Notes on Beilinson s How to glue perverse sheaves

Notes on Beilinson s How to glue perverse sheaves Ryan Reich June 4, 2009 In this paper I provide something o a skeleton key to A.A. Beilinson s How to glue perverse sheaves [1], which I ound hard to understand as well as lacking a discussion o some o its more interesting consequences. In order to maintain readability, I will work in the shea-theoretic context o the classical topology on complex algebraic varieties; or the necessary modiications to étale sheaves, one should consult Beilinson s paper: aside rom the shit in deinitions the only change is some Tate twists. For the D-modules case, read Sam Lichtenstein s undergraduate thesis, [2]. 1 Theoretical preliminaries The topic at hand is perverse sheaves and nearby cycles; I give a summary o the deinitions and necessary properties here. 1.0 Diagram chases Especially in 3, we will need to do some pretty serious diagram chasing. Some people make a big deal about how you can t chase diagrams in a general category because the objects aren t sets, but in act in an abelian category (or even a Quillen-exact category, i one wishes to adapt the results in 3 to the more general context in which Beilinson stated them in [1]) chasing elements is perectly possible. The important act is that an object in an abelian category A can be considered to be a shea on the canonical topology o A. This is, by deinition, the largest Grothendieck topology in which all representable unctors Hom A (, x) are sheaves, and its open covers are precisely the strict universal epimorphisms. Such a map is, in a more general category, a map : u x such that the ibered product u = u x u exists, the coequalizer x = coker(u u) exists, the natural map x x is an isomorphism, and that all o this is also true when we make any base change along a map g : y x, or the induced map x id: u x y y. In an abelian category, however, this is all equivalent merely to the statement that is a surjection. Recall the deinitions o the various constructions on sheaves: 1. Kernels o maps are taken sectionwise; i.e. or a map : F G, ker()(u) = ker((u): F(U) G(U)). Likewise, products and limits are taken sectionwise. 2. Cokernels are locally taken sectionwise: any section s coker()(u) is, on some open cover V o U, o the orm t or t G(V ). Likewise, images, coproducts, and colimits are taken locally. In an abelian category, where all o these constructions exist by assumption, these descriptions are even prescriptive: i one orms the sheaves thus described, they are representable by the objects claimed. Thereore, the ollowing common arguments in diagram chasing are valid: 1. A map : x y is surjective i and only i or every s y, there is some t x such that s = (t). This is code or: or every open set U and every s y(u), there is a surjection V U and a section t y(v ) such that s V = (t). 2. I s y, then s = 0 coker() i and only i s im(). This is code or: i s y(u) and s = 0 coker()(u), then there is some surjection V U and t x(v ) with s V = (t). 1

3. is zero i and only i or all s x, (x) = 0. This is actually the deinition o the zero map. Any other arguments involving elements and some concept related to exactness can also be phrased in this way. Thus, a naïve diagram-chasing argument can be converted into a rigorous one simply by replacing statements like s F with correct ones s F(U) or some open set U, and passing to surjective covers when necessary. The process being mechanical, I will just give the naive proo. 1.1 Derived category and unctors All the action takes place in the derived category; speciically, let X be an algebraic variety and denote by D(X) its derived category o bounded complexes o sheaves with constructible cohomology. By deinition, a map o complexes : K L in D(X) is an isomorphism i its associated map on cohomology sheaves H i (): H i (K ) H i (L ) is or all i. We have a notation or the index-shit: K i+1 = (K[1]) i (technically, the dierential maps also change sign, but we will never need to think about this). The derived category D(X) is a triangulated category, which means merely that in it are a class o triples o complexes and maps A B C A [1] in which two consecutive arrows compose to zero, satisying some axioms we won t need, and with the property that the associated long sequence o cohomology sheaves... H 1 (C ) H 0 (A ) H 0 (B ) H 0 (C ) H 1 (A )... is exact (note that H 0 (A [1]) = H 1 (A )); we say that the H i