BASIC THEORY OF ALGEBRAIC D-MODULES
NICHOLAS SWITALA
I taught a 12-week mini-course on algebraic D-modules at UIC during the autumn of 2016.
After each week, I posted lecture notes. What follows is simply a compilation of these weekly notes,
which means there is more repetition in them than a polished, unified document would tolerate.
I thank Kevin Tucker, Wenliang Zhang, Chris Skalit, Eric Riedl, and especially Matthew Woolf
for comments, questions, and corrections in the weekly seminar. Because of their vigilance, some
false or confusing statements made in the seminar do not appear in these notes. The responsibility
for any remaining errors rests, of course, with me.
Week 1: motivation from local cohomology; Weyl algebras
If R is a commutative Noetherian ring, I = (f1 , . . . , fr ) is an ideal, and M is an R-module,
the local cohomology modules HIi (M ) of M supported at I are the cohomology objects of the Čech
complex
0 → M → ⊕i Mfi → ⊕i,j Mfi fj → · · · → Mf1 ···fr → 0,
and one can show that the modules so obtained are independent of the choice of generators f1 , . . . , fr
for I. (This is the most concrete of myriad equivalent definitions of local cohomology.)
As an example, let R = k[x] where k is a field. For the ideal I, take the principal ideal (x) ⊂ R,
and for the module M , take R itself. In this case, the Čech complex is
δ
0→R→
− Rx → 0,
0 (R) = ker δ = 0,
and the map δ : R → Rx (r 7→ 1r ) is injective, since R is a domain. Therefore H(x)
1 (R) = coker δ = R /R. As a k-space, R /R is identified with the direct sum ⊕
−i of
and H(x)
x
x
i>0 k · x
“inverse polynomials” in x; the R-module structure is defined by the usual distributive laws with
the additional relations xj = 0 for all j ≥ 0.
1 (R) is not a finitely generated R-module. This is a
Observe that, in the above example, H(x)
theme of research in this area: local cohomology modules tend to be huge, and it is frequently useful
to find additional structures on them with respect to which they are “smaller”. One such additional
structure is that of a D-module: a module over a ring D of differential operators.
The basic examples of such rings D are Weyl algebras. Let k be a field of characteristic 0, and
let R = k[x1 , . . . , xn ]. The ring D = D(R, k) of k-linear differential operators on R (called the
nth Weyl algebra and also denoted An ) is the k-subalgebra of Endk (R) generated by {xi , ∂i }ni=1
where xi denotes the endomorphism of R defined by multiplication with xi and ∂i denotes partial
differentiation with respect to xi . Clearly, the xi commute with each other, as do the ∂i ; furthermore,
xi commutes with ∂j if i 6= j. However, due to the product rule for differentiation, ∂i xi = xi ∂i + 1
(here 1 is the identity endomorphism of R).
In particular, D is a non-commutative ring, which contains a copy of R as a commutative subring
of “multiplication” endomorphisms. D is a simple ring (has no nontrivial two-sided ideals), is leftand right-Noetherian, and has the further surprising property that every left (or right) ideal can
be generated by at most two elements. Because D is non-commutative, by a “D-module” we can
1
2
NICHOLAS SWITALA
mean either a left or right module over this ring, and must specify which. By convention, we always
mean left module unless we explicitly say otherwise.
1 (R) was not finitely
Recall our example of local cohomology in the case n = 1: the R-module H(x)
1 (R) = ⊕
−i
generated. Using our concrete identification H(x)
i>0 k · x , we can view this object as a
module over the first Weyl algebra, D = A1 = khx, ∂i/(∂x − x∂ − 1): we only need to say how
d
1 (R) is not only
∂ = dx
acts, and for this, we can just use the quotient rule. As a D-module, H(x)
finitely generated, it is generated by one element: x1 . Indeed, we can differentiate x1 enough times to
get any negative power of x; there will be a scalar numerator, but in characteristic 0, this doesn’t
matter.
This example illustrates two things that happen more generally. First, if I ⊂ R is any ideal,
then HIi (R) (or even HIi (M ), for any D-module M ) can be viewed as a D-module: the partial
derivatives ∂i act via the quotient rule, since the differentials in the Čech complex are just sums
of localization maps. Moreover, the D-modules HIi (R) are finitely generated (even generated by
one element), and of finite length in the category of D-modules. These properties follow from the
fact that HIi (R) is a holonomic D-module, as observed by Lyubeznik in 1993. Lyubeznik used the
holonomy of HIi (R) to deduce strong finiteness properties, for example that HIi (R) has only finitely
many associated prime ideals in R. (A holonomic D-module is one that is as small as possible,
among nonzero, finitely generated D-modules, with respect to a certain measure of dimension.
Holonomic D-modules are sometimes called maximally overdetermined or of Bernstein class; the
term holonomic comes, roughly, from the Greek for “everywhere law-abiding”.)
Some treatments of the theory of Weyl algebras and holonomic modules over them are the book
“A primer of algebraic D-modules” by Coutinho, the first chapter of the notes “Lectures on the
algebraic theory of D-modules” by Milicic, and the first chapter of the book “Rings of differential
operators” by Björk. Chapter 3 of Björk’s book describes the case where R = k[[x1 , . . . , xn ]] is a
formal power series ring, which is similar in many ways to the polynomial case, though harder.
Also useful for learning about the power series case are the early papers and Ph.D. thesis of van
den Essen. (Rather than listing them all, in an act of shameless self-promotion, I refer you to
the bibliography of my own paper, “Van den Essen’s theorem on the de Rham cohomology of a
holonomic D-module over a formal power series ring”.)
The goal of this mini-course is to cover the sophisticated (sheafified) versions of some of the
highlights of a course on Weyl algebras. It is possible to read many research papers on local
cohomology having only worked with D-modules in the concrete settings of polynomial or power
series rings, and the learning curve from such concrete settings to D-modules over smooth schemes
can be steep.
This mini-course will be based on the first two or three chapters of the book “D-modules,
perverse sheaves, and representation theory” by Hotta, Takeuchi, and Tanisaki, hereafter “HTT”.
We will discuss the theory of DX -modules, where X is a smooth scheme of pure dimension over
an algebraically closed field k of characteristic 0, with structure sheaf OX . (HTT assumes k = C
throughout, but they only begin using this assumption in chapter 4.) Eventually, we will need to
make an additional mild assumption on X, namely that every coherent OX -module is the quotient
of a locally free OX -module. Since X is already assumed to be smooth, this assumption (the
“resolution property”) amounts to requiring that X have affine diagonal (the intersection of any
two open affines in X is affine). HTT, and other sources on D-modules, usually assume that X
is quasi-projective in order to guarantee the resolution property. This discussion only becomes
relevant when functors between derived categories are considered.
