, F )[ n ] ∈ std D ( X ) ≤− n +1 , andsince n >
1, we have Hom( IC ( X, A ) , IC ( X, F )[ n ]) = 0 . Now, let H = ker g , and consider the exact sequence · · · → Hom( IC ( X, H )[1] , IC ( X, F )[ n ]) → Hom( IC ( X, E ) , IC ( X, F )[ n ]) → Hom( IC ( X, A ) , IC ( X, F )[ n ]) → · · · . We see that f must be the image of some morphism f ′ : IC ( X, H )[1] → IC ( X, F )[ n ],that is, f = f ′ ◦ d , where d : IC ( X, E ) → IC ( X, H )[1] comes from the distinguishedtriangle associated to the short exact sequence 0 → H → A → E →
0. Now, f ′ [ −
1] : IC ( X, H ) → IC ( X, F )[ n −
1] is in the image of IC ( X, · ) by the inductivehypothesis, and d is in its image by Remark 5.5, so f is in its image as well.It remains to prove the corresponding fact for R H om . For the remainder of theproof, we assume that we are working in the nonequivariant setting (or that G istrivial). As remarked in [7] immediately preceding Lemma 2, R H om commuteswith the forgetful functor from an equivariant category to the nonequivariant one,so we lose nothing by making this assumption.In the nonequivariant setting, all of the preceding arguments also apply to anyopen set V ⊂ X . In particular, since D IC ( X, E ) | V ≃ D IC ( V, E | V ∩ U ) for any object E ∈ M ± ( U ), we have, for any n ∈ Z , an isomorphismHom D ( V ) ( IC ( X, E ) | V , IC ( X, F )[ n ] | V ) ≃ Hom D ( V ∩ U ) ( E | V ∩ U , F [ n ] | V ∩ U ) . Now, for any two objects A , B ∈ D ( X ), there is a canonical morphism j ∗ R H om D ( X ) ( A , B ) → R H om D ( U ) ( j ∗ A , j ∗ B ) . Let us take A = IC ( X, E ) and B = IC ( X, F ). Of course, we then have j ∗ A ≃ E and j ∗ B ≃ F . Now, by adjointness, the above morphism gives rise to a canonicalmorphism φ : R H om D ( X ) ( IC ( X, E ) , IC ( X, F )) → Rj ∗ R H om D ( U ) ( E , F ) . To show that φ is in fact an isomorphism, it suffices to show that it induces isomor-phisms on all hypercohomology groups over all open sets. For any open set V ⊂ X ,we have H n ( R Γ( V, R H om D ( X ) ( IC ( X, E ) , IC ( X, F )))) ≃ Hom D ( V ) ( IC ( X, E ) | V , IC ( X, F )[ n ] | V ) ≃ Hom D ( V ∩ U ) ( E | V ∩ U , F [ n ] | V ∩ U ) ≃ H n ( R Γ( V, Rj ∗ R H om ( E , F ))) . Thus, φ is an isomorphism. (cid:3) Theorem 5.6.
If all objects in M ± ( U ) are short, then D IC ( X, · ) : D ± ( U ) → D ± ( X ) is an equivalence of categories, with inverse given by j ∗ . Moreover, for anytwo objects E , F ∈ D ± ( U ) , there are natural isomorphisms Hom D ( X ) ( D IC ( X, E ) , D IC ( X, F )) ≃ Hom D ( U ) ( E , F )(9) R H om D ( X ) ( D IC ( X, E ) , D IC ( X, F )) ≃ Rj ∗ R H om D ( U ) ( E , F )(10) Proof.
If all objects in M ± ( U ) are short, the isomorphism (6) holds for all objects E , F ∈ M ± ( U ). As observed in the proof of [4, Proposition 3.1.16], the realizationfunctor is an equivalence of categories if and only if (6) holds for all objects in M ± ( U ). Since IC ( X, · ) : D ± ( U ) → DM ± ( X ) is an equivalence of categories, we seethat D IC ( X, · ) = real ◦ IC ( X, · ) is as well. By Lemma 5.2, its inverse must be j ∗ .Once we know that D IC ( X, · ) is an equivalence of categories, (9) is immediate.We deduce (10) from it by an argument identical to that given for (7) above. (cid:3) For applications of this result, we must consider certain categories whose objectsare dual to coherent sheaves. Given a perversity p , let M p, ±∗ ( U ) = D ( M ¯ p, ± ( U )) ⊂ M p, ± ( U ) . Corollary 5.7.
