NICHOLAS SWITALA. Week 1: motivation from local cohomology; Weyl algebras

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1 BASIC THEORY OF ALGEBRAIC D-MODULES NICHOLAS SWITALA I taught a 12-week mn-course on algebrac D-modules at UIC durng the autumn of After each week, I posted lecture notes. What follows s smply a complaton of these weekly notes, whch means there s more repetton n them than a polshed, unfed document would tolerate. I thank Kevn Tucker, Wenlang Zhang, Chrs Skalt, Erc Redl, and especally Matthew Woolf for comments, questons, and correctons n the weekly semnar. Because of ther vglance, some false or confusng statements made n the semnar do not appear n these notes. The responsblty for any remanng errors rests, of course, wth me. Week 1: motvaton from local cohomology; Weyl algebras If R s a commutatve Noetheran rng, I = (f 1,..., f r ) s an deal, and M s an R-module, the local cohomology modules HI (M) of M supported at I are the cohomology objects of the Čech complex 0 M M f,j M f f j M f1 f r 0, and one can show that the modules so obtaned are ndependent of the choce of generators f 1,..., f r for I. (Ths s the most concrete of myrad equvalent defntons of local cohomology.) As an example, let R = k[x] where k s a feld. For the deal I, take the prncpal deal (x) R, and for the module M, take R tself. In ths case, the Čech complex s 0 R δ R x 0, and the map δ : R R x (r r 1 ) s njectve, snce R s a doman. Therefore H0 (x)(r) = ker δ = 0, and H(x) 1 (R) = coker δ = R x/r. As a k-space, R x /R s dentfed wth the drect sum >0 k x of nverse polynomals n x; the R-module structure s defned by the usual dstrbutve laws wth the addtonal relatons x j = 0 for all j 0. Observe that, n the above example, H(x) 1 (R) s not a fntely generated R-module. Ths s a theme of research n ths area: local cohomology modules tend to be huge, and t s frequently useful to fnd addtonal structures on them wth respect to whch they are smaller. One such addtonal structure s that of a D-module: a module over a rng D of dfferental operators. The basc examples of such rngs D are Weyl algebras. Let k be a feld of characterstc 0, and let R = k[x 1,..., x n ]. The rng D = D(R, k) of k-lnear dfferental operators on R (called the nth Weyl algebra and also denoted A n ) s the k-subalgebra of End k (R) generated by {x, } n =1 where x denotes the endomorphsm of R defned by multplcaton wth x and denotes partal dfferentaton wth respect to x. Clearly, the x commute wth each other, as do the ; furthermore, x commutes wth j f j. However, due to the product rule for dfferentaton, x = x + 1 (here 1 s the dentty endomorphsm of R). In partcular, D s a non-commutatve rng, whch contans a copy of R as a commutatve subrng of multplcaton endomorphsms. D s a smple rng (has no nontrval two-sded deals), s leftand rght-noetheran, and has the further surprsng property that every left (or rght) deal can be generated by at most two elements. Because D s non-commutatve, by a D-module we can 1

2 2 NICHOLAS SWITALA mean ether a left or rght module over ths rng, and must specfy whch. By conventon, we always mean left module unless we explctly say otherwse. Recall our example of local cohomology n the case n = 1: the R-module H(x) 1 (R) was not fntely generated. Usng our concrete dentfcaton H(x) 1 (R) = >0k x, we can vew ths object as a module over the frst Weyl algebra, D = A 1 = k x, /( x x 1): we only need to say how = d dx acts, and for ths, we can just use the quotent rule. As a D-module, H1 (x)(r) s not only fntely generated, t s generated by one element: 1 x. Indeed, we can dfferentate 1 x enough tmes to get any negatve power of x; there wll be a scalar numerator, but n characterstc 0, ths doesn t matter. Ths example llustrates two thngs that happen more generally. Frst, f I R s any deal, then HI (R) (or even H I (M), for any D-module M) can be vewed as a D-module: the partal dervatves act va the quotent rule, snce the dfferentals n the Čech complex are just sums of localzaton maps. Moreover, the D-modules HI (R) are fntely generated (even generated by one element), and of fnte length n the category of D-modules. These propertes follow from the fact that HI (R) s a holonomc D-module, as observed by Lyubeznk n Lyubeznk used the holonomy of HI (R) to deduce strong fnteness propertes, for example that H I (R) has only fntely many assocated prme deals n R. (A holonomc D-module s one that s as small as possble, among nonzero, fntely generated D-modules, wth respect to a certan measure of dmenson. Holonomc D-modules are sometmes called maxmally overdetermned or of Bernsten class; the term holonomc comes, roughly, from the Greek for everywhere law-abdng.) Some treatments of the theory of Weyl algebras and holonomc modules over them are the book A prmer of algebrac D-modules by Coutnho, the frst chapter of the notes Lectures on the algebrac theory of D-modules by Mlcc, and the frst chapter of the book Rngs of dfferental operators by Björk. Chapter 3 of Björk s book descrbes the case where R = k[[x 1,..., x n ]] s a formal power seres rng, whch s smlar n many ways to the polynomal case, though harder. Also useful for learnng about the power seres case are the early papers and Ph.D. thess of van den Essen. (Rather than lstng them all, n an act of shameless self-promoton, I refer you to the bblography of my own paper, Van den Essen s theorem on the de Rham cohomology of a holonomc D-module over a formal power seres rng.) The goal of ths mn-course s to cover the sophstcated (sheaffed) versons of some of the hghlghts of a course on Weyl algebras. It s possble to read many research papers on local cohomology havng only worked wth D-modules n the concrete settngs of polynomal or power seres rngs, and the learnng curve from such concrete settngs to D-modules over smooth schemes can be steep. Ths mn-course wll be based on the frst two or three chapters of the book D-modules, perverse sheaves, and representaton theory by Hotta, Takeuch, and Tansak, hereafter HTT. We wll dscuss the theory of D X -modules, where X s a smooth scheme of pure dmenson over an algebracally closed feld k of characterstc 0, wth structure sheaf O X. (HTT assumes k = C throughout, but they only begn usng ths assumpton n chapter 4.) Eventually, we wll need to make an addtonal mld assumpton on X, namely that every coherent O X -module s the quotent of a locally free O X -module. Snce X s already assumed to be smooth, ths assumpton (the resoluton property ) amounts to requrng that X have affne dagonal (the ntersecton of any two open affnes n X s affne). HTT, and other sources on D-modules, usually assume that X s quas-projectve n order to guarantee the resoluton property. Ths dscusson only becomes relevant when functors between derved categores are consdered. Besdes coverng the basc constructons and operatons on D-modules (most mportantly, pushforward and pull-back), our goals wll be to cover the proofs of Kashwara s theorem (f Z X