Besides covering the basic constructions and operations on D-modules (most importantly, pushforward and pull-back), our goals will be to cover the proofs of Kashiwara’s theorem (if Z ,→ X
BASIC THEORY OF ALGEBRAIC D-MODULES
3
is a closed immersion, the category of DZ -modules is equivalent to the category of DX -modules
supported on Z) and Bernstein’s inequality, which is the fundamental result on dimensions of
DX -modules necessary to define the category of holonomic modules.
Some other references for this material include Bernstein’s lecture notes “Algebraic theory of
D-modules”, chapters VI and VII of the book “Algebraic D-modules” by Borel et al., and Jonathan
Wang’s Cambridge Part III essay, “Introduction to D-modules and representation theory”. There is
a riotous surfeit of differing notations for the basic functors: when the standard references disagree,
which is nearly always, we will use the notation in HTT.
Week 2: basic definitions; coherence and quasi-coherence
Let k be an algebraically closed field of characteristic 0, and let X be a smooth scheme of pure
dimension n over k, with structure sheaf OX . The cotangent sheaf Ω1X is a locally free OX -module
of rank n, and hence so is its OX -dual, the tangent sheaf ΘX = HomOX (Ω1X , OX ). (In other sources,
ΘX is frequently denoted TX .) By the universal property of Ω1X as a sheaf of Kähler differentials,
ΘX is identified with the sheaf of k-linear derivations (or “vector fields”) on X:
ΘX ' Derk (OX ) = {δ ∈ Endk (OX ) | δ(f g) = δ(f )g + f δ(g) ∀f, g ∈ OX }
where we abuse notation by writing f ∈ OX when f is a local section of OX . Both OX (as “multiplications”) and ΘX (as derivations) are subsheaves of Endk (OX ).
Definition. DX , the sheaf of k-linear differential operators on X, is the k-subalgebra of Endk (OX )
generated by OX and ΘX .
Let x ∈ X be given. Since X is smooth, there is an open affine neighborhood x ∈ U ⊂ X and
sections x1 , . . . , xn ∈ OX (U ), ∂1 , . . . , ∂n ∈ ΘX (U ) such that
• the ∂i commute with each other (of course, the xi always commute with each other);
• ∂i (xj ) = δi,j (Kronecker delta) for all i, j;
• the ∂i generate ΘX (U ) over OX (U ).
The xi (resp. ∂i ) can be chosen to be lifts of a regular system of parameters for the regular local
ring OX,x (resp. lifts of the dual basis to the differentials dx1 , . . . , dxn ∈ Ω1X,x ). We call {xi , ∂i }ni=1 a
(local) coordinate system on U . The xi are often called étale coordinates, since they define an étale
morphism U → Ank . Over U , the sheaf DX takes the form of a Weyl algebra with respect to the xi
and ∂i : that is, we have
DX |U = ⊕α1 ,...,αn ≥0 OU · ∂1α1 · · · ∂nαn .
A DX -module is just a sheaf of left modules over the sheaf of non-commutative rings DX . For
example, OX is a DX -module in an obvious way. Observe that DX is locally free, and therefore
quasi-coherent, as an OX -module. Often, but not always, we will restrict attention to DX -modules
that are quasi-coherent as OX -modules. We will work with the following categories:
• Mod(DX ), the category of all left DX -modules;
• Modqc (DX ), the category of all left DX -modules that are quasi-coherent as OX -modules;
• Modc (DX ), the category of all left DX -modules that are coherent as DX -modules (that is,
quasi-coherent as OX -modules and locally finitely generated over DX );
op
op
• Mod(Dop
X ), Modqc (DX ), and Modc (DX ), the analogues of the above for right DX -modules.
Since DX is quasi-coherent (but not coherent) over OX , a quasi-coherent DX -module is just a
DX -module that is quasi-coherent over OX . On the other hand, a coherent DX -module need not
be coherent over OX . If it is, then it must be a vector bundle:
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NICHOLAS SWITALA
Proposition. If M is a DX -module that is coherent over OX , then M is locally free over OX .
Proof. Since M is coherent, it suffices to check that Mx is a free OX,x -module for all x ∈ X. Fix
x ∈ X and local coordinates {xi , ∂i } on an open affine neighborhood U of x, where the xi are
lifts of a set of generators of the maximal ideal mx ⊂ OX,x . By Nakayama’s
P lemma, we can find
generators s1 , . . . , sm for Mx over OX,x whose images s1 , . . . , sm in Mx / i xi Mx form a basis for
this vector space over OX,x /mx = k. We claim that s1 , . . . , sm are free
P generators for Mx . Suppose,
for contradiction, that there is a nontrivial dependence relation m
i=1 fi si = 0 (fi ∈ OX,x ) such
that the minimum of the orders of the fi is as small as possible (here the order of f ∈ OX,x is
max{l | f ∈ mlx }). Observe that this minimal order must be positive, since if any fi were a unit, we
could reduce the given dependence relation modulo mx to obtain a nontrivial k-linear dependence
relation among the si . Relabeling if necessary, we may assume f1 realizes the minimal order. Choose
j such that the order of ∂j (f1 ) is strictly less than that of f1 (in down-to-earth terms, choose a
parameter xj that occurs in the lowest-degree term of f1 , and differentiate with respect to it). Then
we have
m
m
X
X
0 = ∂j (0) = ∂j (
fi si ) =
∂j (fi )si + fi ∂j (si ),
i=1
i=1
where the rightmost expression can be expanded as an OX,x -linear combination of s1 , . . . , sm whose
coefficient of minimal order is of strictly smaller order than f1 . This contradiction finishes the
proof.
There is another, more general, definition of DX , in which we construct DX recursively as a
filtered sheaf of rings. Let F0 DX = OX , and for all l > 0, define
Fl DX = {δ ∈ Endk (OX ) | [δ, f ] ∈ Fl−1 DX ∀f ∈ OX };
finally, set DX = ∪l≥0 Fl DX . (Here, [δ, f ] denotes the commutator δf − f δ.) By induction on l + m,
it is easy to prove that, for all l, m ≥ 0, we have
• Fl DX · Fm DX ⊂ Fl+m DX (key formula: [δδ 0 , f ] = δ[δ 0 , f ] + [δ, f ]δ 0 ), and
• [Fl DX , Fm DX ] ⊂ Fl+m−1 DX (key formula: [[δ, δ 0 ], f ] = [[δ, f ], δ 0 ] + [δ, [δ 0 , f ]]).
This definition does not require us to make any assumptions (e.g. smoothness) on X. When X is
smooth, the sheaf DX constructed above agrees with our earlier definition. In general, we always
have F1 DX = OX ⊕ΘX (given δ ∈ F1 DX , we associate with it the pair (δ(1) ∈ OX , δ −δ(1) ∈ ΘX )).
