How to glue perverse sheaves

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1 How to glue perverse sheaves A.A. Beilinson The aim o this note [0] is to give a short, sel-contained account o the vanishing cycle constructions o perverse sheaves; e.g., or the needs o [1]. It diers somewhat rom the alternative approaches oered by MacPherson Vilonen [6] and Verdier [8, 9], which justiies, possibly, its publication. The text ollows closely the notes written down by S. I. Gel and in I am much obliged to him. I am also much obliged to J. Bernstein: the construction o the unctor Ψ in n o 2 below was ound in a joint work with him in spring We will consider the algebraic situation only; or notations see [1, 1]. 1. The monodromy Jordan block In this preliminary we will ix the notation concerning the standard local systems on A 1 \ {0} with unipotent monodromy o a single Jordan block Let us start with the classical topology situation, so in this n o, k = C. For an integer i, as usual, Zi := 2π 1 i Z = Z1 i, Z1 = π 1 A 1 \ {0}C, 1. Consider the group algebra Z[Z1]; or l Z1, let l denote l as an invertible element in Z[Z1]. Let I = Z[Z1] t 1, where t is a generator o Z1 be the augmentation ideal. Let A 0 be the I-adic completion o Z[Z1]: A 0 = Z[ t 1], A i = t 1 i A 0 = A 1 i i 0. The graded ring Gr A 0 = i 0 Ai /A i+1 is canonically isomorphic to the polynomial ring Z Z1 Z2. Put A i = {x A i A i = xa 0 } = {x A i x mod A i+1 generates A i /A i+1 = Zi}; one has A i A j = A i+j, so i 0 Ai is a multiplicative system. Let A A 0 be the corresponding localization o A 0 ; one has A = A 0 et 1 = Z t 1. The ring A has a natural Z-iltration A i = [A 1 ] i A 0 = t 1 i A 0 ; or i 0 these A i coincide with the ones above; one has Gr A = i Z Ai /A i+1 polynomial ring. Deine a Z-bilinear pairing, : A A Z 1 by the ormula = Zi, a Laurent, g, t = Res e t=1 g d log t where g g is the canonical involution o the ring A, l = l 1 = l. This pairing is skew-symmetric and Z1-invariant:, g = g,, l, lg =, g ; 1

2 it is compatible with the iltration and non-degenerate: A 1 = A 1, A a,b := A a /A b Hom A b /A a, Z 1 and the pairing induced on Gr A is: S i, S i 1 = 1 i S i S i 1, where S j Zj. Let J be the local system o invertible A-modules on A 1 \{0}C such that the iber o J over 1 A 1 C is A, and the monodromy action o l π 1 A 1 \ {0}C, 1 is the multiplication by l. We have a canonical iltration J i on J, such that J i 1 = A i, J i 1 /J i+1 1 = Zi, together with a non-degenerate skew-symmetric pairing, : J J Z 1, compatible with the iltration, that coincides on J 1 with the above,. Put J a,b = J a /J b. We may consider J = lim J a,b as iltered A-objects in the category { lim local systems on A 1 \ {0}C } or the deinition o lim see the Appendix, A These deinitions have an obvious étale version: just replace Zi above by Z/l n i and repeat 1.1 word-or-word. We get the ring Aét = lim A a,b in lim{sheaves on Spec két } and the Aét -object Jét = lim J a,b ét in lim{sheaves on A 1 \ {0}ét }. In the same way we get Q l - and mixed variants The holonomic counterpart o the above is as ollows. Put A hol = ks, deine the pairing, : A hol A hol k by the ormula A i hol = s i k[[s]]; s, gs = Res s=0 sg sds. This pairing has the same properties as the one above invariance: s, g +, sg = 0. For integers a b let J a,b hol be a D-module on A 1 \ {0} k such that J a,b hol O as an O-module and α = sα dx x or Aa,b hol here x is the parameter on A 1 \ {0} = G m. Put J hol = lim J a,b hol. This is a iltered A hol-object in lim M hol G m. Clearly, Gr J = O Gm s i recall that the Tate twist is the identity unctor in the holonomic situation For any π A 1 we have isomorphisms σ π : J a,b J a+n,b+n n, σ π x = π n x π n, where π = π mod A 2. In the holonomic or in the Q l -situation, we have a canonical choice σ o σ π : in the holonomic case, put σ = σ s = multiplication by s n ; in the Q l -case put σ = σ log t, where t is a generator o Q l 1. In what ollows I will consider the J a,b as ordinary sheaves on A 1 \ {0}, so J a,b lives in MA 1 \ {0}[ 1] DA 1 \ {0}. 2. The unipotent nearby cycles unctor Ψ un Let X be a scheme, and OX a ixed unction. Put Y := 1 0 i X j U := X \ Y, J a,b := U J a,b so, in the holonomic case, J is the D-module generated by s. For any M MU consider M J = lim M J a,b in lim MU. The ring A acts on M J via J, and the pairing, deines a canonical isomorphism DM J = DM J 1 compatible with the A-action. 2