Suppose the dualizing complexes on U and X have the followingproperties: with respect to some perversity p , ω U is a short object in M p, ± ( U ) , and ω X ≃ IC p ( X, ω U ) . Then, with respect to the dual perversity ¯ p , we have IC ¯ p ( X, E ) ≃ Rj ∗ E for all E ∈ M ¯ p, ±∗ ( U ) .Proof. Let E be an object of M ¯ p, ±∗ ( U ). Then D E = R H om D ( U ) ( E , ω U ) is an objectin M p, ± ( U ). It follows from [7, Lemma 5(a)] that D IC p ( X, D E ) = R H om ( IC p ( X, D E ) , ω X ) ≃ IC ¯ p ( X, E ) . But we also have R H om D ( X ) ( IC p ( X, D E ) , ω X ) ≃ R H om D ( X ) ( IC p ( X, D E ) , IC p ( X, ω U )) ≃ Rj ∗ R H om D ( U ) ( D E , ω U ) . Since R H om D ( U ) ( D E , ω U ) ≃ E , we see that IC ¯ p ( X, E ) ≃ Rj ∗ E . (cid:3) Corollary 5.8.
Suppose X is a Gorenstein scheme. (1) If p + ( x ) = codim ¯ x for all x ∈ U G - gen , then there is an isomorphism offunctors IC ( X, · ) ≃ Rj ∗ . (2) If F is a Cohen–Macaulay sheaf on U , then IC ( X, F ) ≃ Rj ∗ F with respectto any perversity. ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 21
Proof.
On a Gorenstein scheme, we may take ω X ≃ O X . A Gorenstein scheme is,in particular, Cohen–Macaulay, so by Lemma 3.4, ω X ≃ IC ( X, ω U ) with respectto every perversity. Corollary 5.7 now tells us that on M ±∗ ( U ), IC ( X, · ) ≃ Rj ∗ forevery perversity.For part (1) of the corollary, we must simply show that M p, ±∗ ( U ) = M p, ± ( U ),or, equivalently, that M ¯ p, ± ( U ) = M ¯ p, ± ( U ). The assumption that p + ( x ) = codim ¯ x implies that ¯ p − ( x ) = 0 for all x ∈ U G -gen . It follows that M ¯ p, ± ( U ) ⊂ M p − ( U ) = Coh ( U ), as desired.For part (2), we first note that IC ( X, F ) is defined with respect to any perversityby Lemma 3.4. In particular, we see that F ∈ M c, ± ( U ), so DF ∈ M s, ± ( U ) = M s, ± ( U ). Since DF ∈ Coh ( U ), F is in M ±∗ ( U ) with respect to any perversity, andthe result follows by Corollary 5.7. (cid:3) Corollary 5.9.
Let X be a Gorenstein scheme, U ⊂ X an open subscheme, and ρ : ˜ U → U a finite morphism. If ρ admits an S -extension, let ˜ X be the schemethus obtained. (1) If ˜ U is Cohen–Macaulay, then ˜ X is as well. (2) If ρ ∗ O ˜ U is isomorphic to its own Serre–Grothendieck dual, then ˜ X isGorenstein. In particular, part of the content of this corollary is the assertion that if either˜ U is Cohen–Macaulay or ρ ∗ O ˜ U is self-dual, then ρ necessarily admits an S -extension. Proof.