are cohomological. I : A B is given, there always exists a (unique up to non-unique isomorphism) triangle whose third term C = Cone() is the cone o. A unctor between two triangulated categories is triangulated i it sends triangles in one to triangles in the other. In D(X) we also have some standard constructions o shea theory. For any two complexes there is the total tensor product A B obtained by taking in degree n the direct sum o all products A i B j with i + j = n (and some dierentials that are irrelevant) and its derived unctor A L B, with H i (A L B ) = Tor i (A, B ), which is a triangulated unctor. We also have the unctor Hom(A, B ), where Hom(A, B ) i (U) = Hom(A U, B [i] U ), and its derived unctor R Hom(A, B ), with H i R Hom(A, B ) = Ext i (A, B ), which is triangulated. O course, these two have an adjunction: R Hom(A L B, C ) = R Hom(A, R Hom(B, C )). For any Zariski-open subset U X with inclusion map j, there are triangulated unctors j!, j : D(U) D(X) and j = j! : D(X) D(U); i i is the inclusion o its complement Z, then there are likewise maps i!, i : D(X) D(Z) and i = i! : D(Z) D(X). (Again, technically j! and j are derived unctors and should be denoted Lj! and Rj, but we will never have occasion to use the plain versions so we elide this extra notation.) They satisy a number o important relations, o which I will only use one here: there is a natural triangle or any complex A j! (A ) j (A ) i i j (A ) There is also a triangulated duality unctor D: D(X) D(X) op which interchanges! and, in that Dj (A ) = j! (DA ), etc., and is an involution In act, i we set D = DC, then D(A ) = R Hom(A, D ). The only thing we need to know about D is that it exchanges! and ; i.e. that Dj! F = j DF, etc. For any map : X Y o varieties, we have!, as well (also!,, and none o them are equal), with the same relationships to D, and the useul identity! R Hom(F, G) = R Hom( F,! G). 2

1.2 Perverse sheaves Within D(X) there is an abelian category M(X) o perverse sheaves which has nicer properties than the category o actual sheaves. It is speciied by means o a t-structure, namely, a pair o ull subcategories p D(X) 0 and p D(X) 0, also satisying some conditions I won t use, and such that M(X) = p D(X) 0 p D(X) 0. There are truncation unctors τ 0 : D(X) p D(X) 0 and likewise or τ 0, itting into a distinguished triangle or any complex K : τ 0 K K τ >0 K (where τ >0 = τ 1 = [ 1] τ 0 [1]). This triangle is unique with respect to the property that the irst term is in p D(X) 0 and the third is in p D(X) >0. They have the obvious properties implied by the notation: τ a τ b = τ a i a b, and likewise or τ?. Furthermore, there are perverse cohomology unctors p H i : D(X) M(X), where o course p H i (A ) = p H 0 (A [i]) and p H 0 = τ 0 τ 0 = τ 0 τ 0 ; these are cohomological just like the ordinary cohomology unctors. The abelian category structure o M(X) is more or less determined by the act that i we have a map : F G o perverse sheaves (this is the notation I will be using; we will not think o perverse sheaves as complexes), then ker = p H 1 Cone() coker = p H 0 Cone(). I X is connected o dimension D, then the category M(X) is closed under a translate o the duality unctor, D[d], but not necessarily under the six unctors deined or an open/closed pair o subvarieties. However, it is true that j (F), i! (F) p D 0 and j! (F), i (F) p D 0, while j (F), i (F) M; we say these unctors are right, let, or just t-exact. Furthermore, when j is an aine morphism (the primary example being when Z is a Cartier divisor), both j! and j are t-exact, and thus their restriction to M(U) is exact with values in M(X). There is also a middle extension unctor j!, deined so that j! (F) is the image o p H 0 (j! F) in p H 0 (j F) along the natural map j! j ; it is the unique perverse shea such that i j! F p D <0 and i! j! F p D >0, but or me the most useul property is that when j is an aine, open immersion, then we have a sequence o perverse sheaves i j! F[ 1] j! F j! F j F i! j! F[1]; i.e. i j! F[ 1] = ker(j! F j F) and i! j! F[1] = coker(j! F j F) are both perverse sheaves. Perverse sheaves have good category-theoretic properties: M(X) is both artinian and noetherian, so every perverse shea has inite length. Finally, we will use the shea-theoretic act that i L is a locally constant shea on X, then F L is perverse whenever F is. Note that since L is locally ree, it is lat, and thereore F L = F L L. 1.3 Nearby cycles I we have a map : X A 1 such that Z = 1 (0), the nearby cycles unctor RΨ : D(U) D(Z) is deined. Namely, let u: G m G m be the universal cover o G m = A 1 \ {0}, let v : Ũ = U G m Gm U be its pullback, orming a diagram Z i X j U {0} A 1 G m v u Ũ G m and set RΨ (A ) = i Rj Rv v A : D(U) D(Z). 3