Over an open affine U ⊂ X with a coordinate system {xi , ∂i }, we have
Fl DU = ⊕α1 +···+αn ≤l OU · ∂1α1 · · · ∂nαn ,
so each Fl DX is a locally free OX -module of finite rank. The filtration {Fl DX }l≥0 is called the
order filtration (sometimes “degree filtration”) on DX , and the elements of Fl DX are differential
operators of order ≤ l.
Finally, observe that if we pass to the associated graded sheaf of rings
grF DX = ⊕∞
l=0 Fl DX /Fl−1 DX
(F−1 DX = 0), we obtain a sheaf of commutative rings due to the relation [Fl DX , Fm DX ] ⊂
Fl+m−1 DX . (In fact, the sheaf grF DX can be identified with the symmetric algebra of the tangent sheaf ΘX .) We will discuss grF DX in more detail later; it serves a useful role in allowing us
to apply techniques of commutative algebra to obtain results about the non-commutative DX .
BASIC THEORY OF ALGEBRAIC D-MODULES
5
Week 3: side-changing operations
Like last week, we let X be a smooth scheme of pure dimension n over an algebraically closed
field k of characteristic 0. We let OX , ΘX , and Ω1X be the structure, tangent, and cotangent sheaves
of X respectively, and we let DX be the k-subalgebra of Endk (OX ) generated by OX and ΘX . For
the rest of this mini-course, these notations and hypotheses are fixed, although we may occasionally
need to impose additional conditions on X.
Suppose M is a OX -module. Since ΘX generates DX over OX , in order to give M a structure of
left or right DX -module, it is enough to specify how the derivations δ ∈ ΘX act on M , as long as the
relations in and between OX and ΘX are respected. Recall that ΘX is a sheaf of Lie algebras: the
commutator [δ, δ 0 ] = δδ 0 −δ 0 δ of two derivations is again a derivation, and this operation satisfies the
Lie algebra axioms. Of course, if δ ∈ ΘX is a derivation and f ∈ OX , then f δ is again a derivation.
We also have the relations [δ, f ] = δ(f ) for all δ ∈ ΘX and f ∈ OX . If we specify elements δ · m ∈ M
for all δ ∈ ΘX and m ∈ M in such a way that, for all δ, δ 0 ∈ ΘX , f ∈ OX , and m ∈ M , we have
• [δ1 , δ2 ] · m = δ1 · (δ2 · m) − δ2 · (δ1 · m),
• (f δ) · m = f (δ · m), and
• (δf ) · m = f (δ · m) + δ(f )m,
then we obtain a structure of left DX -module on M . To obtain a structure of right DX -module on
M in a similar way, we begin by specifying elements m · δ for all δ ∈ ΘX and m ∈ M such that
the obvious right-to-left analogues of the first two conditions are satisfied; the replacement for the
third condition is
m · (f δ) = f (m · δ) − δ(f )m,
since f δ = δf − δ(f ).
The above recipe for imposing DX -structures on OX -modules can be used to build new DX modules from old ones using the tensor and Hom operations over OX . For example, suppose that
M 0 (resp. N ) is a right (resp. left) DX -module. Then the OX -module M 0 ⊗OX N becomes a right
DX -module using the formula
(m0 ⊗ n) · δ = m0 · δ ⊗ n − m0 ⊗ δ · n
for all δ ∈ ΘX , m0 ∈ M 0 , and n ∈ N . To be fully honest, every time a DX -structure is defined
by specifying how the derivations act, we need to check all the relations as above. As a sample
calculation, we check the third condition:
(m0 ⊗ n) · (f δ) = m0 · (f δ) ⊗ n − m0 ⊗ (f δ) · n
= (f (m0 · δ) − δ(f )m0 ) ⊗ n − m0 ⊗ f (δ · n)
= f (m0 · δ ⊗ n − m0 ⊗ δ · n) − δ(f )(m0 ⊗ n)
= (m0 ⊗ n) · (δf ) − δ(f )(m0 ⊗ n).
As another example, if N 0 is a right DX -module, then the OX -module HomOX (M 0 , N 0 ) becomes a
left DX -module using the formula
(δ · ϕ)(m0 ) = ϕ(m0 · δ) − ϕ(m0 ) · δ
for ϕ ∈ HomOX (M 0 , N 0 ), m0 ∈ M 0 , and δ ∈ ΘX .
Many, but not all, combinations of left and right DX -modules constructed using tensor and Hom
over OX can be given either a left or right DX -structure. A way to remember this is to use Oda’s
rule, which says that if X is a smooth curve of genus g and L is a line bundle on X, then L can be
given a left (resp. right) DX -module structure if and only if the degree of L is 0 (resp. 2g − 2). An
even simpler mnemonic is “left = 0, right = 1, ⊗ = +, and Hom = target minus source”, where
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NICHOLAS SWITALA
addition is not understood modulo 2: the result of the addition or subtraction must be 0 or 1 if
the resulting module is to support a left or right DX -module structure. Therefore, if M and N are
left DX -modules and M 0 and N 0 are right DX -modules, M ⊗OX N can be given a structure of left
DX -module (“0 + 0 = 0”), whereas M 0 ⊗OX N 0 cannot be given a structure of either left or right
DX -module (“1 + 1 = 2”), and neither can HomOX (M 0 , N ) (“0 − 1 = −1”).
The standard example of a left DX -module is the structure sheaf
Vn OX1 . Its counterpart, the standard example of a right DX -module, is the canonical sheaf ωX =
ΩX , an invertible (locally free
of rank 1) OX -module. This sheaf is denoted ΩX in HTT; this will be one of our few deviations from
the notation of that book. We are going to use the canonical sheaf to set up the side-changing operations, which are quasi-inverse functors defining an equivalence of categories between Mod(DX ) and
Mod(Dop
X ). The side-changing operations have simple descriptions in local coordinates. However, it
is useful to define them first using global, sheaf-theoretic constructions and only then to calculate
what they do in coordinates, rather than giving an a priori coordinate-dependent definition and
then providing an independence proof. (We will continue to develop pieces of DX -module theory
in this order.)
The right DX -module structure on ωX is defined by means of the Lie derivative. We have an
isomorphism
n
n
n
^
^
^
HomOX (ΘX , OX ) ' HomOX ( ΘX , OX )
ωX =
Ω1X =
of
VnOX -modules. Let δ ∈ ΘX and ω ∈ ωX be given, and identify ω with an OX -linear homomorphism
ΘX → OX . Then
V the Lie derivative Lieδ (ω) of the form ω along the derivation δ is the OX -linear
homomorphism n ΘX → OX (that is, element of ωX ) defined by
Lieδ (ω)(δ1 ∧ · · · ∧ δn ) = δ(ω(δ1 ∧ · · · ∧ δn )) −
n
X
ω(δ1 ∧ · · · ∧ [δ, δi ] ∧ · · · ∧ δn )
i=1
where δ1 , . . . , δn ∈ ΘX . If we set ω · δ = −Lieδ (ω) for ω ∈ ωX and δ ∈ ΘX , the axioms above for a
right DX -module structure on ωX are satisfied.