3 2.1. Key Lemma The canonical arrow α: j! M J j M J in lim MX is an isomorphism. Proo. The lemma would ollow i or π A 1 we could ind a certain N 0 and a compatible system o morphisms β a,b : j M J a,b j! M J a,b such that α a,b β a,b = π N = β a,b α a,b where the α a,b are the a, b-components o α; then α 1 = π N lim β a,b. To do this, it suices to show that all ker α a,b, coker α a,b are annihilated by some π m independent o a, b: just take N = 2m and β a,b = β! β in the commutative diagram j! M J a,b j M J a,b β π m coim α a,b = im α a,b π m β! j! M J a,b j M J a,b By duality we may consider coker s only. In the holonomic case the desired act ollows rom the lemma on b-unctions see [2, 3.8], since M J = M s s: take m = lu, where u runs over a inite set o generators o M and lu is the number o integral roots o the b-unction o u. In the constructible case one should use the initeness theorem or the usual nearby cycles unctor RΨ see [4, 3]. Note that or any F in DX we have a distinguished triangle i j F RΨ un F 1 t RΨ un F in DY, where RΨ un is the part o RΨ on which monodromy acts in a unipotent way, and t is a generator o the monodromy group Z l 1. Thereore Cone j! M J a,b j M J a,b = i j M J a,b = Cone RΨ un M J a,b 1 1 t RΨ un M J a,b [ 1], where t acts on RΨ un M J a,b 1 = RΨ un M A a,b via t t. Since A a,b is a Z l 1-module with one generator, the power o 1 t that annihilates RΨ un M annihilates the cone also, and we are done by the initeness theorem or RΨ see [4, 3] Put Π M := j! M J = j M J ; clearly Π : MU lim MX is an exact unctor since so are j! and j, and, deines a canonical isomorphism D Π M = Π D M1. On Π, there are two admissible iltrations: Π! M = j! M J Π M = j M J ; one has Π! Π, Gr Π! M = j! M, Gr Π M = j M. By 2.1, or a b any Π a,b M := Πa M/Π b! M belongs to MX lim MX. Clearly the Π a,b : MU MX are exact unctors, Π = lim D Π a,b M = Π b, a D M1; and see 1.4 we have isomorphisms σ π : Π a,b Π a+n,b+n n. 1 Πa,b ; one has The ollowing constructions are quite parallel to the Lax-Phillips scheme in scattering theory: the multiplication by s is time translation, Π 0!, Π/Π0 are in- and out-spaces, etc. 3

4 2.3. We will need the ollowing particular Π a,b -unctors. The irst is the unctor Ψun or simply Ψ or short, o unipotent nearby cycles, and its relatives Ψ i : Ψ un := Π 0,0, Ψi := Π i,i These take values in MY MX, and we have Ψ i extension unctor, and the corresponding Ξ i : We have canonical exact sequences Ξ := Π 0,1, Ξi := Π i,i+1 σ π Ψ i. D = D Ψ i 1. The second is Ξ, the maximal σ π Ξ i. 0 j! Ma α Ξ a M β Ψ a M 0 0 Ψ a+1 M β+ Ξ a M α+ j Ma 0, which are interchanged by duality. Here α + α = α is the canonical morphism j! j, and β β + = β : Ψ 1 Ψ 0 is the canonical arrow Π 1,1 Π0,0 ; under the isomorphism σ π : Ψ 0 Ψ 1 1 the arrow becomes multiplication by π Vanishing cycles and the gluing unctor Let M X be a perverse shea on X, M U its restriction to U. Consider the ollowing complex, j! M U α,γ Ξ M U M X α +, γ + j M U where α ±, γ ± are the only arrows that coincide with id MU on U. Put Φ M X := kerα +, γ + / imα, γ. Clearly Φ M X is supported on Y and Φ : MX MY is an exact unctor since α is injective and α + is surjective. We have canonical arrows given by the ormulas Ψ 1 M U u Φ M X v Ψ M U uψ = β + ψ, 0, vξ, m = β ξ; clearly v u = β β + = β. Deine a vanishing cycles data or, or -data or short, to be a quadruple M U, M Y, u, v, with M U MU, M Y MY, and Ψ 1 M U u v M Y Ψ M U such that v u = β. The -data orm an abelian category M U, Y in the obvious way. Put F M X := M U, Φ M X, u, v ; clearly this deines an exact unctor F : MX M U, Y. Conversely, let M U, M Y, u, v be -data. Consider the complex Ψ 1 M U β+,u β, v Ξ M U M Y Ψ M U. Put G M U, M Y, u, v = kerβ, v/ imβ +, u : this deines an exact unctor G : M U, Y MX G is exact since β + is mono and β is epi. Call G the gluing unctor. 4