For part (1), it follows from Proposition 3.5, Lemma 3.9, and Corollary 5.8that the S -extension ˜ X exists and is locally Cohen–Macaulay at least at all pointsof ˜ X G -gen . This reasoning can be repeated in the nonequivariant category to obtaina nonequivariant S -extension that is in fact Cohen–Macaulay. The latter varietymust coincide with ˜ X by Corollary 4.7.Henceforth, assume that ρ ∗ O ˜ U ≃ D ( ρ ∗ O ˜ U ). Evidently, ρ ∗ O ˜ U ∈ s − D ( U ) ≤ ,and since the dual perversity to s − is c + , we have ρ ∗ O ˜ U ∈ c + D ( U ) ≥ as well.It follows that the intermediate extension of ρ ∗ O ˜ U is defined with respect to anyperversity; furthermore, by Corollary 5.8, it is independent of perversity. Let F = IC ( X, ρ ∗ O ˜ U ). By [7, Lemma 5], D F ≃ IC ( X, D ( ρ ∗ O ˜ U )), and hence F ≃ D F .Now, by Proposition 3.5 and Lemma 3.9, we know that ˜ X = Spec F is Cohen–Macaulay. In particular, given a point x ∈ X and a point y ∈ ρ − ( x ), we know thatthe local ring O y, ˜ X is a finite Cohen–Macaulay extension of the Gorenstein localring O x,X . According to [12, Theorem 3.3.7], O y, ˜ X is Gorenstein if and only if(11) O y, ˜ X ≃ Hom O x,X ( O y, ˜ X , O x,X ) . Consider the fact that i ∗ x F = i ∗ x ρ ∗ O ˜ X ≃ M y ∈ ρ − ( x ) O y, ˜ X . Obviously, (11) implies that(12) i ∗ x F ≃ Hom O x,X ( i ∗ x F , O x,X ) . Conversely, if (12) holds, then by considering the action of each O y, ˜ X on each sideof this isomorphism, we see that (11) must hold as well. Thus, (11) and (12) are equivalent. On the other hand, by [27, Proposition 7.24(iii)] (for instance),Hom O x,X ( i ∗ x F , O x,X ) ≃ i ∗ x H om ( F , O X ) ≃ i ∗ x ( D F ) . Now, (12) is true for all x because it is equivalent to the statement that i ∗ x F ≃ i ∗ x D F .Therefore, (11) is true for all y , so ˜ X is Gorenstein. (cid:3) We conclude this section with the statement of a purely ring-theoretic version ofthe preceding result. The authors are not aware of a direct proof of this statementin the setting of commutative algebra. Note that the implicit hypothesis that X satisfies condition (Q) is not needed here because only affine schemes are involved. Corollary 5.10.
Let A be a Gorenstein domain. Let K be a finite extension ofthe fraction field of A , and let B be the integral closure of A in K . Let I ⊂ A be aradical ideal of codimension at least , and let T be a set of generators for I . (1) If B f is Cohen–Macaulay for all f ∈ T , then B is Cohen–Macaulay.(Equivalently, B is Cohen–Macaulay if B P is for all prime (resp. maximal)ideals P of B lying over prime (resp. maximal) ideals of A not containing I .) (2) If B f ≃ Hom A f ( B f , A f ) for all f ∈ T , then B is Gorenstein. Perverse Constructible Sheaves
We assume henceforth that X and G are separated schemes over S = Spec k for some field k and that U is smooth. In this section, we establish some resultson ordinary (constructible) perverse sheaves on X which we will need in studyingspecial pieces.Fix a prime number ℓ different from the characteristic of k , and let D ( X ) be thebounded G -equivariant derived category of constructible ¯ Q l -sheaves on X (in thesense of Bernstein–Lunts). By an abuse of notation, we use D to denote the Verdierduality functor in this category: here D = R H om ( · , a ! ¯ Q l ), where a : X → Spec k isthe structure morphism.Let ( D ( X ) ≤ , D ( X ) ≥ ) be the perverse t -structure on D ( X ) with respect to themiddle perversity: D ( X ) ≤ = { F ∈ D ( X ) | dim supp H − i ( F ) ≤ i } ,D ( X ) ≥ = { F ∈ D ( X ) | dim supp H − i ( D F ) ≤ i } . Let M ( X ) be the heart of this t -structure. There is an intermediate extensionfunctor M ( U ) → M ( X ). Given an equivariant local system E on U , we denote byIC( X, E ) the object of D ( X ) such that IC( X, E )[dim X ] ∈ M ( X ) is the intermedi-ate extension of E [dim X ] ∈ M ( U ).In addition, let ( std D ( X ) ≤ , std D ( X ) ≥ ) denote the standard t -structure on D ( X ). Note that std D ( X ) ≤− dim X ⊂ D ( X ) ≤ and std D ( X ) ≥− dim X ⊃ D ( X ) ≥ . Proposition 6.1. If X is irreducible and IC( X, ¯ Q l ) is a sheaf, then in fact we have IC( X, ¯ Q l ) ≃ ¯ Q l (i.e., X is rationally smooth).Proof. Recall that there is a distinguished triangleIC( X, ¯ Q l )[dim X ] → Rj ∗ ¯ Q l [dim X ] → F → IC( X, ¯ Q l )[dim X + 1] , where F is supported on Z , and F | Z lies in D ( Z ) ≥ . In particular, this impliesthat F ∈ std D ( X ) ≥− dim Z . Taking the long exact sequence cohomology sequence ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 23 associated to the above distinguished triangle, we see that H k (IC( X, ¯ Q l )[dim X ]) ≃ H k ( Rj ∗ ¯ Q l [dim X ]) for all k < − dim Z . If we take k = − dim X , we find that H (IC( X, ¯ Q l )) ≃ H ( Rj ∗ ¯ Q l ).Since IC( X, ¯ Q l ) is assumed to be a sheaf, we have H (IC( X, ¯ Q l )) ≃ IC( X, ¯ Q l ).On the other hand, we have H ( Rj ∗ ¯ Q l ) ≃ j ∗ ¯ Q l ≃ ¯ Q l , where the last isomorphismholds because X is assumed to be irreducible. (cid:3) Proposition 6.2.