If M is a left DX -module, then by the formula given earlier, ωX ⊗OX M is a right DX -module.
This is the left-to-right side-changing operation. To construct its quasi-inverse, consider the dual
⊗−1
sheaf ωX
= HomOX (ωX , OX ). Let M be a right DX -module. We have an OX -module isomorphism
⊗−1
⊗OX M = HomOX (ωX , OX ) ⊗OX M ' HomOX (ωX , M ),
ωX
and we know by another formula given earlier that any Hom between two right DX -modules is a
left DX -module. This is the right-to-left side-changing operation, and in fact we have
⊗−1
ωX
⊗OX (ωX ⊗OX M ) ' M
as left DX -modules (and an analogous statement for right modules), that is, the two side-changing
operations are quasi-inverse functors.
Finally, we describe the effect of the side-changing operation in local coordinates. Suppose
that X is affine with coordinates {xi , ∂i }. In this case, the canonical sheaf ωX is globally trivial:
∼
OX −
→ ωX via 1 7→ dx1 ∧ · · · ∧ dxn . (We simply write dx for the top form dx1 ∧ · · · ∧ dxn .) Therefore,
given any left DX -module M , the underlying OX -modules M and ωX ⊗OX M are isomorphic. To
describe the right DX -action on ωX ⊗OX M , it suffices to specify how the derivations ∂i act. The
key observationVhere is that Lie∂i (dx) = 0 for all i. Indeed, since dx is the dual basis element to
∂1 ∧ · · · ∧ ∂n ∈ n ΘX , the first term in the Lie derivative, ∂i (dx(∂1 ∧ · · · ∧ ∂n )), is ∂i applied to a
constant and hence vanishes, and the remaining terms vanish because the ∂j all commute with ∂i .
Therefore, if m ∈ M , we have (by our rule for the right DX -action on the tensor product of the
BASIC THEORY OF ALGEBRAIC D-MODULES
7
right DX -module ωX with the left DX -module M )
(dx ⊗ m) · ∂i = −Lie∂i (dx) ⊗ m − dx ⊗ ∂i · m = −dx ⊗ ∂i · m,
so if f ∈ OX , we have
(dx ⊗ m) · (f ∂i ) = (dx ⊗ f m) · ∂i = −dx ⊗ ∂i · (f m).
From this calculation, it is easy to see how any element of DX acts on M on the right. Under the
isomorphism M ' ωX ⊗OX M , the element m corresponds to dx ⊗ m, and using this identification,
we see that the right action of f ∂i on m is the same as the left action of −∂i f on m. In general, we
can define the right DX -action on M by m · δ = δ t · m, where for any differential operator δ ∈ DX
(not just derivations), we define the “transpose” (or formal adjoint) δ t of δ by setting
(xα1 1 · · · xαnn ∂1β1 · · · ∂nβn )t = (−1)β1 +···βn ∂1β1 · · · ∂nβn xα1 1 · · · xαnn
and extending by linearity. Since DX is a Weyl algebra with respect to the xi and ∂i , this defines the
formal adjoint for all differential operators: the derivations are moved to the left past the variables,
and each derivation contributes a sign.
Week 4: naive pullback and pushforward
Let f : X → Y be a morphism of smooth schemes over k. We are going to describe the
“naive” (non-derived) inverse and direct image functors for DX - and DY -modules. The story is
simpler in the case of the inverse image. If M is a left DY -module, its O-module inverse image,
f ∗ M = OX ⊗f −1 OY f −1 M , can be given a structure of left DX -module (here f −1 M is the sheaftheoretic inverse image), as follows. Corresponding to the scheme morphism f , we have a map
f ∗ Ω1Y → Ω1X of OX -modules. Taking the OX -dual, we obtain a map ΘX → f ∗ ΘY , which we denote
df and refer to as the differential of f . Given δ ∈ ΘX and p ⊗ m ∈ f ∗ M (where, abusively, we write
m for an element of f −1 M ), we define
δ · (p ⊗ m) = δ(p) ⊗ m + p · df (δ)(1 ⊗ m),
a “chain-rule-type” action, which makes sense because df (δ) ∈ f ∗ ΘY acts on f ∗ M .
As an example, suppose X = An and Y = Am are affine spaces over k, and let f : X → Y be a
morphism, which must come from a ring map f # : k[y1 , . . . , ym ] → k[x1 , . . . , xn ]. Write k[~y ] (resp.
k[~x]) for these coordinate rings, and let Fj = f # (yj ) for j = 1, . . . , m. If M is a left DY -module,
the action of ∂xi ∈ DX on p ⊗ m ∈ f ∗ M = k[~x] ⊗k[~y] M defined in the previous paragraph becomes
∂xi · (p ⊗ m) = ∂xi (p) ⊗ m +
m
X
j=1
p·
∂Fj
⊗ ∂yj · m,
∂xi
and it is perhaps easier to see here the resemblance to the chain rule. (In Coutinho’s book, there
is a careful proof that this action respects the relations [∂xi , xj ] = δi,j in DX .) More generally,
if f : X → Y is an arbitrary morphism of smooth schemes, dim Y = m, and {yj , ∂j } are local
coordinates on Y , then we have
δ · (p ⊗ m) = δ(p) ⊗ m + p ·
m
X
δ(yj ◦ f ) ⊗ ∂j · m
j=1
in these coordinates. Here yj ◦ f makes literal sense as a regular function on X if X and Y are, for
example, affine varieties; but in the general case of an abstract morphism of schemes, we must use
the sheaf map OY → f∗ OX to make sense of it. (Borel’s book takes the above as the definition of
the inverse image operation on DY -modules, and then sketches a proof that the action so defined
is independent of the chosen local coordinates on Y .)
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NICHOLAS SWITALA
If we apply the inverse image operation to the left DY -module DY itself, the resulting object,
f ∗ DY = OX ⊗f −1 OY f −1 DY , is a (DX , f −1 DY )-bimodule: the left DX -action comes from the “chain
rule” as above, and the right f −1 DY -action is just right multiplication on the right tensor factor.
We denote this bimodule by DX→Y . Observe that by the associativity of tensor products, we have
f ∗ M = OX ⊗f −1 OY f −1 M ' OX ⊗f −1 OY (f −1 DY ⊗f −1 DY f −1 M ) ' DX→Y ⊗f −1 DY f −1 M
as left DX -modules. We may therefore express the inverse image operation as M 7→ DX→Y ⊗f −1 DY
f −1 M , which makes it clear that this operation is a functor f ∗ : Mod(DY ) → Mod(DX ) that is
right-exact and preserves quasi-coherence. The complicated nature of the left DX -action on f ∗ M
is “quarantined” in the first tensor factor.