3 BASIC THEORY OF ALGEBRAIC D-MODULES 3 s a closed mmerson, the category of D Z -modules s equvalent to the category of D X -modules supported on Z) and Bernsten s nequalty, whch s the fundamental result on dmensons of D X -modules necessary to defne the category of holonomc modules. Some other references for ths materal nclude Bernsten s lecture notes Algebrac theory of D-modules, chapters VI and VII of the book Algebrac D-modules by Borel et al., and Jonathan Wang s Cambrdge Part III essay, Introducton to D-modules and representaton theory. There s a rotous surfet of dfferng notatons for the basc functors: when the standard references dsagree, whch s nearly always, we wll use the notaton n HTT. Week 2: basc defntons; coherence and quas-coherence Let k be an algebracally closed feld of characterstc 0, and let X be a smooth scheme of pure dmenson n over k, wth structure sheaf O X. The cotangent sheaf Ω 1 X s a locally free O X-module of rank n, and hence so s ts O X -dual, the tangent sheaf Θ X = Hom OX (Ω 1 X, O X). (In other sources, Θ X s frequently denoted T X.) By the unversal property of Ω 1 X as a sheaf of Kähler dfferentals, Θ X s dentfed wth the sheaf of k-lnear dervatons (or vector felds ) on X: Θ X Der k (O X ) = {δ End k (O X ) δ(fg) = δ(f)g + fδ(g) f, g O X } where we abuse notaton by wrtng f O X when f s a local secton of O X. Both O X (as multplcatons ) and Θ X (as dervatons) are subsheaves of End k (O X ). Defnton. D X, the sheaf of k-lnear dfferental operators on X, s the k-subalgebra of End k (O X ) generated by O X and Θ X. Let x X be gven. Snce X s smooth, there s an open affne neghborhood x U X and sectons x 1,..., x n O X (U), 1,..., n Θ X (U) such that the commute wth each other (of course, the x always commute wth each other); (x j ) = δ,j (Kronecker delta) for all, j; the generate Θ X (U) over O X (U). The x (resp. ) can be chosen to be lfts of a regular system of parameters for the regular local rng O X,x (resp. lfts of the dual bass to the dfferentals dx 1,..., dx n Ω 1 X,x ). We call {x, } n =1 a (local) coordnate system on U. The x are often called étale coordnates, snce they defne an étale morphsm U A n k. Over U, the sheaf D X takes the form of a Weyl algebra wth respect to the x and : that s, we have D X U = α1,...,α n 0O U α 1 1 αn n. A D X -module s just a sheaf of left modules over the sheaf of non-commutatve rngs D X. For example, O X s a D X -module n an obvous way. Observe that D X s locally free, and therefore quas-coherent, as an O X -module. Often, but not always, we wll restrct attenton to D X -modules that are quas-coherent as O X -modules. We wll work wth the followng categores: Mod(D X ), the category of all left D X -modules; Mod qc (D X ), the category of all left D X -modules that are quas-coherent as O X -modules; Mod c (D X ), the category of all left D X -modules that are coherent as D X -modules (that s, quas-coherent as O X -modules and locally fntely generated over D X ); Mod(D op X ), Mod qc(d op X ), and Mod c(d op X ), the analogues of the above for rght D X-modules. Snce D X s quas-coherent (but not coherent) over O X, a quas-coherent D X -module s just a D X -module that s quas-coherent over O X. On the other hand, a coherent D X -module need not be coherent over O X. If t s, then t must be a vector bundle:

4 4 NICHOLAS SWITALA Proposton. If M s a D X -module that s coherent over O X, then M s locally free over O X. Proof. Snce M s coherent, t suffces to check that M x s a free O X,x -module for all x X. Fx x X and local coordnates {x, } on an open affne neghborhood U of x, where the x are lfts of a set of generators of the maxmal deal m x O X,x. By Nakayama s lemma, we can fnd generators s 1,..., s m for M x over O X,x whose mages s 1,..., s m n M x / x M x form a bass for ths vector space over O X,x /m x = k. We clam that s 1,..., s m are free generators for M x. Suppose, for contradcton, that there s a nontrval dependence relaton m =1 f s = 0 (f O X,x ) such that the mnmum of the orders of the f s as small as possble (here the order of f O X,x s max{l f m l x}). Observe that ths mnmal order must be postve, snce f any f were a unt, we could reduce the gven dependence relaton modulo m x to obtan a nontrval k-lnear dependence relaton among the s. Relabelng f necessary, we may assume f 1 realzes the mnmal order. Choose j such that the order of j (f 1 ) s strctly less than that of f 1 (n down-to-earth terms, choose a parameter x j that occurs n the lowest-degree term of f 1, and dfferentate wth respect to t). Then we have m m 0 = j (0) = j ( f s ) = j (f )s + f j (s ), =1 where the rghtmost expresson can be expanded as an O X,x -lnear combnaton of s 1,..., s m whose coeffcent of mnmal order s of strctly smaller order than f 1. Ths contradcton fnshes the proof. There s another, more general, defnton of D X, n whch we construct D X recursvely as a fltered sheaf of rngs. Let F 0 D X = O X, and for all l > 0, defne =1 F l D X = {δ End k (O X ) [δ, f] F l 1 D X f O X }; fnally, set D X = l 0 F l D X. (Here, [δ, f] denotes the commutator δf fδ.) By nducton on l + m, t s easy to prove that, for all l, m 0, we have F l D X F m D X F l+m D X (key formula: [δδ, f] = δ[δ, f] + [δ, f]δ ), and [F l D X, F m D X ] F l+m 1 D X (key formula: [[δ, δ ], f] = [[δ, f], δ ] + [δ, [δ, f]]). Ths defnton does not requre us to make any assumptons (e.g. smoothness) on X. When X s smooth, the sheaf D X constructed above agrees wth our earler defnton. In general, we always have F 1 D X = O X Θ X (gven δ F 1 D X, we assocate wth t the par (δ(1) O X, δ δ(1) Θ X )). Over an open affne U X wth a coordnate system {x, }, we have F l D U = α1 + +α n lo U α 1 1 αn n, so each F l D X s a locally free O X -module of fnte rank. The fltraton {F l D X } l 0 s called the order fltraton (sometmes degree fltraton ) on D X, and the elements of F l D X are dfferental operators of order l. Fnally, observe that f we pass to the assocated graded sheaf of rngs gr F D X = l=0 F ld X /F l 1 D X (F 1 D X = 0), we obtan a sheaf of commutatve rngs due to the relaton [F l D X, F m D X ] F l+m 1 D X. (In fact, the sheaf gr F D X can be dentfed wth the symmetrc algebra of the tangent sheaf Θ X.) We wll dscuss gr F D X n more detal later; t serves a useful role n allowng us to apply technques of commutatve algebra to obtan results about the non-commutatve D X.