Since i and v are exact, indeed we have RΨ = R(i j v v ) is the right derived unctor o a let-exact unctor Ψ rom sheaves on U to sheaves on Z. The undamental group π 1 (G m ) acts on any j F via deck transormations o G m and thereore acts on Ψ and RΨ. There is a natural map, obtained rom (v, v )-adjunction, rom i F Ψ (j F), on whose image π 1 (G m ) acts trivially. We set, by deinition, i F Ψ (j F) Φ (F) 0 where Φ (F) is the vanishing cycles shea, and or the generator t π 1 (G m ) we get a map Var(t): Φ (F) Ψ (F) actoring 1 t on Ψ (F) through the quotient. Using some homological algebra tricks the above sequence induces a natural distinguished triangle i A RΨ (j A ) RΦ (A ) where RΦ is (morally) the right derived unctor o Φ. I beore taking the derived unctor we single out the subshea Ψ un (F) on which 1 t is locally nilpotent (in act, by the constructibility theorem o SGA7 it is actually nilpotent), we get the unctor o unipotent nearby cycles. The unit o (v, v )-adjunction i F Ψ (j F) actors through Ψ un (F) since π 1(G m ) acts trivially on i F, and the sequence i F Ψ un (j F) 1 t Ψ un (j F) induces a distinguished triangle (replacing F with the complex j A ) i j A RΨ un (A ) 1 t RΨ un (A ) (1) I L is any locally constant shea on G m with underlying vector space L, then RΨ un (A L) = RΨ un (A ) L, where t acts on the tensor product by acting on each actor (this ollows since L is trivialized on G m ). 2 The unipotent nearby cycles unctor We start o with a nice result about nearby cycles. Theorem 2.1. The unctor RΨ un [ 1] sends p D(U) 0 to p D(Z) 0 and takes M(U) to M(Z). Proo. Since j is aine and an open immersion, j and j! are t-exact, so or any A p D(U) 0, i j K = i Cone(j! A j A ) is in p D(Z) 0. I we apply the long exact sequence o perverse cohomology to equation (1), we thereore get H 3 (RΨ un A ) 1 t H 3 (RΨ un A ) H 2 (i j A ) H 2 (RΨ un A ) 1 t H 2 (RΨ un A ) H 1 (i j A ) H 1 (RΨ un A ) 1 t H 1 (RΨ un A ) H 0 (i j A ) 0 H 0 (RΨ un A ) 1 t H 0 (RΨ un A ) 0 For i 1, the map H i (RΨ un M) Hi (RΨ un M) is both given by a nilpotent operator and is injective, thereore zero. For i = 0 it is nilpotent and surjective, so also zero. It ollows that RΨ un (A ) p D(Z) <0, as promised. Now let M M(U) be a perverse shea. Then i j M p D(Z) [ 1,0] since it consists o the kernel and cokernel o the map j! M j M. This means that or i 2 in the above sequence, all the maps 1 t are injective and nilpotent, hence zero. Thus RΨ un (M) p D(Z) 1, as desired. We proceed to give a construction o RΨ un. Let La be the vector space o dimension a 0 together with the action o a matrix J a = [δ ij + δ i,j 1 ], a unipotent Jordan block o dimension n. Let L a be the locally constant shea on G m with underlying space L a and monodromy action where a generator t o π 1 (G m ) acts by J a ; we consider it a complex in cohomological degree zero. Lemma 2.2. Let A D(U); then D(A L a ) = D(A ) L a. 4

Proo. First we prove a related act: i we set Ľa = Hom(L a, C), then there is an isomorphism Ľa = L a. Indeed, L a has a given basis e 1,..., e a associated to J, and we ix the basis e a,..., e 1 o dual vectors or Ľa, sending J to J t, and thus inducing the desired map o local systems. In general, then, we compare: D(F) L a = R Hom(F, D ) L L a D(F L a ) = R Hom(F L L a, D ) = R Hom(F, DL a ). There is a natural map rom the ormer to the latter induced by ( L, R Hom)-adjunction: R Hom(F, D ) R Hom( L a, R Hom(F, D L a )) = R Hom(F, R Hom( L a, D L a )). Note that D =! D (one on X, the other on G m ) by deinition and D L a =! DL a. By the property o!, we have R Hom( L a,! DL a ) =! R Hom(L a, DL a ) and thereore the desired map is the one induced on R Hom(F, ) rom the one obtained by applying! to a certain map D R Hom(L a, DL a ) = D(L a L L a ). This map, in turn, is obtained by irst replacing the L with (since L a is locally ree), replacing one copy o L a with its dual, and applying D to the trace map L a Ľa C. The map D(F) L a D(F L a ) is an isomorphism locally since it would be an isomorphism i L a were replaced with a constant shea. Thus, it is an isomorphism. For each a, b 0 there is a natural exact sequence 0 L a ga,b L a+b