Consider the following simple example: let X = An and Y = An+1 , and let i : X ,→ Y be the
closed immersion defined by the surjective ring map i# : k[~x, y] → k[~x] that sends y to 0. The
bimodule DX→Y is, by definition, k[~x] ⊗k[~x,y] DY , which is isomorphic to the quotient DY /y · DY
of DY by its right ideal y · DY . Another way to describe this bimodule is as the tensor product
DX ⊗k k[∂y ], which decomposes (as a left DX -module) into a direct sum of infinitely many copies of
DX , indexed by the powers of ∂y . It follows that, for any left DY -module M , i∗ M ' M/y · M as left
DX -modules. More generally, if i : X ,→ Y is any closed immersion between smooth schemes over
k, we can choose local coordinates {yi , ∂i }ni=1 on Y such that yn−c+1 = · · · = yn = 0 are defining
equations for X as a closed subscheme of Y , where c is the codimension of X in Y . With respect
to these coordinates,
DX→Y ' DX ⊗k k[∂n−c+1 , . . . , ∂n ]
as left DX -modules. In particular, if c > 0, DX→Y is a free left DX -module of infinite rank.
The previous example shows that the inverse image functor does not, in general, preserve coherence, because tensoring with an infinite-rank DX -module does not produce a coherent DX -module
in general. The example also shows that, in contrast to the case of inverse images, we cannot define
a direct image functor for D-modules that agrees with the usual f∗ on the underlying O-modules.
To see this, consider again the surjection i# : k[~x, y] → k[~x] defining a closed immersion of affine
spaces X ,→ Y . Let M be a left DX -module. The functor i∗ corresponds to restriction of scalars,
so y acts as 0 on the OY -module i∗ M . In order to make i∗ M a left DY -module, we would need to
define the action of ∂y on M in a manner respecting the relations in DY . This is impossible: in DY ,
we have the relation ∂y · y − y · ∂y = 1, but if y acts as 0 on M , so must ∂y · y − y · ∂y .
Given a morphism f : X → Y and a left DX -module M , our goal is to build a left DY -module
out of M and f∗ in some way; the previous paragraph shows that we cannot simply take f∗ M .
It turns out to be easier to see what to do if we begin with a right DX -module M . Recall that
DX→Y is a (DX , f −1 DY )-bimodule. Since M is a right DX -module, we can form the tensor product
M ⊗DX DX→Y . Right multiplication on the second tensor factor gives this product a right f −1 DY module structure. If we then apply f∗ , we get a right f∗ f −1 DY -module f∗ (M ⊗DX DX→Y ), which
becomes a right DY -module via the adjunction unit DY → f∗ f −1 DY . By using the side-changing
operations, we can define a similar operation for a left DX -module M : the sequence of operations
M 7→ ωX ⊗OX M 7→ f∗ ((ωX ⊗OX M ) ⊗DX DX→Y ) 7→ ωY⊗−1 ⊗OY f∗ ((ωX ⊗OX M ) ⊗DX DX→Y )
produces a left DX -module. Now recall that the projection formula says that if F is any OX -module
and L is a line bundle (or any vector bundle) on Y , we have
f∗ F ⊗OY L ' f∗ (F ⊗OX f ∗ L) = f∗ (F ⊗f −1 OY f −1 L)
as OY -modules. If we apply the projection formula with L = ωY⊗−1 and F = (ωX ⊗OX M )⊗DX DX→Y ,
and use the commutativity and associativity of tensor products, we obtain an isomorphism
ωY⊗−1 ⊗OY f∗ ((ωX ⊗OX M ) ⊗DX DX→Y ) ' f∗ ((ωX ⊗OX DX→Y ⊗f −1 OY f −1 ωY⊗−1 ) ⊗DX M )
BASIC THEORY OF ALGEBRAIC D-MODULES
9
of left DX -modules. Therefore we can write the direct image operation as a functor Mod(DX ) →
Mod(DY ) defined by M 7→ f∗ (DY ←X ⊗DX M ), where DY ←X is the (f −1 DY , DX )-bimodule
DY ←X = ωX ⊗OX DX→Y ⊗f −1 OY f −1 ωY⊗−1 .
We call DX→Y and DY ←X the transfer bimodules associated with f . Because the naive candidate
for a direct image functor just defined mixes a left exact functor (f∗ ) with a right exact functor
(⊗), we will only ever work with its derived version, to avoid difficulties with homological algebra
and properties such as the composition rule.
Week 5: good filtrations; structure of DX and gr DX
Recall that DX = ∪l Fl DX is a filtered sheaf of rings on X, via the order (or degree) filtration. Since [Fl DX , Fm DX ] ⊂ Fl+m−1 DX for all l and m, the associated graded sheaf, grF DX =
⊕∞
l=0 Fl DX /Fl−1 DX , is a sheaf of commutative rings on X. If {xi , ∂i } are local coordinates on an
open affine U ⊂ X, let ξi be the image of ∂i in F1 DU /F0 DU ⊂ grF DU (called the principal symbol
of ∂i ); then grF DU ' OU [ξ1 , . . . , ξn ].
We are going to discuss filtrations on left DX -modules. For now, we assume that X is affine,
and let DX be the filtered ring DX (X). We simply write gr DX for grF DX . A left DX -module M is
called a filtered DX -module if it is provided with an increasing, exhaustive filtration F by additive
subgroups, {Fp M }∞
p=0 (that is, we assume that Fp M ⊂ Fp+1 M for all p, and that ∪p Fp M = M ),
such that Fl DX · Fp M ⊂ Fl+p M for all l and p. The following theory, of course, works for more
general filtered rings (not just DX ), as well as for more general filtered modules where we allow
nonzero Fp M for p < 0 (however, Fp M must be zero for sufficiently negative p).
If M is a filtered DX -module, its associated graded module, grF M = ⊕∞
p=0 Fp M/Fp−1 M , is a
F
module over gr DX . If gr M is a finitely generated gr DX -module, we say that F is a good filtration
on M .
Proposition. There exists a good filtration F on a left DX -module M if and only if M is finitely
generated over DX .
Proof. The “if” direction is easy: given a finite set of generators of M over DX , it is clear how to
define a filtration such that the classes of these generators in F0 M = F0 M/F−1 M ⊂ grF M generate
grF M over gr DX . For the “only if” direction, let m1 ∈ Fp1 M, . . . , mk ∈ Fpk M be such that the
classes mi ∈ Fpi M/Fpi −1 M generate grF M over gr DX (just pick any finite set of generators for
grF M and split each generator up into its homogeneous
P components). We claim that m1 , . . . , mk
generate M over DX . It suffices to show that Fp M ⊂ i DX · mi for all p. We use induction on p;
since Fp M = 0 for negative p, the base case is obvious. Let p and m ∈ Fp M \ Fp−1 M be given, and
assume the statement for smaller values of p. By assumption, the class m ∈ Fp M/Fp−1 M can be
P
written m = i δi mi where δi ∈ gr DX . This equality still holds if we replace δi by its homogeneous
component of degree
P p − pi . After doing so, choose lifts δi ∈ DX of δi , and apply the induction
hypothesis to m − i δi mi ∈ Fp−1 M .