5 BASIC THEORY OF ALGEBRAIC D-MODULES 5 Week 3: sde-changng operatons Lke last week, we let X be a smooth scheme of pure dmenson n over an algebracally closed feld k of characterstc 0. We let O X, Θ X, and Ω 1 X be the structure, tangent, and cotangent sheaves of X respectvely, and we let D X be the k-subalgebra of End k (O X ) generated by O X and Θ X. For the rest of ths mn-course, these notatons and hypotheses are fxed, although we may occasonally need to mpose addtonal condtons on X. Suppose M s a O X -module. Snce Θ X generates D X over O X, n order to gve M a structure of left or rght D X -module, t s enough to specfy how the dervatons δ Θ X act on M, as long as the relatons n and between O X and Θ X are respected. Recall that Θ X s a sheaf of Le algebras: the commutator [δ, δ ] = δδ δ δ of two dervatons s agan a dervaton, and ths operaton satsfes the Le algebra axoms. Of course, f δ Θ X s a dervaton and f O X, then fδ s agan a dervaton. We also have the relatons [δ, f] = δ(f) for all δ Θ X and f O X. If we specfy elements δ m M for all δ Θ X and m M n such a way that, for all δ, δ Θ X, f O X, and m M, we have [δ 1, δ 2 ] m = δ 1 (δ 2 m) δ 2 (δ 1 m), (fδ) m = f(δ m), and (δf) m = f(δ m) + δ(f)m, then we obtan a structure of left D X -module on M. To obtan a structure of rght D X -module on M n a smlar way, we begn by specfyng elements m δ for all δ Θ X and m M such that the obvous rght-to-left analogues of the frst two condtons are satsfed; the replacement for the thrd condton s m (fδ) = f(m δ) δ(f)m, snce fδ = δf δ(f). The above recpe for mposng D X -structures on O X -modules can be used to buld new D X - modules from old ones usng the tensor and Hom operatons over O X. For example, suppose that M (resp. N) s a rght (resp. left) D X -module. Then the O X -module M OX N becomes a rght D X -module usng the formula (m n) δ = m δ n m δ n for all δ Θ X, m M, and n N. To be fully honest, every tme a D X -structure s defned by specfyng how the dervatons act, we need to check all the relatons as above. As a sample calculaton, we check the thrd condton: (m n) (fδ) = m (fδ) n m (fδ) n = (f(m δ) δ(f)m ) n m f(δ n) = f(m δ n m δ n) δ(f)(m n) = (m n) (δf) δ(f)(m n). As another example, f N s a rght D X -module, then the O X -module Hom OX (M, N ) becomes a left D X -module usng the formula for ϕ Hom OX (M, N ), m M, and δ Θ X. (δ ϕ)(m ) = ϕ(m δ) ϕ(m ) δ Many, but not all, combnatons of left and rght D X -modules constructed usng tensor and Hom over O X can be gven ether a left or rght D X -structure. A way to remember ths s to use Oda s rule, whch says that f X s a smooth curve of genus g and L s a lne bundle on X, then L can be gven a left (resp. rght) D X -module structure f and only f the degree of L s 0 (resp. 2g 2). An even smpler mnemonc s left = 0, rght = 1, = +, and Hom = target mnus source, where

6 6 NICHOLAS SWITALA addton s not understood modulo 2: the result of the addton or subtracton must be 0 or 1 f the resultng module s to support a left or rght D X -module structure. Therefore, f M and N are left D X -modules and M and N are rght D X -modules, M OX N can be gven a structure of left D X -module ( = 0 ), whereas M OX N cannot be gven a structure of ether left or rght D X -module ( = 2 ), and nether can Hom OX (M, N) ( 0 1 = 1 ). The standard example of a left D X -module s the structure sheaf O X. Its counterpart, the standard example of a rght D X -module, s the canoncal sheaf ω X = n Ω 1 X, an nvertble (locally free of rank 1) O X -module. Ths sheaf s denoted Ω X n HTT; ths wll be one of our few devatons from the notaton of that book. We are gong to use the canoncal sheaf to set up the sde-changng operatons, whch are quas-nverse functors defnng an equvalence of categores between Mod(D X ) and Mod(D op X ). The sde-changng operatons have smple descrptons n local coordnates. However, t s useful to defne them frst usng global, sheaf-theoretc constructons and only then to calculate what they do n coordnates, rather than gvng an a pror coordnate-dependent defnton and then provdng an ndependence proof. (We wll contnue to develop peces of D X -module theory n ths order.) The rght D X -module structure on ω X s defned by means of the Le dervatve. We have an somorphsm n n ω X = Ω 1 X = HomOX (Θ X, O X ) Hom ( n OX ΘX, O X ) of O X -modules. Let δ Θ X and ω ω X be gven, and dentfy ω wth an O X -lnear homomorphsm n Θ X O X. Then the Le dervatve Le δ (ω) of the form ω along the dervaton δ s the O X -lnear homomorphsm n Θ X O X (that s, element of ω X ) defned by Le δ (ω)(δ 1 δ n ) = δ(ω(δ 1 δ n )) n ω(δ 1 [δ, δ ] δ n ) where δ 1,..., δ n Θ X. If we set ω δ = Le δ (ω) for ω ω X and δ Θ X, the axoms above for a rght D X -module structure on ω X are satsfed. If M s a left D X -module, then by the formula gven earler, ω X OX M s a rght D X -module. Ths s the left-to-rght sde-changng operaton. To construct ts quas-nverse, consder the dual sheaf ω 1 X = Hom OX (ω X, O X ). Let M be a rght D X -module. We have an O X -module somorphsm ω 1 X =1 O X M = Hom OX (ω X, O X ) OX M Hom OX (ω X, M), and we know by another formula gven earler that any Hom between two rght D X -modules s a left D X -module. Ths s the rght-to-left sde-changng operaton, and n fact we have ω 1 X O X (ω X OX M) M as left D X -modules (and an analogous statement for rght modules), that s, the two sde-changng operatons are quas-nverse functors. Fnally, we descrbe the effect of the sde-changng operaton n local coordnates. Suppose that X s affne wth coordnates {x, }. In ths case, the canoncal sheaf ω X s globally trval: O X ωx va 1 dx 1 dx n. (We smply wrte dx for the top form dx 1 dx n.) Therefore, gven any left D X -module M, the underlyng O X -modules M and ω X OX M are somorphc. To descrbe the rght D X -acton on ω X OX M, t suffces to specfy how the dervatons act. The key observaton here s that Le (dx) = 0 for all. Indeed, snce dx s the dual bass element to 1 n n Θ X, the frst term n the Le dervatve, (dx( 1 n )), s appled to a constant and hence vanshes, and the remanng terms vansh because the j all commute wth. Therefore, f m M, we have (by our rule for the rght D X -acton on the tensor product of the