ga+b, b L b 0; (2) that is, or any r Z, g a,r sends L a to the irst r coordinates o L a+r i r 0, and to the quotient L a ( r) given by collapsing the irst r coordinates i r 0 (that is, r 0). This sequence respects the action o J? on the terms and under the identiication o Lemma 2.2, the (a, b) sequence is dual to the (b, a) sequence; that is, Dg a,r = g a+r, r. These have various ormal identities: 1. When r 0, we have g a+r, a g a,r = 0; 2. When r 0 and a + r 0, we have g a+r, r g a,r = (1 t) r ; 3. When r and s have the same sign, and a + r 0, we have g a,r+s = g a+r,s g a,r. Let M M(U), so M L a is also perverse and has an action o the monodromy via J a acting on the second actor, which we will call simply t. The maps id g a,r between these will be called g a,r M, or just ga,r again i there is no need to mention M, and we have Dg a,r M = ga+r, r DM. Note that since the L a are locally ree, the g a,r M are all injective when r 0 and surjective when r 0. We can prove a strong uniormity lemma based on the monodromy action: Lemma 2.3. Let α a : j! (M L a ) j (M L a ). annihilates both ker α a and coker α a or all a. There exists an integer N such that (1 t) N 5

Proo. We consider again the triangle (1); i we could ind such an N annihilating Ψ un (M L a ), then it would annihilate i j (M L a ) as well, and hence its perverse cohomologies ker α a and coker α a. We know that Ψ un (M L a ) = Ψ un (M) La and that 1 t is nilpotent on Ψ un (M), say (1 t)n = 0, so it suices to show that this power kills the tensor product as well. Indeed, the short exact sequences (2) with (a, b) = (1, a 1): 0 L 1 L a L a 1 0 induce short exact sequences o perverse sheaves respecting the monodromy action: 0 Ψ un Thus the claim is proved by induction. (M) Ψ un (M) L a Ψ un (M) L a 1 0. Corollary 2.4. There exists a map β a : j! (M L a ) j (M L a ) such that α a β a = β a α a = (1 t) 2N. Proo. Consider the diagram j! (M L a ) j (M L α! a ) β coim α a = im α a (1 t) N β! α j! (M L a ) j (M L a ) (1 t) N where α α! = α a, and set β a = β! β. Note that β! and β exist and β! α! = α β = (1 t) N because (1 t) N annihilates ker α a and coker α a, respectively. We claim that α! β! = β α = (1 t) N as well. Indeed, consider the latter and apply α to both sides: α β α = (1 t) N α = α (1 t) N, since α a, and hence α and α!, respect the monodromy action. Since α is injective, β α = (1 t) N, as desired. Likewise or α! β!, applying it to α! and using surjectivity. Thereore, α a β a = α α! β! β = α (1 t) N β = (1 t) N α β = (1 t) 2N. Likwise or β a α a. Let r Z with a + r 0, and set α a,r = j (g a,r ) α a = α a+r j! (g a,r ), so α a,r : j! (M L a ) j (M L a+r ). Setting β a,r = j! (g a+r, r ) β a+r = β a g (g a+r, r ), we can generalize Corollary 2.4: Corollary 2.5. We have: 1. When r 0, α a,r β a,r = (1 t) 2N+r ; 2. When r 0, β a,r α a,r = (1 t) 2N+r ; 3. For any r, β a+2r, r α a,r = (1 t) 2(N+r) j! (g a,2r ) and α a+r,r β a+r, r = (1 t) 2(N+r) j (g a,2r ). Proposition 2.6. For a 0, the natural maps j! (g a,1 ) and j (g a+r,1 ) respectively induce isomorphisms ker(α a,r ) ker(α a+1,r ) and coker(α a,r ) coker(α a+1,r ). When r 0, the ormer are isomorphisms or all a, and when r 0, the latter are. These statements also hold with α replaced by β and r negated. Proo. The latter statement is obvious: when r 0, g a,r and hence j (g a,r ) is injective, so α a,r has the same kernel as α a ; likewise or r 0 and j! (g a,r ) being surjective. 6

Suppose r 0; then, applying Corollary 2.5, we have β a,r α a,r = (1 t) 2N+r, so (1 t) 2N+r annihilates ker(α a,r ). Likewise, when r 0, α a,r β a,r = (1 t) 2N+r, so (1 t) 2N+r annihilates coker(α a,r ). Using the natural maps j! (g a,1 ) and j (g a+r,1 ) we get a commutative square j! (M L a ) α a,r j (M L a+r ) j! (g a,1 ) j! (M L a+1 ) j (g a+r,1 ) α a+1,r j (M L a+r+1 ) showing that j! (g a,1 ) induces a map on kernels and j (g a+r,1 ) on cokernels. Since the j! (g a,1 ) are injective, we get a long sequence o inclusions o kernels: ker α a 1,r ker α a,r ker α a+1,r. Each kernel is annihilated by (1 t) 2N+r, whose kernel is (or a 2N +r) the perverse shea j! (M L 2N+r ). Since perverse sheaves have inite length, this chain must have a maximum, whence the claim or kernels. For cokernels, they are all quotients j (M L 2N+r ), and stabilized since their kernels are a similar increasing subsequence. To obtain the results or β, just swap it with α in all the arguments and negate r; we