Let f : Y → X be a finite morphism of irreducible varieties. Let V = f − ( U ) , and assume that f ∗ ( ¯ Q l | V ) is a local system on U . If IC(
X, f ∗ ( ¯ Q l | V )) is a sheaf, then Y is rationally smooth.Proof. Since f is finite (and hence affine and proper), f ∗ is exact and t -exact by [4,Corollaire 2.2.6], and in particular, f ∗ IC( Y, ¯ Q l ) is an intersection cohomology com-plex on X , namely, it is IC( X, f ∗ ( ¯ Q l | V )). This complex is, by assumption, actuallya sheaf. Now, f ∗ kills no nonzero sheaf, so the fact that f ∗ IC( Y, ¯ Q l ) is a sheaf impliesthat IC( Y, ¯ Q l ) itself is a sheaf. By Proposition 6.1, Y is rationally smooth. (cid:3) Since the morphism obtained by S -extension of a finite morphism is also finite,the same argument as above gives us the following result relating intersection coho-mology complexes on a scheme obtained by S -extension with those on the originalscheme. This fact will be a vital step in the calculations of Section 7, as anticipatedby Lusztig in his original formulation of Conjecture 1.1 [24, § Proposition 6.3.
Let ρ : ˜ X → X be the S -extension of a finite morphism ρ :˜ U → U ⊂ X . Let E be a local system on ˜ U , and assume that ρ ∗ E is a local systemon U . Then we have ρ ∗ IC( ˜
X, E ) ≃ IC(
X, ρ ∗ E ) . (cid:3) We close this section with the following result expressing the size of fibers of thenormalization map in terms of intersection cohomology.
Proposition 6.4.
Let X be an irreducible variety with rationally smooth normal-ization ¯ X , and let ν : ¯ X → X be the normalization morphism. Then for any x ∈ X , | ν − ( x ) | = dim H x (IC( X, ¯ Q l )) . If X is also rationally smooth, then X is unibranch.Proof. Since ν is a finite morphism, it is exact and t -exact. This and the fact that ν is birational imply that ν ∗ ¯ Q l ≃ ν ∗ IC( ¯ X, ¯ Q l ) ≃ IC(
X, ν ∗ ¯ Q l ) ≃ IC( X, ¯ Q l ). Takingstalks at x gives ν ∗ ( ¯ Q l | ν − ( x ) ) ≃ IC( X, ¯ Q l ) x , and hence, H x (IC( X, ¯ Q l )) ≃ ¯ Q | ν − ( x ) | l .The formula for the fiber size follows by taking dimensions. Finally, if X is rationallysmooth, then IC( X, ¯ Q l ) ≃ ¯ Q l , so | ν − ( x ) | = 1. (cid:3) Remark . This proposition is known, but we have provided a proof for lack ofa suitable reference. The statement (without the assumption that ¯ X is rationallysmooth) is given without proof in [5, 5E].7. The Geometry of ˜ P In this section, we prove Theorem 1.3. The field k is now assumed to be alge-braically closed of good characteristic for the group G . We begin by observing that IC ( P, ρ ∗ O ˜ C ) is defined. Indeed, ˜ C is open dense in P , and its complement hascodimension at least 2. Also, the stalk of ρ ∗ O ˜ C at x ∈ C is just the direct sumof | F | copies of O C ,x . Since C is smooth, ρ ∗ O ˜ C certainly satisfies the conditionof Lemma 3.10. We may thus define ˜ P by S -extension:˜ P = Spec IC ( P, ρ ∗ O ˜ C ) . Now, ˜ C is regular and S (because it is smooth), and its complement in ˜ P (whichhas codimension at least 2) is S . By Serre’s criterion, ˜ P is normal. The first partof Theorem 1.3 is then immediate from Theorem 4.5.The remainder of Theorem 1.3 is given by Propositions 7.1–7.4 below. Proposition 7.1. (1)