7 rght D X -module ω X wth the left D X -module M) so f f O X, we have BASIC THEORY OF ALGEBRAIC D-MODULES 7 (dx m) = Le (dx) m dx m = dx m, (dx m) (f ) = (dx fm) = dx (fm). From ths calculaton, t s easy to see how any element of D X acts on M on the rght. Under the somorphsm M ω X OX M, the element m corresponds to dx m, and usng ths dentfcaton, we see that the rght acton of f on m s the same as the left acton of f on m. In general, we can defne the rght D X -acton on M by m δ = δ t m, where for any dfferental operator δ D X (not just dervatons), we defne the transpose (or formal adjont) δ t of δ by settng (x α 1 1 xαn n β 1 1 βn n ) t = ( 1) β 1+ β n β 1 1 βn n x α 1 1 xαn n and extendng by lnearty. Snce D X s a Weyl algebra wth respect to the x and, ths defnes the formal adjont for all dfferental operators: the dervatons are moved to the left past the varables, and each dervaton contrbutes a sgn. Week 4: nave pullback and pushforward Let f : X Y be a morphsm of smooth schemes over k. We are gong to descrbe the nave (non-derved) nverse and drect mage functors for D X - and D Y -modules. The story s smpler n the case of the nverse mage. If M s a left D Y -module, ts O-module nverse mage, f M = O X f 1 O Y f 1 M, can be gven a structure of left D X -module (here f 1 M s the sheaftheoretc nverse mage), as follows. Correspondng to the scheme morphsm f, we have a map f Ω 1 Y Ω1 X of O X-modules. Takng the O X -dual, we obtan a map Θ X f Θ Y, whch we denote df and refer to as the dfferental of f. Gven δ Θ X and p m f M (where, abusvely, we wrte m for an element of f 1 M), we defne δ (p m) = δ(p) m + p df(δ)(1 m), a chan-rule-type acton, whch makes sense because df(δ) f Θ Y acts on f M. As an example, suppose X = A n and Y = A m are affne spaces over k, and let f : X Y be a morphsm, whch must come from a rng map f # : k[y 1,..., y m ] k[x 1,..., x n ]. Wrte k[ y] (resp. k[ x]) for these coordnate rngs, and let F j = f # (y j ) for j = 1,..., m. If M s a left D Y -module, the acton of x D X on p m f M = k[ x] k[ y] M defned n the prevous paragraph becomes x (p m) = x (p) m + m j=1 p F j x yj m, and t s perhaps easer to see here the resemblance to the chan rule. (In Coutnho s book, there s a careful proof that ths acton respects the relatons [ x, x j ] = δ,j n D X.) More generally, f f : X Y s an arbtrary morphsm of smooth schemes, dm Y = m, and {y j, j } are local coordnates on Y, then we have m δ (p m) = δ(p) m + p δ(y j f) j m n these coordnates. Here y j f makes lteral sense as a regular functon on X f X and Y are, for example, affne varetes; but n the general case of an abstract morphsm of schemes, we must use the sheaf map O Y f O X to make sense of t. (Borel s book takes the above as the defnton of the nverse mage operaton on D Y -modules, and then sketches a proof that the acton so defned s ndependent of the chosen local coordnates on Y.) j=1

8 8 NICHOLAS SWITALA If we apply the nverse mage operaton to the left D Y -module D Y tself, the resultng object, f D Y = O X f 1 O Y f 1 D Y, s a (D X, f 1 D Y )-bmodule: the left D X -acton comes from the chan rule as above, and the rght f 1 D Y -acton s just rght multplcaton on the rght tensor factor. We denote ths bmodule by D X Y. Observe that by the assocatvty of tensor products, we have f M = O X f 1 O Y f 1 M O X f 1 O Y (f 1 D Y f 1 D Y f 1 M) D X Y f 1 D Y f 1 M as left D X -modules. We may therefore express the nverse mage operaton as M D X Y f 1 D Y f 1 M, whch makes t clear that ths operaton s a functor f : Mod(D Y ) Mod(D X ) that s rght-exact and preserves quas-coherence. The complcated nature of the left D X -acton on f M s quarantned n the frst tensor factor. Consder the followng smple example: let X = A n and Y = A n+1, and let : X Y be the closed mmerson defned by the surjectve rng map # : k[ x, y] k[ x] that sends y to 0. The bmodule D X Y s, by defnton, k[ x] k[ x,y] D Y, whch s somorphc to the quotent D Y /y D Y of D Y by ts rght deal y D Y. Another way to descrbe ths bmodule s as the tensor product D X k k[ y ], whch decomposes (as a left D X -module) nto a drect sum of nfntely many copes of D X, ndexed by the powers of y. It follows that, for any left D Y -module M, M M/y M as left D X -modules. More generally, f : X Y s any closed mmerson between smooth schemes over k, we can choose local coordnates {y, } n =1 on Y such that y n c+1 = = y n = 0 are defnng equatons for X as a closed subscheme of Y, where c s the codmenson of X n Y. Wth respect to these coordnates, D X Y D X k k[ n c+1,..., n ] as left D X -modules. In partcular, f c > 0, D X Y s a free left D X -module of nfnte rank. The prevous example shows that the nverse mage functor does not, n general, preserve coherence, because tensorng wth an nfnte-rank D X -module does not produce a coherent D X -module n general. The example also shows that, n contrast to the case of nverse mages, we cannot defne a drect mage functor for D-modules that agrees wth the usual f on the underlyng O-modules. To see ths, consder agan the surjecton # : k[ x, y] k[ x] defnng a closed mmerson of affne spaces X Y. Let M be a left D X -module. The functor corresponds to restrcton of scalars, so y acts as 0 on the O Y -module M. In order to make M a left D Y -module, we would need to defne the acton of y on M n a manner respectng the relatons n D Y. Ths s mpossble: n D Y, we have the relaton y y y y = 1, but f y acts as 0 on M, so must y y y y. Gven a morphsm f : X Y and a left D X -module M, our goal s to buld a left D Y -module out of M and f n some way; the prevous paragraph shows that we cannot smply take f M. It turns out to be easer to see what to do f we begn wth a rght D X -module M. Recall that D X Y s a (D X, f 1 D Y )-bmodule. Snce M s a rght D X -module, we can form the tensor product M DX D X Y. Rght multplcaton on the second tensor factor gves ths product a rght f 1 D Y - module structure. If we then apply f, we get a rght f f 1 D Y -module f (M DX D X Y ), whch becomes a rght D Y -module va the adjuncton unt D Y f f 1 D Y. By usng the sde-changng operatons, we can defne a smlar operaton for a left D X -module M: the sequence of operatons M ω X OX M f ((ω X OX M) DX D X Y ) ω 1 Y OY f ((ω X OX M) DX D X Y ) produces a left D X -module. Now recall that the projecton formula says that f F s any O X -module and L s a lne bundle (or any vector bundle) on Y, we have f F OY L f (F OX f L) = f (F f 1 O Y f 1 L) as O Y -modules. If we apply the projecton formula wth L = ω 1 Y and F = (ω X OX M) DX D X Y, and use the commutatvty and assocatvty of tensor products, we obtan an somorphsm ω 1 Y OY f ((ω X OX M) DX D X Y ) f ((ω X OX D X Y f 1 O Y f 1 ω 1 Y ) DX M)