have only used Corollary 2.5, which is symmetric under this transormation. Departing slightly rom Beilinson s notation, we denote these stable kernels ker α,r or r 0 and coker α,r or r 0; when r = 0 we drop it. Lemma 2.7. Suppose that in Proposition 2.6 the stable cokernels coker(α a,r ), coker(β a,r ) obtain or a L. Then in this range the ollowing natural maps are isomorphisms: im(α a+1,r )/ im(α a,r ) j (M L a+1+r )/j (M L a+r ) im(β a+1,r )/ im(β a,r ) j! (M L a+1 )/j! (M L a ). In particular, (α a,r ) 1 (im(1 t) n ) im(1 t) n and the same or β, or n a L. Proo. This is o course merely a general algebraic act: suppose we have objects A 1 A 2 in an abelian category and subobjects B i A i with B 1 B 2. Then the ollowing three conditions are equivalent: 1. The inclusion B 2 A 2 induces an inclusion B 2 /B 1 A 2 /A 1 ; 2. The inclusion A 1 A 2 induces an inclusion A 1 /B 1 A 2 /B 2 ; 3. The natural map B 1 B 2 A1 A 2 is an isomorphism. Furthermore, B 2 /B 1 A 2 /A 1 i and only i A 1 /B 1 A 2 /B 2. For the proo, assume by symmetry that the irst map is injective (resp. surjective), and chase the ollowing diagram: 0 0 0 B 1 A 1 A 1 /B 1 0 0 B 2 A 2 A 2 /B 2 0 B 1 /B 1 A 2 /A 1 0 0 7

Proposition 2.8. For r 0, there is an isomorphism coker α,r = ker α, r. Proo. Starting with the map β a o Corollary 2.4, and or a M, let γ a,r = (1 t) a 2N β a+r, so γ a : j (M L a+r ) j! (M L a+r ). By deinition o α a,r, we have β a α a,r = (1 t) 2N j! (g a,r ), so γ a,r α a,r = j! (g a,r ) (1 t) a = 0 since (1 t) a annihilates L a. Likewise, α a+r, r β a = (1 t) 2N j (g a+r, r ), so α a+r, r γ a,r = 0 or the same reason. So, in act γ a,r : coker α a,r ker α a+r, r. The kernel o γ a,r is by deinition (β a+r ) 1 im(1 t) 2N+r mod im α a,r, but since α a,r β a,r = (1 t) 2N+r, to show that γ a,r is injective it suices to show that (β a+r ) 1 im(1 t) 2N+r im(1 t) 2N+r, which ollows rom Lemma 2.7 when a 0. For surjectivity, using the act that (1 t) 2N r annihilates ker(α a+r, r ) (so ker(α a+r, r ) im(1 t) a 2N ), a quick diagram chase shows that it suices that (α a+r, r ) 1 im(1 t) 2N im(β a+r ), or since β a+r α a+r = (1 t) 2N, that (α a,r ) 1 im(1 t) 2N im(1 t) 2N, which is again true or a 0 by Lemma 2.7. Because they are equal, we will give a single name to the stable kernel and cokernel: Π r (M) = ker(α, r ) = coker(α,r ). The deinitions o α a and g a,r easily imply that DΠ r (M) = Π r (DM). Theorem 2.9. There is an isomorphism RΨ un (M)[ 1] = Π 0 (M), so that RΨun commutes with D. Proo. This is really just a more speciic version o the argument in Lemma 2.3. Consider again the triangle (1) with A = M L a. Since we know that RΨ un (M)[ 1] is a perverse shea and that i j (M L a ) p D [ 1,0] (X), it suices to show that ker(1 t) = RΨ un (M)[ 1]. Since RΨ un (M L a )[ 1] = RΨ un (M)[ 1] L a = a i=1 RΨ un (M)[ 1] (a), where the i th coordinate o t is t (i) + t (i+1). Thus, i c i is the i th coordinate (unction) o ker(1 t), we have c i+1 = (1 t 1 (i) )c i, or by induction, c i = (1 t 1 ) i 1 c 1 and (1 t 1 ) a c 1 = 0. Since 1 t is nilpotent on RΨ un (M), or a suiciently large, this ormula or c i in act deines ker(1 t) to be the isomorphic image o RΨ un (M)[ 1], as desired. Note that by rearranging some deinitions (and doing the completely analogous proo or the cokernel), we could have given Theorem 2.9 right ater Lemma 2.2 and proven the r = 0 cases o Proposition 2.6 and Proposition 2.8 as a side eect. However, we need the r = 1 case or the gluing constructions. Note also the corollary: Corollary 2.10. Suppose (1 t) N annihilates RΨ un (M). Then we have (and conversely, o course) RΨ un (M)[ 1] = i j! (M L N )[ 1] = i! j! (M L N )[1]. Nearby cycles and the perverse t-structure One thing Beilinson does not at all address in his paper is the interaction o RΨ un with the entire perverse t-structure. However, this an easy consequence o two general lemmas. To this end, let T be a triangulated unctor on a t-category C and let the core o C be the abelian category A. We will assume that the objects o C are bounded above, meaning that C = b Z C b, and that C is nondegenerate, meaning that b Z C b = 0. These assumptions hold or the perverse t-structure. Lemma 2.11. Suppose T is right t-exact and that T A A; then T is t-exact. 8