P
The proof of the “only if” direction shows that Fp M = p≥pi (Fp−pi DX ) · mi for all p. That is,
all good filtrations arise from shifts of the filtration on DX after choosing generators for M . This
fact can be used to compare two good filtrations on the same module.
Proposition. Let M be a finitely generated left DX -module. Let F and G be good filtrations on
M . There exists an integer a such that Fp−a M ⊂ Gp M ⊂ Fp+a M for all p.
10
NICHOLAS SWITALA
Proof. By symmetry, it suffices to assume only that F is good and to show the first containment. By the previous result, there exist m1 , . . . , mk ∈ M and p1 , . . . , pk ≥ 0 such that Fp M =
P
p≥pi (Fp−pi DX ) · mi for all p. Since ∪q Gq M = M , for all i we can choose qi such that mi ∈ Gqi M .
Let a = max{qi − pi }. Then we have
X
X
X
Fp M =
(Fp−pi DX ) · mi ⊂
Fp−pi DX · Gqi M ⊂
Gp−pi +qi M ⊂ Gp+a M,
p≥pi
p≥pi
p≥pi
as claimed (up to a shift).
We say that the filtrations F and G are neighboring if the integer a in the proposition can be
taken to be 1. The proposition implies that any two good filtrations on a left DX -module can be
connected by a chain of pairs of neighboring filtrations, which will be useful later in proofs.
The fact that gr DX is a commutative ring can be used to reduce proofs of properties of the
non-commutative ring DX to proofs involving its commutative approximation gr DX . Perhaps the
easiest example of this strategy is the following:
Proposition. The ring DX is left and right Noetherian.
Proof. Let I ⊂ DX be a left ideal. Make I into a filtered DX -module by setting Fl I = I ∩ Fl DX for
all l. Then gr I is an ideal in the Noetherian commutative ring gr DX , hence is finitely generated;
it follows that I is finitely generated over DX . The proof for right ideals is exactly the same.
Another result (whose proof is more involved) about DX that is proved by reducing everything
to the setting of the commutative gr DX involves global dimensions. Recall that the left (resp. right)
global dimension of a ring A is the supremum of the set of projective dimensions of left (resp. right)
A-modules. If A is left and right Noetherian, its left and right global dimensions coincide, and we
speak simply of its global dimension. The global dimension of the commutative ring gr DX is 2n.
Proposition. The global dimension of DX is ≤ 2n.
(In fact, it is exactly 2n, but this requires much more work to prove.) The idea behind the
2n+1
(M, N ) = 0 for all
proof of this weaker result is the following. It suffices to prove that ExtD
X
finitely generated left DX -modules M and N . Fix good filtrations on M and N . We know that
Ext2n+1
gr DX (gr M, gr N ) = 0, because the global dimension of gr DX is 2n. There is a filtration on
2n+1
2n+1
ExtDX (M, N ) such that gr Ext2n+1
DX (M, N ) is isomorphic to a subquotient of Extgr DX (gr M, gr N ) =
0, which completes the proof. The key to this last step is showing that for any good filtered DX module M , there is a resolution F• → M by filtered finite free DX -modules that descends to a
resolution gr F• → gr M .
Week 6: resolutions; derived pullback and pushforward
Beginning now, we add to our list of permanent assumptions about our scheme X (and other
schemes Y , Z that we will map to and from X) that it be separated and finite type over k. In
particular, X is now quasi-compact. What is more, X has the resolution property: any coherent
OX -module is a quotient of a locally free OX -module. Totaro proved that for a smooth scheme X
of finite type over k, the resolution property is equivalent to X having affine diagonal (a weaker
condition than separated). HTT assume that X is quasi-projective, in which case the resolution
property is easy to see.
If M ∈ Mod(DX ), M has an injective resolution M → I • and a flat resolution F • → M by
left DX -modules: this is a general fact for left modules over any sheaf of rings. Last week, we
sketched a proof that the ring of sections DX (U ) has global dimension ≤ 2n for any open U ⊂ X.
BASIC THEORY OF ALGEBRAIC D-MODULES
11
It follows that M has bounded injective and flat resolutions. Suppose furthermore that M is a
quasi-coherent DX -module. If F M is an OX -linear surjection from a locally free OX -module,
then DX ⊗OX F DX ⊗OX M M is a DX -linear surjection, and DX ⊗OX F is a locally free
DX -module. It follows that every M ∈ Modqc (DX ) has a resolution by locally free DX -modules,
and therefore a bounded resolution by locally projective DX -modules (using the finiteness of global
dimension).
If M is coherent as a DX -module, it has a resolution by locally free DX -modules of finite rank.
To see this, we must replace M in the proof above by a coherent OX -submodule M 0 ⊂ M that
generates M (globally) over DX . Such a thing exists by the following argument: take a finite open
affine cover {Ui } of X such that M |Ui is generated over DUi by a coherent OUi -submodule Mi00
00
(such a cover exists by the definition
P of 0coherence), extend each Mi to a coherent OX -submodule
0
0
Mi of M , then simply take M = i Mi .
We now introduce derived categories of DX -modules, the correct setting for the inverse and
direct image operations. Recall that the derived category D(DX ) = D(Mod(DX )) is obtained
by taking the Abelian category of complexes of left DX -modules, forming its quotient by chain
homotopy equivalences, and finally inverting all quasi-isomorphisms (maps of complexes inducing
isomorphisms on all cohomology objects). A single DX -module M is viewed as an object in this
category by considering the complex which is M in degree zero and 0 elsewhere (its sole nonzero
cohomology object is H 0 (M ) = M ). The category D(DX ) is no longer Abelian; its consolation prize
is a triangulated category structure. A morphism M • → N • in D(DX ) need not be induced by a
single map of complexes: instead, such morphisms are equivalence classes of “roofs” M • ← P • → N •
where the map P • → M • (but not necessarily the other map) is a quasi-isomorphism. Standard
references on derived categories are chapter 1 of Hartshorne’s Residues and Duality and chapter 10
of Weibel’s Introduction to Homological Algebra. Appendices B and C of HTT include an excellent
summary of the theory, but with most proofs omitted.