9 BASIC THEORY OF ALGEBRAIC D-MODULES 9 of left D X -modules. Therefore we can wrte the drect mage operaton as a functor Mod(D X ) Mod(D Y ) defned by M f (D Y X DX M), where D Y X s the (f 1 D Y, D X )-bmodule D Y X = ω X OX D X Y f 1 O Y f 1 ω 1 Y. We call D X Y and D Y X the transfer bmodules assocated wth f. Because the nave canddate for a drect mage functor just defned mxes a left exact functor (f ) wth a rght exact functor ( ), we wll only ever work wth ts derved verson, to avod dffcultes wth homologcal algebra and propertes such as the composton rule. Week 5: good fltratons; structure of D X and gr D X Recall that D X = l F l D X s a fltered sheaf of rngs on X, va the order (or degree) fltraton. Snce [F l D X, F m D X ] F l+m 1 D X for all l and m, the assocated graded sheaf, gr F D X = l=0 F ld X /F l 1 D X, s a sheaf of commutatve rngs on X. If {x, } are local coordnates on an open affne U X, let ξ be the mage of n F 1 D U /F 0 D U gr F D U (called the prncpal symbol of ); then gr F D U O U [ξ 1,..., ξ n ]. We are gong to dscuss fltratons on left D X -modules. For now, we assume that X s affne, and let D X be the fltered rng D X (X). We smply wrte gr D X for gr F D X. A left D X -module M s called a fltered D X -module f t s provded wth an ncreasng, exhaustve fltraton F by addtve subgroups, {F p M} p=0 (that s, we assume that F pm F p+1 M for all p, and that p F p M = M), such that F l D X F p M F l+p M for all l and p. The followng theory, of course, works for more general fltered rngs (not just D X ), as well as for more general fltered modules where we allow nonzero F p M for p < 0 (however, F p M must be zero for suffcently negatve p). If M s a fltered D X -module, ts assocated graded module, gr F M = p=0 F pm/f p 1 M, s a module over gr D X. If gr F M s a fntely generated gr D X -module, we say that F s a good fltraton on M. Proposton. There exsts a good fltraton F on a left D X -module M f and only f M s fntely generated over D X. Proof. The f drecton s easy: gven a fnte set of generators of M over D X, t s clear how to defne a fltraton such that the classes of these generators n F 0 M = F 0 M/F 1 M gr F M generate gr F M over gr D X. For the only f drecton, let m 1 F p1 M,..., m k F pk M be such that the classes m F p M/F p 1M generate gr F M over gr D X (just pck any fnte set of generators for gr F M and splt each generator up nto ts homogeneous components). We clam that m 1,..., m k generate M over D X. It suffces to show that F p M D X m for all p. We use nducton on p; snce F p M = 0 for negatve p, the base case s obvous. Let p and m F p M \ F p 1 M be gven, and assume the statement for smaller values of p. By assumpton, the class m F p M/F p 1 M can be wrtten m = δ m where δ gr D X. Ths equalty stll holds f we replace δ by ts homogeneous component of degree p p. After dong so, choose lfts δ D X of δ, and apply the nducton hypothess to m δ m F p 1 M. The proof of the only f drecton shows that F p M = p p (F p p D X ) m for all p. That s, all good fltratons arse from shfts of the fltraton on D X after choosng generators for M. Ths fact can be used to compare two good fltratons on the same module. Proposton. Let M be a fntely generated left D X -module. Let F and G be good fltratons on M. There exsts an nteger a such that F p a M G p M F p+a M for all p.

10 10 NICHOLAS SWITALA Proof. By symmetry, t suffces to assume only that F s good and to show the frst contanment. By the prevous result, there exst m 1,..., m k M and p 1,..., p k 0 such that F p M = p p (F p p D X ) m for all p. Snce q G q M = M, for all we can choose q such that m G q M. Let a = max{q p }. Then we have F p M = p p (F p p D X ) m p p F p p D X G q M p p G p p +q M G p+a M, as clamed (up to a shft). We say that the fltratons F and G are neghborng f the nteger a n the proposton can be taken to be 1. The proposton mples that any two good fltratons on a left D X -module can be connected by a chan of pars of neghborng fltratons, whch wll be useful later n proofs. The fact that gr D X s a commutatve rng can be used to reduce proofs of propertes of the non-commutatve rng D X to proofs nvolvng ts commutatve approxmaton gr D X. Perhaps the easest example of ths strategy s the followng: Proposton. The rng D X s left and rght Noetheran. Proof. Let I D X be a left deal. Make I nto a fltered D X -module by settng F l I = I F l D X for all l. Then gr I s an deal n the Noetheran commutatve rng gr D X, hence s fntely generated; t follows that I s fntely generated over D X. The proof for rght deals s exactly the same. Another result (whose proof s more nvolved) about D X that s proved by reducng everythng to the settng of the commutatve gr D X nvolves global dmensons. Recall that the left (resp. rght) global dmenson of a rng A s the supremum of the set of projectve dmensons of left (resp. rght) A-modules. If A s left and rght Noetheran, ts left and rght global dmensons concde, and we speak smply of ts global dmenson. The global dmenson of the commutatve rng gr D X s 2n. Proposton. The global dmenson of D X s 2n. (In fact, t s exactly 2n, but ths requres much more work to prove.) The dea behnd the proof of ths weaker result s the followng. It suffces to prove that Ext 2n+1 D X (M, N) = 0 for all fntely generated left D X -modules M and N. Fx good fltratons on M and N. We know that Ext 2n+1 gr D X (gr M, gr N) = 0, because the global dmenson of gr D X s 2n. There s a fltraton on Ext 2n+1 D X (M, N) such that gr Ext 2n+1 D X (M, N) s somorphc to a subquotent of Ext 2n+1 gr D X (gr M, gr N) = 0, whch completes the proof. The key to ths last step s showng that for any good fltered D X - module M, there s a resoluton F M by fltered fnte free D X -modules that descends to a resoluton gr F gr M. Week 6: resolutons; derved pullback and pushforward Begnnng now, we add to our lst of permanent assumptons about our scheme X (and other schemes Y, Z that we wll map to and from X) that t be separated and fnte type over k. In partcular, X s now quas-compact. What s more, X has the resoluton property: any coherent O X -module s a quotent of a locally free O X -module. Totaro proved that for a smooth scheme X of fnte type over k, the resoluton property s equvalent to X havng affne dagonal (a weaker condton than separated). HTT assume that X s quas-projectve, n whch case the resoluton property s easy to see. If M Mod(D X ), M has an njectve resoluton M I and a flat resoluton F M by left D X -modules: ths s a general fact for left modules over any sheaf of rngs. Last week, we sketched a proof that the rng of sectons D X (U) has global dmenson 2n for any open U X.