Proo. We will show that T commutes with all truncations. Suppose we have an object x C b, so that there is a distinguished triangle τ <b x x τ b x where by deinition, τ b x = H b (x)[ b] A[ b]. By hypothesis on T, we have T (x) C b, T (τ <b x) C <b, and T (H b x[ b]) A[ b] C b. Since T is triangulated, there is a triangle T (τ <b x) T (x) T (H b x[ b]) and thereore, by uniqueness o the truncation triangle, it must be that T (τ <b x) = τ <b T (x). This is under the hypothesis that x C b ; since then τ <b x C b 1 and since τ <b 1 τ <b = τ <b 1, we can apply truncationsby-one repeatedly and conclude that or all n, we have τ n T (x) = T (τ n x). Now suppose we have any x, and or any n orm the distinguished triangle τ <n x x τ n to which we apply T. Since T (τ <n x) = τ <n T (x), the cone o the resulting triangle τ <n T (x) T (x) T (τ n x) must be isomorphic to τ n T (x), by uniqueness o cones and the truncation triangle or T (x). τ n T (x) = T (τ n x). Since then T commutes with all trunctions, it is a ortiori t-exact. Thus, Lemma 2.12. Suppose that T is t-exact and that T vanishes on A; then T = 0. Proo. T commutes with cohomology, so or any x we have H n (T x) = T H n (x) = 0. Thus, by nondegeneracy, we must have T (x) = 0. In the irst lemma, take T = i RΨ un [ 1]; by Theorem 2.1, it satisies the hypothesis o Lemma 2.11, and thereore is t-exact. By deinition, the Verdier duality unctor D is (contravariantly) t-exact or the perverse t-structure, so taking T = D i RΨ un [ 1] i RΨ un [ 1] D in Lemma 2.12 we get T = 0 on all o D(X) by Theorem 2.9. That is: Theorem 2.13. The unipotent nearby cycles RΨ un 3 Vanishing cycles and gluing commute with Verdier duality. Since Π 0 = RΨ un [ 1], we will reer to it simply as Ψ (M), and Π 1 as Ξ, what Beilinson calls the maximal extension unctor. Proposition 3.1. There are two natural exact sequences exchanged by duality and M DM: where α + α = α and β β + = 1 t. 0 j! (M) α Ξ (M) β Ψ (M) 0 0 Ψ (M) β+ Ξ (M) α+ j (M) 0, Proo. Since α a,r = α a+r j! (g a,r ) we have im(α a,r ) im(α a+r ) and thus we obtain a natural surjection coker(α,r ) coker(α ). The kernel o this map is im(α a,r )/ im(α a ) or a 0; this receives a natural surjection, via α a+r, rom coker(1 t) r inside j! (M L a+r ), which is injective by Lemma 2.7. In particular, or r = 1 we get the irst short exact sequence. Likewise, since α a+r, r = j (g a+r, r ) α a+r, we have ker(α ) ker(α, r ), where the quotient is ker(α a+r, r )/ ker(α a+r ) or a 0; it maps, via α a+r, injectively to ker(g a+r, r ) = im(1 t) a, and surjectively by Lemma 2.7. For r = 1 we get the second short exact sequence. 9

The duality claim ollows rom the act that D(α a,r M ) = αa+r, r DM and that D is contravariant and exact. Since j Ψ = 0 and j (α a ) = id, we conclude that j (α + α ) = id as well; by adjunction, α + α = α. Observe that the constructions above take place in j!, (M L a+1 ). Recall the isomorphism γ a,r = (1 t) a 2N β a+r o Proposition 2.8; what we must really prove is that β (γ a,1 ) 1 β + = (1 t)(γ a+1,0 ) 1. First we show that β + γ a+1,0 β = (1 t)γ a,1, which is immediately obvious rom the deinition o γ a,r (note that the symbolic ormula given above is not the isomorphism o coker(α a,r ) with ker(α a+r, r ), but that this isomorphism is canonically induced by it. This induction is precisely the contribution o the β ± ). Thereore, multiplying on the let by β (γ a,1 ) 1, we get: β (γ a,1 ) 1 β + γ a+1,0 β = β (1 t) = (1 t) β. Since β is surjective, we may remove it rom the right, and we are done ater multiplying by (γ a+1,0 ) 1. The remainder o the paper is simply what Beilinson calls linear algebra. Let M = j F or a perverse shea F M(X). From the maps in these two exact sequences we can orm a complex: j! j (F) (α,γ ) Ξ j (F) F (α+, γ+) j j (F), (3) where γ : j! j (F) F and γ + : M j j (F) are deined by the let- and right-adjunctions (j!, j ) and (j, j ) and the property that j (γ ) = j (γ + ) = id. Proposition 3.2. The complex (3) is in act a complex; let Φ (F) be its cohomology shea. Then Φ is an exact unctor whose values are supported on Z and there are maps u, v such that v u = 1 t as in the ollowing diagram: Ψ j (F) u Φ (F) v Ψ j (F). Proo. That (3) is a complex amounts to showing that γ + γ = α = α + α, which is true by deinition o the γ ± and adjunction. To show that it is exact, suppose we have 0 F 1 F 2 F 3 0, so that we get a short exact sequence o complexes 0 C (F 1 ) C (F 2 ) C (F 3 ) 0, where by C (F) we have denoted the complex (3) padded with zeroes on both sides. Note that since α is injective and α + surjective, C (F) ails to be exact only at the middle