The variety ˜ P is endowed with natural actions of F and G , and these actions commute. If we regard F as acting trivially on P , then ρ is both G and F -equivariant. (2) The variety ˜ P is rationally smooth. Moreover, if char k = 0 , then ˜ P isGorenstein.Proof. First, note that ρ ∗ O ˜ C naturally has the structure of a ( G × F )-equivariantsheaf (where F acts trivially on P ), so IC ( P, ρ ∗ O ˜ C ) acquires one as well. Applyingthe Spec construction to an equivariant sheaf produces a scheme carrying an groupaction and an equivariant morphism.Next, the rational smoothness of ˜ P follows from Proposition 6.2, together withthe fact that IC( P, E ) is a sheaf for any equivariant local system E on C . (See [24,Proposition 0.7(c)].)Finally, observe that because ˜ P is normal, the canonical morphism ρ : ˜ P → P factors through the normalization ¯ P of P :˜ P ¯ ρ / / ˜ ρ ¯ P ν / / P By invoking Proposition 3.5, we see that ˜ P can also be constructed as Spec IC ( ¯ P , (¯ ρ | ˜ C ) ∗ O ˜ C ) . Now, ¯ P is Gorenstein in characteristic 0 by the theorem of Hinich–Panyushev [19,26], so ˜ P is Gorenstein as well by Corollary 5.9. (Note that ¯ P satisfies condition (Q),since it is normal and quasiaffine.) (cid:3) Proposition 7.2.
Each special piece P is unibranch.Proof. This follows from Proposition 6.4, the preceding result, and the fact that P is rationally smooth. (cid:3) Proposition 7.3.
The morphism ¯ ρ : ˜ P → ¯ P is the algebraic quotient of the F -action while ρ : ˜ P → P is the topological quotient. In particular, F acts transitivelyon the fibers of ρ .Proof. Since P is unibranch, P is homeomorphic to ¯ P , and it suffices to show that¯ P ≃ ˜ P /F .The functor IC is an equivalence of categories between appropriate categories ofsheaves on C and P , and it accordingly preserves finite limits. In particular, itpreserves F -fixed objects, so IC ( P, ρ ∗ O ˜ C ) F ≃ IC ( P, ( ρ ∗ O ˜ C ) F ) ≃ IC ( P, O C ).The result now follows, since we have ˜ P /F ≃ Spec IC ( P, ρ ∗ O ˜ C ) F and ¯ P ≃ Spec IC ( P, O C ). (cid:3) Proposition 7.4.
For each parabolic subgroup H ⊂ F , we have ρ − ( C H ) ≃ ˜ C H . We prove this proposition in two steps. First, in Lemma 7.5, we obtain a generaldescription of the varieties ρ − ( C H ) in terms of unknown F -stabilizers. This de-scription will suffice to prove the proposition when F is abelian, and in particular, ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 25 for the classical groups. Then, in Lemma 7.6, we show by case-by-case consider-ations that the F -stabilizers of points in ρ − ( C H ) are in fact conjugates of H forthe exceptional groups. Lemma 7.5.