11 BASIC THEORY OF ALGEBRAIC D-MODULES 11 It follows that M has bounded njectve and flat resolutons. Suppose furthermore that M s a quas-coherent D X -module. If F M s an O X -lnear surjecton from a locally free O X -module, then D X OX F D X OX M M s a D X -lnear surjecton, and D X OX F s a locally free D X -module. It follows that every M Mod qc (D X ) has a resoluton by locally free D X -modules, and therefore a bounded resoluton by locally projectve D X -modules (usng the fnteness of global dmenson). If M s coherent as a D X -module, t has a resoluton by locally free D X -modules of fnte rank. To see ths, we must replace M n the proof above by a coherent O X -submodule M M that generates M (globally) over D X. Such a thng exsts by the followng argument: take a fnte open affne cover {U } of X such that M U s generated over D U by a coherent O U -submodule M (such a cover exsts by the defnton of coherence), extend each M to a coherent O X -submodule M of M, then smply take M = M. We now ntroduce derved categores of D X -modules, the correct settng for the nverse and drect mage operatons. Recall that the derved category D(D X ) = D(Mod(D X )) s obtaned by takng the Abelan category of complexes of left D X -modules, formng ts quotent by chan homotopy equvalences, and fnally nvertng all quas-somorphsms (maps of complexes nducng somorphsms on all cohomology objects). A sngle D X -module M s vewed as an object n ths category by consderng the complex whch s M n degree zero and 0 elsewhere (ts sole nonzero cohomology object s H 0 (M) = M). The category D(D X ) s no longer Abelan; ts consolaton prze s a trangulated category structure. A morphsm M N n D(D X ) need not be nduced by a sngle map of complexes: nstead, such morphsms are equvalence classes of roofs M P N where the map P M (but not necessarly the other map) s a quas-somorphsm. Standard references on derved categores are chapter 1 of Hartshorne s Resdues and Dualty and chapter 10 of Webel s Introducton to Homologcal Algebra. Appendces B and C of HTT nclude an excellent summary of the theory, but wth most proofs omtted. Varants on the basc derved category D(D X ) nclude D b (D X ) (resp. D + (D X ), D (D X )), where only bounded (resp. bounded below, bounded above) complexes are consdered, and D qc(d X ) (resp. D c (D X )), where only complexes wth quas-coherent (resp. coherent) cohomology objects are consdered (here stands for b, +,, or no superscrpt). It follows from the dscusson above and standard derved category technques that every object of D b (D X ) s represented by a bounded complex of flats and a bounded complex of njectves, and every object of D b qc(d X ) s represented by a bounded complex of locally-projectves. Let f : X Y be any morphsm. Recall that we assocated wth f a par of transfer bmodules: D X Y = f D Y = O X f 1 O Y f 1 D Y s a (D X, f 1 D Y )-bmodule, where the left D X -acton s the chan-rule-type acton defned durng Week 4, and the rght f 1 D Y -acton s just rght multplcaton on the second tensor factor; by applyng sde-changng operatons, we obtan whch s a (f 1 D Y, D X )-bmodule. D Y X = ω X OX D X Y f 1 O Y f 1 ω 1 Y, Defnton. The nverse mage functor Lf : D b (D Y ) D b (D X ) s defned by Lf (M ) = D X Y L f 1 D Y f 1 M, and the drect mage functor f : Db (D X ) D b (D Y ) s defned by f M = Rf (D Y X L D X M ). Observe that these are the same as the nave operatons defned n Week 4, except that all occurrences of tensor product or pushforward have been replaced wth ther derved versons. (There s a varant of f defned for derved categores of rght modules.) The functor f 1 s exact, so does

12 12 NICHOLAS SWITALA not need to be derved. To compute Lf (M ), we replace M wth a bounded flat resoluton F and then form the tensor product D X Y f 1 D Y f 1 F. (Alternatvely, we could replace D X Y wth a resoluton by flat rght f 1 D Y -modules.) Snce the drect mage functor s a composton of a left derved functor and a rght derved functor, we would n general need frst to replace M wth a bounded flat resoluton F, and then replace D Y X DX F wth a bounded njectve resoluton before applyng f. As a complex of O X -modules, Lf M s naturally somorphc to O X L f 1 O Y f 1 M, usng assocatvty of the derved tensor product. Snce quas-coherence of a D X -module smply means quas-coherence of the underlyng O X -module, t s farly straghtforward to see that Lf preserves quas-coherence, that s, restrcts to a functor Dqc(D b Y ) Dqc(D b X ). However, Lf does not, n general, preserve coherence: we have Lf D Y = D X Y, and we have seen before that f f s a nontrval closed mmerson, D X Y s a locally free left D X -module of nfnte rank. Fnally, we have a composton rule for the nverse mage: f X f Y g Z, then the functors L(g f) and Lf Lg from D b (D Z ) to D b (D X ) are naturally somorphc. Ths s easy to prove usng assocatvty of the derved tensor product and the fact that sheaf-theoretc nverse mage (f 1 ) commutes wth tensor product (note that snce D Y s locally free over O Y, f 1 D Y s certanly flat over f 1 O Y, and so D X Y = O X f 1 O Y f 1 D Y = O X L f 1 O Y f 1 D Y ). The analogous statements for the drect mage f are all true, but wth dfferent and more nvolved proofs. There s a composton rule ( g f and g f are naturally somorphc) for whch t s necessary to use the derved verson of the drect mage, whereas the composton rule for nverse mages s even true for the nave verson. The drect mage does not preserve coherence, even for open mmersons; however, t preserves coherence f f s proper, n partcular for closed mmersons. Fnally, drect mage does preserve quas-coherence: f restrcts to a functor Db qc(d X ) Dqc(D b Y ). However, t s not clear that D Y X L D X M has an O X -module structure, and so ths proof does not mmedately reduce to a proof for the O-module categores. Instead, we wll factor a general morphsm f nto manageable peces. In fact, ths strategy s how we wll understand drect and nverse mages more generally. The precedng dscusson s about as far as we wll go wth arbtrary morphsms f. Gven such a morphsm f : X Y, we can factor t as X Γ f X Y p 2 Y, where the frst map s the graph Γ f = (d X, f) of f, and the second map s projecton on the second factor. Snce X and Y are smooth, p 2 s smooth. Snce X and Y are separated, Γ f, whch s a base change of the dagonal f, s a closed mmerson. We therefore focus our attenton on the specal cases of closed mmersons and projectons (more generally, smooth morphsms). Closed mmersons wll be gven prde of place because of ther mportance for Kashwara s theorem. We remark that, by the composton rule, t wll suffce to show that f preserves quas-coherence n case f s a closed mmerson or a projecton n order to conclude t for arbtrary f. We frst ndcate what happens n the easest case of all: when j : U X s an open mmerson. In ths case, j 1 D X s just D U, from whch t follows that D U X and D X U are both smply D U. Therefore Lj s just j 1, and j s just Rj. Next we consder closed mmersons, whch wll occupy us for some tme. Let : X Y be a closed mmerson, where dm X = r and dm Y = n. As we saw n Week 4, we can choose local coordnates {y j, yj } n j=1 on Y such that y r+1 = = y n = 0 are local defnng equatons for the mmerson. Wrte x j = y j for j = 1,..., r: then {x j, xj } r j=1 are local coordnates on X. In these coordnates, D X Y D X k k[ r+1,..., n ] as left D X -modules. Recall that D X Y