term. Thereore we have a long exact sequence o cohomology sheaves: (0 = H 1 C (F 3 )) Φ (F 1 ) Φ (F 2 ) Φ (F 3 ) (0 = H 1 (C (F 1 )))... which shows that Φ is unctorial and an exact unctor. I we apply j to (3), it becomes simply j F (id,id) j F j F (id, id) j F which is actually exact, so j Φ (F) = 0; i.e. Φ (F) is supported on Z. Finally, to deine u and v, let pr: Ξ j (F) F Ξ j (F), and set u = (β +, 0) in coordinates, and v = β pr. Then v u = β β + = 1 t by Proposition 3.1. Deine a vanishing cycles gluing data or to be a quadruple (F U, F Z, u, v) modeled on Proposition 3.2; or any F M(X), we have F (F) = (j F, Φ (F), u, v) as there. Let M (U, Z) be the category o such data, and F : M(X) M (U, Y ) is a unctor. Conversely, given a vanishing cycles data, we can orm the complex Ψ (F U ) (β+,u) (β, v) Ξ (F U ) F Z Ψ (F U ) (4) since v u = 1 t = β β +, and let G (F U, F Y, u, v) be its cohomology shea. We could prove Theorem 3.6 directly, but some o the ormalism in Beilinson s paper is too elegant to cut. Thereore, we make two urther deinitions (which are a little dierent rom the terminology he uses): 10

Deinition 3.3. Let a diad be a complex o perverse sheaves on X o the orm ( ) D (L) = (a L,b L ) = F L A B (R) = (a R,b R ) F R in which a L is injective and a R is surjective (so it is exact on the ends). Let the category o diads be denoted M 2 (X). Let a triad be a short exact sequence o the orm S = (c,d (0 1 F,d2 ) A B 1 B 2 (c+,d1 +,d2 + ) ) F + 0 in which both (c, d i ): F A B i are injections and (c +, d i +): A B i F + are surjections. Let the category o triads be denoted M 3 (X); it has a natural relection unctor r : M 3 (X) M 3 (X) which invokes the natural symmetry 1 2, and is an involution. We can deine a map T : M 2 (X) M 3 (X) by setting ( ) T (D) = ker(r) (ι A,ι B,h) A B H(D ) (π A,π B, k) coker(l), where: 1. ι = (ι A, ι B ): ker(r) A B is the natural inclusion, 2. π = (π A, π B ): A B coker(l) is the natural projection, 3. h: ker(r) H(D ) is the natural projection onto H(D ) = ker(r)/f L, 4. k : H(D ) coker(l) is the natural inclusion o H(D ) = ker(coker(l) F R ). These our maps are related: we have πι = kh. Deine the inverse T 1 by the ormula T 1 (S) = (ker(d 2 ) (c,d1 ) ) A B 1 coker(c, d 1 ). Lemma 3.4. The unctors T, T 1 are mutually inverse equivalences o M 2 (X) with M 3 (X). Proo. First we have to show that T (D ) M 3 (X) at all. The conditions that (ι A, ι B ) = ι be injective and (π A, π B ) = π be surjective are true by deinition. To show that (ι A, h) is injective we do a diagram chase: i x ker(r) and ι A (x) = h(x) = 0, then x im(l): x = (a L (y), b L (y)) with y F L, and a L (y) = ι A (x) = 0. But a L is injective, so y = 0 and hence x = 0. Likewise, we show that (π A, k) is surjective: suppose that x coker(l), and pick a lit (y 1, y 2 ) A B, with π A (y 1 ) + π B (y 2 ) = x. Since a R is surjective, there is some z A such that a R (z) = b R (y 2 ), and thereore (R)( z, y 2 ) = 0: ( z, y 2 ) ker(r). By deinition, kh( z, y 2 ) = π( z, y 2 ) = π B (y 2 ) π A (z), so x = π A (y 1 + z) kh(z, y 2 ), as desired. Now we have to check exactness in the middle. First o all, T (D ) is a complex, since π ι = k h. We prove exactness: suppose (x, y) ker(π, k), so π(x) = k(y) coker(l), with x A B and y H(D ). Pick z ker(r) such that y = h(z), so π(x) = k(y) = πι(z); thus, x ι(z) ker(π) = im(l) ker(r) and thus x ker(r), or more precisely, there is w ker(r) such that x = ι(w). Also, k(y) = π(x) = πι(w) = kh(w), so since k is injective, y = h(w); thus, (x, y) = (ι, h)(w), as desired. Likewise, we must check that T 1 (S) M 2 (X). That it is a complex is obvious, so we must show that c is injective on ker(d 2 ) and that A alone surjects onto coker(c, d 1 ). The ormer ollows rom the act that (c, d 2 ) is assumed to be injective. For the latter, let (x 1, x 2 ) A B 1 ; since (c +, d 2 +) is surjective, there is some (y 1, y 3 ) A B 2 such that c + (y 1 ) + d 2 +(y 3 ) = d 1 +(x 2 ). That is, (y 1, x 2, y 3 ) ker(c +, d 1 +, d 2 +), and hence in im(c, d 1, d 2 ). In particular, (y 1, x 2 ) im(c, d 1 ), so (x 1, x 2 ) and (x 1 + y 1, 0) have the same image in coker(c, d 1 ), as desired. It is easy to check that T 1 T = id: its F L is ker(h) = im(l) = F L ; its A and B are indeed A and B, and its F R is coker(ι) = F R ; one checks quickly that the maps are right as well. Conversely, T T 1 = id: 11