Let H be a parabolic subgroup of F . Each connected component of ρ − ( C H ) is isomorphic to ( ˜ C H ) ◦ . Let K H be the stabilizer in F of some closedpoint of ρ − ( C H ) . Then | K H | = | H | , and there is a subgroup L H ⊂ N F ( K H ) suchthat | L H | = | N F ( H ) | and ρ − ( C H ) ≃ ( ˜ C H ) ◦ × L H F . Moreover, if H is conjugateto a subgroup of another parabolic H ′ , then K H is conjugate to a subgroup of K H ′ .Proof. Let E denote either the regular representation of F or, by abuse of notation,the corresponding local system on C . We will calculate IC( P, E ) | C H in a way thatreflects the structure of ρ − ( C H ) and then compare with the known calculationsfollowing [24] to prove the result.Consider the commutative diagram ρ − ( C H ) (cid:31) (cid:127) ˜ ı H / / ρ (cid:15) (cid:15) ˜ P ρ (cid:15) (cid:15) C H (cid:31) (cid:127) i H / / P From Proposition 6.3, we have ρ ∗ IC( ˜
P , ¯ Q l ) ≃ IC(
P, ρ ∗ ¯ Q l | C ). Moreover, because ρ | ρ − ( C ) is a principal F -bundle, ρ ∗ ¯ Q l | C = E . On the other hand, IC( ˜ P , ¯ Q l ) ≃ ¯ Q l as ˜ P is rationally smooth, so we have ρ ∗ ¯ Q l ≃ IC(
P, E ) , and hence IC( P, E ) | C H ≃ ( ρ ∗ ¯ Q l ) | C H . Now, since ρ is proper, we know that ρ ∗ ¯ Q l | C H ≃ ρ ∗ ( ¯ Q l | ρ − ( C H ) ). We seek to understand ρ ∗ ( ¯ Q l | ρ − ( C H ) ).Choose a point x ∈ C H and a point ˜ x ∈ ρ − ( x ). Since the map ρ : ρ − ( C H ) → C H is finite and G -equivariant, the stabilizer in G of ˜ x , which we denote G ˜ x , mustbe a finite-index subgroup of the stabilizer G x of x . The connected componentof ρ − ( C H ) containing ˜ x , which will be denoted B , must be isomorphic to thehomogeneous space G/G ˜ x . Then, since F acts transitively on the fiber ρ − ( x ),and the actions of F and G commute, it follows that every connected componentof ρ − ( C H ) is isomorphic to G/G ˜ x . Let L H be the subgroup of F that preserves B (without necessarily fixing ˜ x ). The preceding discussion shows that ρ − ( C H )is isomorphic to B × L H F (where a ∈ F acts on a pair ( b, f ) ∈ B × L H F by a · ( b, f ) = ( b, f a − )). In particular, the number of connected components of ρ − ( C H )is [ F : L H ].Let K H be the stabilizer in F of ˜ x . Since the actions of F and G commute, itfollows that K H is also the F -stabilizer of every other point in B . This implies that K H is a normal subgroup of L H . Now, the group L H /K H acts simply transitivelyon ρ − ( x ) ∩ B , so this is the group of deck transformations of B over C H . Let A ′ ( C H ) = L H /K H . We also have A ′ ( C H ) ≃ G x /G ˜ x , which is the quotient of A ( C H ) ≃ G x / ( G x ) ◦ by G ˜ x / ( G x ) ◦ .The local system ( ρ | B ) ∗ ¯ Q l on C H corresponds to the regular representationof A ′ ( C H ), and the full local system ρ ∗ ( ¯ Q l | ρ − ( C H ) ) is then clearly just the di-rect sum of [ F : L H ] copies of the regular representation of A ′ ( C H ). It is easilychecked that the action of L H /K H on the space E K H of K H -invariant vectors in E is also the direct sum of [ F : L H ] copies of its regular representation. Thus,IC( P, E ) | C H ≃ E K H as an A ( C H )-representation. Set Q ⊂ A ( C H ) equal to thekernel of this representation. Since E K H is a faithful representation of A ′ ( C H ), weobtain A ′ ( C H ) ≃ A ( C H ) /Q as groups and left A ( C H )-spaces. (In other words, if A ′ ( C H ) is viewed as a quotient of A ( C H ), then A ′ ( C H ) = A ( C H ) /Q .)On the other hand, according to [24, Proposition 0.7], IC( P, E ) | C H is the repre-sentation of A ′′ ( C H ) def = N F ( H ) /H on E H . Following [24, 3], this group is a directfactor of ¯ A ( C H ) and hence naturally a quotient of A ( C H ). The same argumentnow shows that A ′′ ( C H ) ≃ A ( C H ) /Q and hence A ′ ( C H ) ≃ A ′′ ( C H ) as groups and A ( C H )-spaces. Moreover, E H and E K H are isomorphic representations via thisisomorphism. In particular, we have | K H | = | H | , since dim E K H = [ F : K H ] anddim E H = [ F : H ]. It follows immediately that | L H | = | N F ( H ) | . (However, wecannot conclude that K H is conjugate to H ; see Remark 7.7.)The fact that A ′ ( C H ) ≃ A ′′ ( C H ) as homogeneous spaces implies that G ˜ x is pre-cisely the kernel G xF of the canonical map G x → N F ( H ) /H . Thus, B ≃ G x /G xF =( ˜ C H ) ◦ , and ρ − ( C H ) ≃ ( ˜ C H ) ◦ × L H F .Finally, the points of ˜ P fixed by K H form a closed subvariety. If we repeat theabove argument with another parabolic H ′ with H ⊂ H ′ , so that C H ′ ⊂ C H , we seethat K H must be contained in the F -stabilizers of points of B ∩ ρ − ( C H ′ ). Everysuch stabilizer is conjugate to K H ′ , so K H is conjugate to a subgroup of K H ′ . (cid:3) If F is abelian, then | L H | = | N F ( H ) | implies that both groups are in fact equalto F . Thus, ρ − ( C H ) ≃ ( ˜ C H ) ◦ × N F ( H ) F = ˜ C H .It remains to identify the K H ’s and L H ’s for the exceptional groups. There, theonly nontrivial groups F that occur are symmetric groups S n with 2 ≤ n ≤