13 BASIC THEORY OF ALGEBRAIC D-MODULES 13 s a (D X, 1 D Y )-bmodule. As sheaves of k-spaces, D X Y D Y X (usng the smultaneous trvalzatons of ω X and ω Y by dx 1 dx r and dy 1 dy n ); the rght D X - and left 1 D Y - actons on D Y X are the transposes (formal adjonts) of those on D X Y. In partcular, we observe that D Y X s a locally free rght D X -module. The defnton of drect mage nvolves a tensor product over D X wth D Y X. In contrast, the defnton of nverse mage L nvolves a tensor product over 1 D Y wth D X Y, and the latter s not a locally free left 1 D Y -module. Therefore the drect mage s actually smpler n ths case. Snce s affne, s exact, and therefore both derved functors occurrng n the defnton of can be replaced wth ther non-derved versons. If M Mod(D X ) (a sngle D X -module), the complex M therefore has no cohomology n nonzero degrees, and the functor 0 M = H 0 ( M) = (D Y X DX M) s exact. Observe that ths functor clearly preserves quas-coherence, snce D Y X s locally free as a rght D X -module. On the other hand, L M may have nontrval cohomology n nonzero degrees. Week 7: pushforward for closed mmersons; summary for smooth morphsms Last tme, we saw that f : X Y s a closed mmerson, then the functor 0 : Mod(D X ) Mod(D Y ) defned by 0 M = (D Y X DX M) s exact (and preserves quas-coherence). We wrte Mod X qc(d Y ) for the category of all quas-coherent left D Y -modules supported on X: clearly 0 M Mod X qc(d Y ). Theorem. (Kashwara) The functor 0 : Mod qc (D X ) Mod X qc(d Y ) s an equvalence of categores. Indeed, 0 possesses a rght adjont such that ( 0, ) s an adjont equvalence. We wll construct the rght adjont (and prove t s a rght adjont), leavng for next week the proof of equvalence. In fact, N wll be defned for any left D Y -module N, and the two wll be adjont as functors Mod(D X ) Mod(D Y ). Gven any such N, we defne N = Hom 1 D Y (D Y X, 1 N), whch s naturally a left 1 D Y -module and can be vewed as a left D X -module usng the rght D X -structure on D Y X. The functor : Mod(D Y ) Mod(D X ), beng the composton of a sheaf Hom and the exact functor 1, s left exact. A more sophstcated verson of Kashwara s equvalence (stated n terms of a trangulated equvalence between derved categores) uses the rght derved functor R of ths left exact functor. We remark here that f N D b (D Y ), we have R N L N [dm X dm Y ] n D b (D X ). (The proof of ths fact s an explct calculaton usng a locally free left 1 D Y -resoluton of D Y X.) Suppose ψ : D Y X 1 N s an 1 D Y -lnear map. In local coordnates {y, } on Y, we have, as dscussed last tme, an somorphsm D Y X D X k k[ r+1,..., n ] as k-spaces (here r = dm X and n = dm Y ). The left 1 D Y - and rght D X -actons are transposes of those on D X Y. As a left 1 D Y -module, D Y X s generated by 1 1. Let I O Y be the defnng deal sheaf of : X Y (locally generated by y r+1,..., y n ). Then 1 I annhlates 1 1, because t annhlates the left tensor factor (y r+1,..., y n all act as zero on D X ). Snce ψ s 1 D Y -lnear, we have 1 I ψ(1 1) = 0. It follows that the mage of ψ les n 1 Γ X N, where Γ X N s the subsheaf of sectons of N supported on X.

14 14 NICHOLAS SWITALA Ths observaton s crucal for the proof that s rght adjont to 0, whch we gve now. Let M Mod(D X ) and N Mod(D Y ) be gven. We have functoral D Y -module somorphsms Hom DX (M, Hom 1 D Y (D Y X, 1 N)) Hom DX (M, Hom 1 D Y (D Y X, 1 Γ X N)) Hom 1 D Y (D Y X DX M, 1 Γ X N) Hom DY ( (D Y X DX M), Γ X N) Hom DY ( (D Y X DX M), N), where the frst and fourth somorphsms follow from the prevous paragraph, the second somorphsm s a form of Hom adjuncton, and the thrd somorphsm uses the full fathfulness of as well as the dentfcaton 1 Γ X N Γ X N (f we begn wth a sheaf supported on X, pullng back to X and then pushng forward to Y changes nothng). If we take global sectons of both sdes, we obtan functoral bjectve correspondences Hom DX (M, N) = Γ(X, Hom DX (M, Hom 1 D Y (D Y X, 1 N))) = Γ(Y, Hom DX (M, Hom 1 D Y (D Y X, 1 N))) Γ(Y, Hom DY ( (D Y X DX M), N)) = Hom DY ( 0 M, N), so that ( 0, ) form an adjont par, as clamed. Recall that our strategy for studyng the nverse and drect mage functors along general morphsms was to factor such morphsms nto closed mmersons followed by smooth morphsms (specfcally, projectons) usng the graph. Before provng Kashwara s theorem and dscussng ts consequences next tme, we brefly sketch what happens n the smooth case. Let f : X Y be a smooth morphsm. If M Mod(D Y ), then Lf M O X L f 1 O Y f 1 M n D b (O X ). Snce f s smooth, t s n partcular flat, so O X s flat over f 1 O Y. It follows that Lf M has cohomology only n degree zero (that s, up to dentfyng a left D X -module wth a complex concentrated n degree zero, we smply have Lf = f ). To see what happens for the drect mage, we use de Rham complexes. Recall that the (absolute) de Rham complex Ω X on X takes the form 0 O X d Ω 1 X Ω X d Ω +1 X Ωn X 0, where Ω X = Ω 1 X, the map d s the unversal dervaton, and the maps d, called exteror dervatves, are nduced by d. The objects n ths complex are coherent O X -modules, but the maps are merely k-lnear. There s a relatve verson of ths complex: f r s the relatve dmenson dm X dm Y of the smooth morphsm f, then by replacng Ω 1 X = Ω1 X/k wth Ω1 X/Y, we obtan a complex Ω X/Y of length r whose objects are coherent O X-modules but whose maps are f 1 O Y -lnear. If M s an O X -module, a connecton on M s a k-lnear map : M Ω 1 X O X M such that (fm) = df m+f (m) for all m M and f O X. The datum of a connecton on M s equvalent to that of a k-lnear map Θ X End k (M) (δ δ ) such that f δ = fδ and δ f = f δ + δ(f). Recall from Week 3 that ths means M s two-thrds of the way to beng a D X -module : f we want to extend the O X -structure on M to a D X -structure by specfyng how the dervatons δ Θ X act, these are two of the three requred propertes. If the map Θ X End k (M) satsfes the thrd property (that [δ1,δ 2 ] = [ δ1, δ2 ] for all δ 1, δ 2 Θ X ), the connecton s called ntegrable (or flat), and we see that an ntegrable connecton on M and a left D X -module structure on M amount