5 3.1. Proposition The unctors MX F M U, Y G are mutually inverse equivalences o categories. Proo. For M X in MX consider as a diad or diads see the Appendix, A.2; this way we may identiy MX with the category o diads o type having the property that both γ + U and γ U are isomorphisms. In the same way, or M U, M Y, u, v M U, Y take the diad ; in this way we identiy M U, Y with the category o diads o type having the property that M Y is supported on Y. Ater this identiication is done, we see that F and G are just the relection unctors; since r r = id, we are done. Here is the simplest case o the above proposition: 3.2. Corollary Let X be a small disk around 0 in the complex line. Then the category MX o perverse sheaves on X with singularities at 0 only is equivalent to the category C o diagrams V v 0 V 1 u spaces such that both operators id V0 u v and id V1 v u are invertible. o vector Proo. For a vector space V and φ End V, let V, φ 0 V be the maximal subspace on which φ acts in a nilpotent way. Consider the category C o diagrams V 0, V 1, φ, u, v, where V 0, V 1 are vector spaces, φ Aut V 1, and V 1, id V 1 φ 0 u V 0 are such that v u = id φ. The category C is equivalent to C v via the unctor V v 0 V 1 V 0, u v 0, V 1, id V1 v u, u, v. u The category M X \ {0}, {0}, where : X A 1, is equivalent to C, since M{0} = {vector spaces}, MU = {vector spaces with an automorphism monodromy}, and, under this identiication, Ψ V, φ = V, id V φ 0. Now apply 3.1. Remark The end o the proo o 2.1 in act shows that Ψ = Ψ un as deined in 2.3 coincides with the standard RΨ un [ 1]; the same is true or Φ. One may recover all RΨM by applying Ψ un to M?, where? runs through the irreducible local systems on a punctured disk. For example, i M is RS holonomic, then the component o RΨDRM that corresponds to the eigenvalue α C o the monodromy is just DRΨ un M a, where exp2πia = α since Ψ un, obviously, commutes with DR. This act was also ound by Malgrange [7] and Kashiwara [5]. See also [3] where the total nearby cycles unctor or arbitrary holonomic modules not necessarily RS was introduced. 5

6 A. Appendix Here some linear algebra constructions, needed in the main body o the paper, are presented. Below, A will be an exact category in the sense o Quillen; as usual,, denote an admissible monomorphism, resp. epimorphism. I C is any category, then C is its dual. A.1. Monads A monad is a complex o the orm P = α P P α+ P+. Denote by HP := ker α + / im α ObA the cohomology o P. The category o monads à is an exact category: the exact sequences in à are the ones which are componentwise exact; it is easy to see that H : à A is an exact unctor. An exact unctor between exact categories induces one between their categories o monads; these unctors commute with H. Also one has à = Ã. Oten it is convenient to represent monads by somewhat dierent types o diagrams. Namely, let Ã1 be the category o objects together with a 3-step admissible iltration and Ã2 be the category o short exact sequences P 1 = γ P 1 1 γ P 0 0 P 1 ; P 2 = L δ,ε A B δ+,ε+ L + such that δ is an admissible monomorphism, ε an admissible epimorphism. These are exact categories in the same way as Ã. Lemma The categories Ã, Ã1, Ã2 are canonically equivalent. Proo. Here are the corresponding unctors: the unctor à Ã1 maps P above to P ker α + P ; the one Ã1 Ã2 maps P 1 above to P 0 γ 0,ε δ +, γ P 1 P 0 /P 0 1 P 1 /P 1 where ε, δ + are the natural projections; inally, Ã2 à maps P 2 above to ker ε A A/δ L. I leave the proo o the lemma to the reader; note that B in the Ã2-avatar is HP. A.2. Diads and the relection unctor A diad in A is a commutative diagram o the orm α A α + Q = C C + β. B β+ Clearly, diads with componentwise exact sequences orm an exact category A # ; one has A # = A # and exact unctors between A s induce ones betwen A # s. As in the case o monads, we may represent diads by other diagrams. Let A # 1 be the category o monads o the orm Q 1 = C A B C + such that the corresponding arrow C A is an admissible monomorphism, and the one A C + is an admissible epimorphism. Let A # 2 be the category o short exact sequences Q 2 = D γ,δ 1,δ2 A B 1 B 2 γ +,δ 1 +,δ2 + D + γ,δ i such that both D A B i are admissible monomorphisms, and both A B i γ +,δ i + D + are admissible epimorphisms. These are exact categories in the same way as A #. 6