5. Thefollowing lemma gives us the required information about the K H ’s. Lemma 7.6.
Let F = S n with ≤ n ≤ . Let { K H } be a collection of subgroupsof F , where H ranges over the parabolic subgroups of F . Assume that | K H | = | H | and that K H is conjugate to a subgroup of K H whenever H is conjugate to asubgroup of H . Then each K H is conjugate to H .Proof. If F = S there are no nontrivial cases of H to consider.If F = S , we must consider H = S . It is clear that every subgroup of F oforder 2 is conjugate to H .If F = S , then the nontrivial possibilities for H are S , S , and S × S . Thelast one is a Sylow 2-subgroup of F , so every subgroup of order 4 is conjugate to it.Next, it is easy to verify by hand calculation that every subgroup of S generatedby an element of order 3 and another of order 2 either has more than 6 elements oris conjugate to S . Finally, if H = S , we now know that K H must be conjugateto a subgroup of S , so K H is conjugate to S by the preceding paragraph.If f = S , there are five nontrivial parabolic subgroups up to conjugacy. Anotherhand calculation shows that any subgroup generated by an element of order 4 andanother of order 2 either has size different from 24 or is conjugate to S . Next, if H is any of S , S × S , or S , then K H must be conjugate to a subgroup of S , soby the previous paragraph, K H is conjugate to H . Finally, suppose H = S × S .Then K H must contain a subgroup conjugate to S . Again, an easy calculationshows that every subgroup generated by S and an element of order 2 either hassize different from 12 or is conjugate to S × S . (cid:3) ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 27
Remark . The above lemma does not hold in general for F an finite Coxetergroup. For example, suppose F is the Weyl group of type B , generated by simplereflections s and t with ( st ) = 1. The groups h s i and h t i are representatives of thetwo conjugacy classes of nontrivial parabolic subgroups. If we set K h s i = h s i and K h t i = h ( st ) i , the hypotheses of the lemma are satisfied, but evidently K h t i is notconjugate to h t i .We now know that K H is conjugate to H in all cases. Returning to the settingof Lemma 7.5, we see that since L H ⊂ N F ( K H ) ≃ N F ( H ) and | L H | = | N F ( H ) | ,we must in fact have L H = N F ( K H ), so ρ − ( C H ) ≃ ( ˜ C H ) ◦ × N F ( H ) F = ˜ C H . Theproof of Proposition 7.4 is now complete, and hence, so is the proof of Theorem 1.3.Finally, we observe that any smooth variety containing ˜ C as a dense open settogether with a finite morphism to P extending ρ must coincide with ˜ P . Inparticular: Corollary 7.8. If char k = 0 and G is classical, ˜ P is isomorphic to the smoothvariety over P constructed by Kraft–Procesi.Proof. This follows immediately from the theorem and Proposition 4.11. (cid:3) Normality of Special Pieces
Recall that Conjecture 1.1 contains implicitly the additional Conjecture 1.2 thatall special pieces are normal. In the classical types in characteristic 0, this statementfollows from the work of Kraft–Procesi [20]; they show that each special piece P isthe algebraic quotient of ˜ P by F , so by Proposition 7.3, P is normal.In positive characteristic or for the exceptional types in characteristic 0, there isno uniform answer. Of course, some special pieces consist only of a single unipotentclass, so those ones are obviously normal (and even smooth). In other cases, it isknown that the full closure of a special unipotent class in the unipotent variety isnormal. Since a special piece is an open subvariety of its closure, the normalityof the closure implies the normality of the special piece. Normality of closures ofunipotent classes (or, more typically, nilpotent orbits) has been studied extensivelyby a number of authors, so this technique gives information about a large numberof special pieces. In this section, we list the normality results that can be obtainedin this way.The following proposition summarizes the situation for classical groups. Proposition 8.1.