15 BASIC THEORY OF ALGEBRAIC D-MODULES 15 to the same thng. Gven a connecton on M, the maps : Ω X O X M Ω +1 X O X M defned by (ω m) = d ω m ω (m) form a complex DR X (M) = (0 M Ω 1 X OX M Ω n X OX M 0) f and only f s ntegrable. Therefore, for any left D X -module M, we can buld the complex DR X (M), whch s called the de Rham complex of M. Lkewse, we can defne the relatve verson DR X/Y (M). The complex DR X (D X )[ n] s a locally free resoluton of ω X as a rght D X -module, and the complex DR X/Y (D X )[ r] s a locally free resoluton of D Y X as a left f 1 D Y -module. Therefore, f M Mod(D X ), we have f M = Rf (DR X/Y (M)[ r]) n D b (D Y ). From ths, we see at once that f preserves quas-coherence (the objects n the complex DR X/Y (M) are quascoherent f M s), so by our factorng argument, f descends to a functor Db qc(d X ) D b qc(d Y ) for any morphsm f. Week 8: Kashwara s theorem Recall that f : X Y s a closed mmerson, we have the exact functor 0 : Mod(D X ) Mod(D Y ) defned by 0 M = (D Y X DX M), and the left exact functor : Mod(D Y ) Mod(D X ) defned by N = Hom 1 D Y (D Y X, 1 N). We saw last tme that ( 0, ) form an adjont par. Today, we wll prove Kashwara s theorem, stated last tme: f we restrct these functors to 0 : Mod qc (D X ) Mod X qc(d Y ), : Mod X qc(d Y ) Mod qc (D X ) (where the superscrpt X means those D Y -modules supported on X), we obtan an (adjont) equvalence of categores. The same s true wth qc replaced by c; one must smply check that 0 and preserve coherence (we already know they preserve quas-coherence). Proof. Snce the functors form an adjont par, t suffces to show that the unt M 0 M and count 0 N N are somorphsms for all M Mod qc (D X ) and N Mod X qc(d Y ). By the composton rule, we may factor nto a sequence of codmenson-one closed mmersons and thus assume that tself s of codmenson one. To prove that two sheaves are somorphc s a local queston, so we may assume that we have coordnates {y, } n =1 on Y such that y n = 0 s a defnng equaton for X. We wrte y for y n and for n. In these coordnates, the transfer bmodule D Y X takes the form k[ ] k D X. The canoncal bundles of X and Y have been smultaneously trvalzed by these coordnates, so D Y X = O X 1 O Y 1 D Y, wth left 1 D Y -acton gven by the transpose of the obvous rght acton. We frst consder N for arbtrary N Mod(D Y ). Snce O X s just 1 O Y /(y), we see that N = Hom 1 D Y (O X 1 O Y 1 D Y, 1 N) s just the kernel of y 1 D Y actng on 1 N. Let M Mod qc (D X ) be gven. Then 0 M = (D Y X DX M) ((k[ ] k D X ) DX M) (k[ ] k M) = k[ ] k M, so 0 M s the kernel of y actng on 1 (k[ ] k M) = k[ ] k M. It s easy to see that the unt M 0 M sends m to 1 m under ths dentfcaton. We clam that the kernel s precsely 1 k M, from whch t wll follow that the unt s an somorphsm. Wrte the left 1 D Y -module k[ ] k M as j 0 j M, where the powers of appear on the left because the acton s transposed. For any j 0 and any m M, we have y j m = j ym j j 1 m = j j 1 m;

16 16 NICHOLAS SWITALA that s, ym = 0, but powers of can absorb powers of y before they reach m. It follows that y j m = 0 f and only f j = 0, that s, the kernel of y s 1 k M k[ ] k M. (Caveat lector: n Week 6, I stated ncorrectly n the semnar that y annhlates all of k[ ] k M! Ths has now been fxed n the posted notes.) We now consder the count. Let N Mod X qc(d Y ). Let θ be the lnear operator y : N N, and let N j = {n N θn = jn}, for j Z, be the j-egenspace of ths operator. Note that θ : N j N j s an somorphsm (multplcaton by j) f j 0, and y = θ + 1 : N j N j s an somorphsm (multplcaton by j + 1) f j 1. Let n N j, for any j, be gven. The calculaton θ(yn) = y yn = y(y + 1)n = yθn + yn = y(jn) + yn = (j + 1)yn shows that yn N j+1, that s, yn j N j+1. A smlar calculaton shows that N j N j 1. Therefore, f j < 1, the somorphsms θ : N j+1 N j+1 and θ + 1 : N j N j can be factored N j+1 N j y N j+1 and N j y N j+1 N j, and so : N j+1 N j and y : N j N j+1 are both somorphsms for such j. We clam that N = j>0 N j as k-spaces. Before provng ths clam, we show how the remanng part of Kashwara s theorem follows from t. Assumng that N = j>0 N j, we have N = k[ ] k N 1 (snce : N j N j 1 s an somorphsm for all j > 0) and N, the kernel of y actng on 1 N = j>0 1 N j, s 1 N 1 (snce y : 1 N j 1 N j+1 s an somorphsm for j > 1). Therefore 0 N 0 1 N 1 ((k[ ] k D X ) DX 1 N 1 ) (k[ ] k 1 N 1 ) k[ ] k N 1 N, whch completes the proof of Kashwara s theorem modulo the (omtted) verfcaton that the composte somorphsm above concdes wth the count. Fnally, we prove the clamed drect sum decomposton. Ths step s where the quas-coherence of N s essental: snce N s supported on X, ths quas-coherence mples that every n N s annhlated by some power of the defnng deal (y) of X. Therefore t wll be enough to show that ker(n yk N) k j=1n j for all k 1 (then just take the ascendng unon of both sdes). We prove ths last statement by nducton on k. For the base case, f n N s such that yn = 0, then θn = y n = yn n = n, so n N 1. Now assume that y k n = 0 for some k > 1. Snce y k 1 (yn) = 0, the nducton hypothess mples that yn k 1 j=1 N j. Applyng (whch drops the egenvalue by 1), we fnd θn + n = yn k j=2 N j. On the other hand, the element θn + kn s also annhlated by y k 1 : we have y k 1 (θn + kn) = y k n + ky k 1 n = (y k n) = (0). So the nducton hypothess also gves θn + kn k 1 j=1 N j, from whch t follows that the dfference (k 1)n = (θn + kn) (θn + n) belongs to k 1 j=1 N j k j=2 N j k j=1 N j. Snce k > 1, k 1 s nvertble, and the proof s complete. The verson of Kashwara s theorem just proved can be vewed as the base case of a proof by nducton (on cohomologcal length) for a more general derved category verson of the theorem: namely, : Db qc(d X ) Dqc b,x (D Y ) s an equvalence of trangulated categores wth quas-nverse R L [dm X dm Y ], where the target category s the bounded derved category of D Y - modules whose cohomology sheaves are both quas-coherent and supported on X. (Agan, the same s true wth qc replaced by c.) We also remark that f Z X s a closed mmerson where X s smooth but Z s not, we can defne (nspred by Kashwara s theorem) Mod qc (D Z ) to be the subcategory Mod Z qc(d X ). In Denns Gatsgory s notes on geometrc representaton theory, there s a proof that the resultng category s ndependent of the choce of X and embeddng Z X.