Variants on the basic derived category D(DX ) include Db (DX ) (resp. D+ (DX ), D− (DX )), where
∗ (D ) (resp.
only bounded (resp. bounded below, bounded above) complexes are considered, and Dqc
X
∗
Dc (DX )), where only complexes with quasi-coherent (resp. coherent) cohomology objects are considered (here ∗ stands for b, +, −, or no superscript). It follows from the discussion above and
standard derived category techniques that every object of Db (DX ) is represented by a bounded
b (D ) is represented
complex of flats and a bounded complex of injectives, and every object of Dqc
X
by a bounded complex of locally-projectives.
Let f : X → Y be any morphism. Recall that we associated with f a pair of transfer bimodules:
DX→Y = f ∗ DY = OX ⊗f −1 OY f −1 DY
is a (DX , f −1 DY )-bimodule, where the left DX -action is the “chain-rule-type” action defined during
Week 4, and the right f −1 DY -action is just right multiplication on the second tensor factor; by
applying side-changing operations, we obtain
DY ←X = ωX ⊗OX DX→Y ⊗f −1 OY f −1 ωY⊗−1 ,
which is a (f −1 DY , DX )-bimodule.
Definition. The inverse image functor Lf ∗ : Db (DY )R → Db (DX ) is defined by Lf ∗ (M • ) =
DX→Y ⊗Lf −1 DY f −1 M • , and the direct image functor f : Db (DX ) → Db (DY ) is defined by
R
•
L
•
f M = Rf∗ (DY ←X ⊗DX M ).
Observe that these are the same as the “naive” operations defined in Week 4, except that all
occurrences of Rtensor product or pushforward have been replaced with their derived versions. (There
is a variant of f defined for derived categories of right modules.) The functor f −1 is exact, so does
12
NICHOLAS SWITALA
not need to be derived. To compute Lf ∗ (M • ), we replace M • with a bounded flat resolution F •
and then form the tensor product DX→Y ⊗f −1 DY f −1 F • . (Alternatively, we could replace DX→Y
with a resolution by flat right f −1 DY -modules.) Since the direct image functor is a composition of
a left derived functor and a right derived functor, we would in general need first to replace M • with
a bounded flat resolution F • , and then replace DY ←X ⊗DX F • with a bounded injective resolution
before applying f∗ .
As a complex of OX -modules, Lf ∗ M • is naturally isomorphic to OX ⊗Lf −1 OY f −1 M • , using
associativity of the derived tensor product. Since quasi-coherence of a DX -module simply means
quasi-coherence of the underlying OX -module, it is fairly straightforward to see that Lf ∗ preserves
b (D ) → D b (D ). However, Lf ∗ does not, in
quasi-coherence, that is, restricts to a functor Dqc
Y
X
qc
∗
general, preserve coherence: we have Lf DY = DX→Y , and we have seen before that if f is a
nontrivial closed immersion, DX→Y is a locally free left DX -module of infinite rank. Finally, we
f
g
have a composition rule for the inverse image: if X −
→Y →
− Z, then the functors L(g ◦ f )∗ and
∗
∗
b
b
Lf ◦Lg from D (DZ ) to D (DX ) are naturally isomorphic. This is easy to prove using associativity
of the derived tensor product and the fact that sheaf-theoretic inverse image (f −1 ) commutes with
tensor product (note that since DY is locally free over OY , f −1 DY is certainly flat over f −1 OY ,
and so DX→Y = OX ⊗f −1 OY f −1 DY = OX ⊗Lf −1 OY f −1 DY ).
R
The analogous statements for the direct image f are all true, but with different and more
R
R R
involved proofs. There is a composition rule ( g◦f and g ◦ f are naturally isomorphic) for which it
is necessary to use the derived version of the direct image, whereas the composition rule for inverse
images is even true for the “naive” version. The direct image does not preserve coherence, even for
open immersions; however, it preserves coherence if fR is proper, in particular for closed immersions.
b (D ) → D b (D ).
Finally, direct image does preserve quasi-coherence: f restricts to a functor Dqc
X
Y
qc
•
L
However, it is not clear that DY ←X ⊗DX M has an OX -module structure, and so this proof does
not immediately reduce to a proof for the O-module categories. Instead, we will factor a general
morphism f into manageable pieces.
In fact, this strategy is how we will understand direct and inverse images more generally. The
preceding discussion is about as far as we will go with arbitrary morphisms f . Given such a morphism f : X → Y , we can factor it as
Γf
p2
X −→ X × Y −→ Y,
where the first map is the graph Γf = (idX , f ) of f , and the second map is projection on the
second factor. Since X and Y are smooth, p2 is smooth. Since X and Y are separated, Γf , which
is a base change of the diagonal ∆f , is a closed immersion. We therefore focus our attention on
the special cases of closed immersions and projections (more generally, smooth morphisms). Closed
immersions will be given pride of place because of their importance
R for Kashiwara’s theorem. We
remark that, by the composition rule, it will suffice to show that f preserves quasi-coherence in
case f is a closed immersion or a projection in order to conclude it for arbitrary f .
We first indicate what happens in the easiest case of all: when j : U ,→ X is an open immersion.
In this case, j −1 DX is just DU ,Rfrom which it follows that DU →X and DX←U are both simply DU .
Therefore Lj ∗ is just j −1 , and j is just Rj∗ .
Next we consider closed immersions, which will occupy us for some time. Let i : X ,→ Y be
a closed immersion, where dim X = r and dim Y = n. As we saw in Week 4, we can choose
local coordinates {yj , ∂yj }nj=1 on Y such that yr+1 = · · · = yn = 0 are local defining equations
for the immersion i. Write xj = yj ◦ i for j = 1, . . . , r: then {xj , ∂xj }rj=1 are local coordinates on
X. In these coordinates, DX→Y ' DX ⊗k k[∂r+1 , . . . , ∂n ] as left DX -modules. Recall that DX→Y
BASIC THEORY OF ALGEBRAIC D-MODULES
13
is a (DX , i−1 DY )-bimodule. As sheaves of k-spaces, DX→Y ' DY ←X (using the simultaneous
trivializations of ωX and ωY by dx1 ∧ · · · ∧ dxr and dy1 ∧ · · · ∧ dyn ); the right DX - and left i−1 DY actions on DY ←X are the transposes (formal adjoints) of those on DX→Y .
In particular,
we observe that DY ←X is a locally free right DX -module. The definition of direct
R
image i involves a tensor product over DX with DY ←X . In contrast, the definition of inverse
image Li∗ involves a tensor product over i−1 DY with DX→Y , and the latter is not a locally free
left i−1 DY -module. Therefore the direct image is actually simpler in this case.