Let G be a simple algebraic group of classical type over an al-gebraically closed field k of good characteristic. Let C be a special unipotent class,and let P be the corresponding special piece. (1) If char k = 0 or G is of type A n , then P is normal. (2) P = C if and only if ¯ A ( C ) = 1 . In that case, of course, P is normal. (3) If G is of type B n and C is the subregular class, then P is normal. We remark that it is easy to determine whether ¯ A ( C ) = 1 for a given specialclass, using the straightforward combinatorial descriptions of that group given in,say, [3] or [24]. Proof.
As we remarked above, in characteristic 0, the result follows from the workof Kraft–Procesi [20]. In type A n , every unipotent class is special, so every specialpiece consists of a single class. G r o up Smooth Normal Normalinchar. 0 Normal ifBPS conj.is true Unknown G G , 1 G ( a ) F F , F ( a ), F ( a ), C , B , ˜ A , A , A + ˜ A , 1 F ( a ),˜ A E E , E ( a ), D , D ( a ), A + A , D , A , A , A + 2 A , 2 A , A + A ,2 A , A , 1 E ( a ) D ( a ), A E E , E ( a ), E ( a ), E , E ( a ), E ( a ), D ( a ), D + A , A , D , D ( a ) + A , A + A , A + A , ( A ) ′′ , A + A + A , A , A + A , D ,( A + A ) ′′ , A + 3 A ,2 A , A , A + 2 A ,(3 A ) ′′ , 2 A , A , 1 E ( a ) E ( a ), E ( a ), D ( a ), D ( a ), A + A , A D ( a )+ A E E , E ( a ), E ( a ), E ( a ), E ( b ), E ( a ), E ( a ), D ( a ), E ( a ) + A , D ( a ), E , D + A , E ( a ), E ( a ), A + A , A , D , D + A , A + A + A , D ( a ) + A , A + A , A + A , A , A + A , D , A , A + 2 A , 2 A , A , 1 E ( a ) E ( a ), E ( b ), E ( b ), E ( a ), A + 2 A , D ( a )+ A , D ( a )+ A , D ( a ),2 A , A + A , A E ( a ), D ( a ), E ( a ), D ( a ) Table 1.
Normality of special pieces in the exceptional typesNext, it is obvious that P = C if ¯ A ( C ) = 1; the other implication followsfrom [3, Theorem 2.1].Finally, the subregular classes in type B n occur in Thomsen’s list [35, §
9] ofclasses known to have normal closure in any good characteristic. (cid:3)
Remark . Thomsen lists many more classes with normal closures in the classicaltypes, including the subregular class in all types, but it happens that all otherclasses listed by him fall into case (2) of the proposition above.In Table 1, we indicate what is known for special pieces in the exceptional groups.We name a special piece by giving the Bala–Carter label of the special class itcontains. The column labelled “Smooth” lists all special pieces that contain only asingle class (this is easily deduced from, say, the partial order diagram of unipotentclasses in [13, Chapter 13]). Among the remaining special pieces, those with normal
ERVERSE COHERENT SHEAVES AND THE GEOMETRY OF SPECIAL PIECES 29 closure in any good characteristic (following Thomsen [35]) are listed in the nextcolumn, and those known to have normal closure only in characteristic 0 (followingBroer [10] and Sommers [32]) appear in the column after that.Before explaining the last two columns, we remark that in types E and E , thenormality question has not been answered for all nilpotent orbit closures, even incharacteristic 0. However, a number of specific orbits are known to have nonnormalclosures, and Broer, together with Panyushev and Sommers, has conjectured thatall remaining orbits have normal closures (see the Remarks at the end of [11, § Proposition 8.3.
Let G be a simple algebraic group of exceptional type over analgebraically closed field k of good characteristic. (1) If G is of type G , all special pieces are normal. (2) If G is of type F or E and char k = 0 , all special pieces are normal. If char k > , all but two special pieces are known to be normal. (3) If G is of type E (resp. E ), then all but seven (resp. fifteen) special piecesare known to be normal. If char k = 0 and the Broer–Panyushev–Sommersconjecture holds, then all but one (resp. four) special pieces will be knownto be normal. (cid:3) References [1] P. Achar,
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Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803
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