7 Lemma The categories A #, A # 1, and A# 2 are canonically equivalent. Proo. The corresponding unctors are: the one A # A # 1 maps Q above to C α, β A B α+,β+ C + and the one A # 1 A# 2 maps Q 1 above to its Ã2-avatar so B 1 = B, B 2 = HQ 1. The irst unctor is obviously an equivalence o categories; as or the second one, this ollows rom Lemma A.1. Note that in the diagram o type A # 2 one may interchange the objects Bi. This deines the important automorphism r o A # 2, and thus o A#, with r r = id A #; call it the relection unctor. Clearly it transorms a diad Q above into A rq = ker α + coker α, HQ where HQ := HC A B C + is the cohomology o the corresponding monad. A.3. Generalities on lim In what ollows I will make no distinction between an ordered set S and the usual category it deines; so or a category C an S-object in C is a unctor S C; i.e. a set o objects C i, i S, and arrows C i C j deined or i j with the usual compatibilities. Let Z denote the set o integers with the standard order, and Π = {i, j i j} Z Z with the order induced rom Z Z. Let A Π be the category o Π-objects in A; this is an exact category in the usual way: a short sequence X i,j Y i,j Z i,j is exact i any corresponding i, j s sequence in A is exact. Say that an object X i,j o A Π is admissible i or any i j k the corresponding sequence X i,j X i,k X j,k is short exact. Let A Π a A Π be the ull subcategory o admissible objects. In any short exact sequence i two objects are admissible then the third one is, so A Π a is an exact category in the obvious way. Clearly A embeds in A Π a as the ull exact subcategory o objects X i,j with X i,0 = 0, X 1,j = 0. Let φ be any order-preserving map Z Z so that lim i ± φi = ±. Then or any Π-object X i,j we have a Π-object φx i,j := X φi,φj ; clearly X φx is an exact unctor that preserves A Π a, and ĩd is the identity unctor. I φ ψ, i.e. φi ψi or any i, then we have an obvious morphism o unctors φ ψ. Deine the category lim A to be the localization o A Π a with respect to the morphisms φx ψx, X A Π a, and φ ψ as above. The natural unctor lim: A Π a lim A is surjective on isomorphism classes o objects. A morphism lim X i,j lim Y i,j is given by a pair α, α, where α: Z Z is an order-preserving unction and α : X i,j Y αi,αj is a compatible system o morphisms; two pairs α, α and β, β give the same morphism i, g give the same maps X i,j Y max{αi,βi},max{αj,βj}. Say that a short sequence in lim A is exact i it is isomorphic to the lim o some exact sequence in A Π a. A routine veriication o Quillen s axioms shows that this way, lim A becomes an exact category. The unctor lim is exact; it deines a aithul exact embedding A lim A. Any exact unctor between A s induces one between their lim s; we also have lim A = lim A. Remarks a. We have Q lim A = lim QA where Q is Quillen s Q-construction; b. I A 0, then lim A is not abelian; c. One can also take lim s along any ordered sets with any inite subset having upper and lower bounds. 7

8 Reerences [0] A. A. Beĭlinson, How to glue perverse sheaves, K-theory, arithmetic and geometry Moscow, 1984, Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp [1], On the derived category o perverse sheaves, K-theory, arithmetic and geometry Moscow, 1984, Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp [2] J. Bernstein, Algebraic theory o D-modules, available at Bernstein-dmod.pd. [3] P. Deligne. Letter to Malgrange rom [4], Théorèmes de initude en cohomologie l-adique, Lect. Notes Math., vol. 569, 1977, pp [5] M. Kashiwara, Vanishing cycle sheaves and holonomic systems o dierential equations, Algebraic geometry Tokyo/Kyoto, 1982, Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp [6] Robert MacPherson and Kari Vilonen, Elementary construction o perverse sheaves, Invent. Math , no. 2, [7] B. Malgrange, Polynômes de Bernstein-Sato et cohomologie évanescente, Analysis and topology on singular spaces, II, III Luminy, 1981, Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp French. [8] Jean-Louis Verdier, Extension o a perverse shea over a closed subspace, Astérisque , Dierential systems and singularities Luminy, [9], Prolongement des aisceaux pervers monodromiques, Astérisque , French. Dierential systems and singularities Luminy, This version o Beilinson s paper was typeset and corrected by Ryan Reich Harvard University on May 10 th,