its F is ker(a B 1 coker(c, d 1 )) = F since (c, d 1 ) is an injection; its A and B 1 are obviously the original A and B 1 ; we will deal at greater length with B 2 and F +. For B 2, we must show that F / ker(d 2 ) = B 2, or in other words, that d 2 is surjective. This is another diagram chase: i x B 2, then since (c +, d 1 +) is surjective there is some (y 1, y 2 ) A B 1 such that c + (y 1 ) + d 1 +(y 2 ) = d + 2 (x). Thereore (y 1, y 2, x) ker(c +, d 1 +, d 2 +) = im(c, d 1, d 2 ), so in particular x = d 2 +(z), or x = d 2 +( z), or some z F, as desired. For F +, we must show that the smaller sequence 0 ker(d 2 ) (c,d1 ) A B 1 (c+,d1 + ) F + 0 is still exact. The hypotheses on S already show that it is exact at the ends, so we deal only with the middle. First o all, it is a complex, since c + c + d 1 +d 1 = d 2 +d 2 = 0 on ker(d 2 ). Now we do still another diagram chase: i x ker(c +, d 1 +), then (x, 0) ker(c +, d 1 +, d 2 +), so there is some y F such that (c (y), d 1 (y)) = x and d 2 (y) = 0; i.e. y ker(d 2 ), as desired. Corollary 3.5. The relection unctor on a diad T 1 rt (D ) is the complex r(d ) = ( ) ker(a R ) (a L,b L ) A H(D ) (a R,b R ) coker(a L ), where a L is the natural inclusion and a R the natural projection, b L = h (a L, 0), and b R actorizes k through coker(a L ) coker(l). Proo. The d 2 o rt (D ) is ι B, whose kernel (in ker(r)) we show is ker(a R ). Indeed, the map (id, 0) exhibits ker(a R ) ker(ι B ) ker(r); conversely, i (x 1, x 2 ) ker(ι B ) ker(r), then x 2 = 0 and a R (x 1 ) = 0, so x 1 ker(a R ). Likewise, the d 1 o rt (D ) is h, so we must show that coker(ι A, h) = coker(a L ). The map (id, 0) rom A to A H(D ) induces a map A coker(ι A, h); to show that it descends to coker(a L ), we must show that or x F L, (a L (x), 0) = (ι A (y), h(y)) or some y ker(r). In particular, y im(l) since h(y) = 0, so we can take y = (L)(x) = (a L (x), b L (x)). Thus we have a map coker(a L ) coker(ι A, h), which this argument also shows is injective. It is also surjective: i (x 1, x 2 ) A H(D ), then there is some y ker(r) with h(y) = x 2, so (x 1 ι(y), 0) has the same image in coker(ι A, h). The injectivity argument just given shows that coker(a L ) coker(l), so the description o b R makes sense. Indeed, k does actor through coker(a L ), as shown by the surjectivity argument above, and in act the computation there shows that it coincides with the map H(D ) coker(a L ). Similar considerations prove the identities o the other maps. Theorem 3.6. The gluing category M (U, Z) is abelian, and F : M(X) M (U, Z) and G : M (U, Z) M(X) are mutually inverse exact unctors, so M (U, Z) is equivalent to M(X). Proo. That M (U, Z) is abelian amounts to proving that taking coordinatewise kernels and cokernels works. That is, i we have (F U, F Z, u, v) and (F U, F Z, u, v ) with maps a U : F U F U, a Z : F Z F Z and such that the ollowing diagram commutes: Ψ (F U ) u v F Z Ψ (F U ) Ψ (a U ) Ψ (F U ) u F Z a Z Ψ (a U ) v Ψ (F U ) then (ker a U, ker a Z, ũ, ṽ) is a kernel or (a U, a V ), where ũ and ṽ are induced maps; likewise or the cokernel; and we must show that (a U, a V ) is an isomorphism i and only i the kernel and cokernel vanish. The maps ũ and ṽ are constructed rom the natural sequence o kernels (or cokernels) in the above diagram, and the exactness o Ψ, and once they exist it is obvious rom the deinition o morphisms in M (U, Z) that the 12

desired gluing data is a kernel (resp. cokernel). Since M(U) and M(Z) are abelian and kernels and cokernels are taken coordinatewise, the last claim ollows. That F is exact ollows rom the orm taken by kernels and cokernels, and the exactness o Φ ; that G is unctorial and exact is exactly the same proo as that Φ is. All that remains is to prove that they are mutually inverse. We do this by interpreting both M(X) and M (U, Z) as diads in the orm given, respectively, by diagrams (3) and (4). Then Corollary 3.5 shows that these two types are interchanged by the relection unctor, with the correspondence explicitly realizing the two unctors F and G previously given. Reerences [1] A. A. Beilinson, How to glue perverse sheaves, K-theory, arithmetic and geometry (Moscow, 1984), Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 42 51, available at http://math.harvard.edu/~ryanr/beilinson%20--%20how%20to% 20glue%20perverse%20sheaves.pd. [2] Sam Lichtenstein, Vanishing cycles or algebraic D-modules, Bachelor s thesis, Harvard University, 2009, http://math. harvard.edu/~gaitsgde/grad_2009/lichtenstein(2009).pd. 13