Since i is affine, i∗
R
is exact, and therefore both derived functors occurring in the definition of i can beRreplaced with
their non-derived versions. If M ∈ Mod(DX ) (a single DX -module), the complex i M therefore
has no cohomology in nonzero degrees, and the functor
Z 0
Z
M = H 0 ( M ) = i∗ (DY ←X ⊗DX M )
i
i
is exact. Observe that this functor clearly preserves quasi-coherence, since DY ←X is locally free as
a right DX -module. On the other hand, Li∗ M may have nontrivial cohomology in nonzero degrees.
Week 7: pushforward for closed immersions; summary for smooth morphisms
R0
Last time, we saw that if i : X ,→ Y is a closed immersion, then the functor i : Mod(DX ) →
R0
Mod(DY ) defined by i M = i∗ (DY ←X ⊗DX M ) is exact (and preserves quasi-coherence). We
write ModX
qc (DY ) for the category of all quasi-coherent left DY -modules supported on X: clearly
R0
X
i M ∈ Modqc (DY ).
R0
(DY ) is an equivalence of cateTheorem. (Kashiwara) The functor i : Modqc (DX ) → ModX
R0
R 0 \ qc
\
gories. Indeed, i possesses a right adjoint i such that ( i , i ) is an adjoint equivalence.
We will construct the right adjoint (and prove it is a right adjoint), leaving for next week the
proof of equivalence. In fact, i\ N will be defined for any left DY -module N , and the two will be
adjoint as functors Mod(DX ) ↔ Mod(DY ). Given any such N , we define
i\ N = Homi−1 DY (DY ←X , i−1 N ),
which is naturally a left i−1 DY -module and can be viewed as a left DX -module using the right
DX -structure on DY ←X . The functor i\ : Mod(DY ) → Mod(DX ), being the composition of a
sheaf Hom and the exact functor i−1 , is left exact. A more sophisticated version of Kashiwara’s
equivalence (stated in terms of a triangulated equivalence between derived categories) uses the
right derived functor Ri\ of this left exact functor. We remark here that if N • ∈ Db (DY ), we have
Ri\ N • ' Li∗ N • [dim X − dim Y ] in Db (DX ). (The proof of this fact is an explicit calculation using
a locally free left i−1 DY -resolution of DY ←X .)
Suppose ψ : DY ←X → i−1 N is an i−1 DY -linear map. In local coordinates {yi , ∂i } on Y , we have,
as discussed last time, an isomorphism DY ←X ' DX ⊗k k[∂r+1 , . . . , ∂n ] as k-spaces (here r = dim X
and n = dim Y ). The left i−1 DY - and right DX -actions are transposes of those on DX→Y . As a left
i−1 DY -module, DY ←X is generated by 1 ⊗ 1. Let I ⊂ OY be the defining ideal sheaf of i : X ,→ Y
(locally generated by yr+1 , . . . , yn ). Then i−1 I annihilates 1⊗1, because it annihilates the left tensor
factor (yr+1 , . . . , yn all act as zero on DX ). Since ψ is i−1 DY -linear, we have i−1 I · ψ(1 ⊗ 1) = 0. It
follows that the image of ψ lies in i−1 ΓX N , where ΓX N is the subsheaf of sections of N supported
on X.
14
NICHOLAS SWITALA
R0
This observation is crucial for the proof that i\ is right adjoint to i , which we give now. Let
M ∈ Mod(DX ) and N ∈ Mod(DY ) be given. We have functorial DY -module isomorphisms
i∗ HomDX (M, Homi−1 DY (DY ←X , i−1 N )) ' i∗ HomDX (M, Homi−1 DY (DY ←X , i−1 ΓX N ))
' i∗ Homi−1 DY (DY ←X ⊗DX M, i−1 ΓX N )
' HomDY (i∗ (DY ←X ⊗DX M ), ΓX N )
' HomDY (i∗ (DY ←X ⊗DX M ), N ),
where the first and fourth isomorphisms follow from the previous paragraph, the second isomorphism is a form of ⊗ − Hom adjunction, and the third isomorphism uses the full faithfulness of i∗
as well as the identification i∗ i−1 ΓX N ' ΓX N (if we begin with a sheaf supported on X, pulling
back to X and then pushing forward to Y changes nothing). If we take global sections of both sides,
we obtain functorial bijective correspondences
HomDX (M, i\ N ) = Γ(X, HomDX (M, Homi−1 DY (DY ←X , i−1 N )))
= Γ(Y, i∗ HomDX (M, Homi−1 DY (DY ←X , i−1 N )))
' Γ(Y, HomDY (i∗ (DY ←X ⊗DX M ), N ))
Z 0
= HomDY (
M, N ),
i
R0
so that ( i , i\ ) form an adjoint pair, as claimed.
Recall that our strategy for studying the inverse and direct image functors along general morphisms was to factor such morphisms into closed immersions followed by smooth morphisms (specifically, projections) using the graph. Before proving Kashiwara’s theorem and discussing its consequences next time, we briefly sketch what happens in the smooth case.
Let f : X → Y be a smooth morphism. If M ∈ Mod(DY ), then Lf ∗ M ' OX ⊗Lf −1 OY f −1 M in
Db (OX ). Since f is smooth, it is in particular flat, so OX is flat over f −1 OY . It follows that Lf ∗ M
has cohomology only in degree zero (that is, up to identifying a left DX -module with a complex
concentrated in degree zero, we simply have Lf ∗ = f ∗ ).
To see what happens for the direct image, we use de Rham complexes. Recall that the (absolute)
de Rham complex Ω•X on X takes the form
d
di
n
0 → OX −
→ Ω1X → · · · → ΩiX −→ Ωi+1
X → · · · → ΩX → 0,
V
where ΩiX = i Ω1X , the map d is the universal derivation, and the maps di , called exterior derivatives, are induced by d. The objects in this complex are coherent OX -modules, but the maps
are merely k-linear. There is a relative version of this complex: if r is the relative dimension
dim X − dim Y of the smooth morphism f , then by replacing Ω1X = Ω1X/k with Ω1X/Y , we obtain a complex Ω•X/Y of length r whose objects are coherent OX -modules but whose maps are
f −1 OY -linear.
If M is an OX -module, a connection on M is a k-linear map ∇ : M → Ω1X ⊗OX M such that
∇(f m) = df ⊗m+f ∇(m) for all m ∈ M and f ∈ OX . The datum of a connection on M is equivalent
to that of a k-linear map ΘX → Endk (M ) (δ 7→ ∇δ ) such that f ∇δ = ∇f δ and ∇δ f = f ∇δ + δ(f ).
Recall from Week 3 that this means M is “two-thirds of the way to being a DX -module”: if we
want to extend the OX -structure on M to a DX -structure by specifying how the derivations δ ∈ ΘX
act, these are two of the three required properties. If the map ΘX → Endk (M ) satisfies the third
property (that ∇[δ1 ,δ2 ] = [∇δ1 , ∇δ2 ] for all δ1 , δ2 ∈ ΘX ), the connection ∇ is called integrable (or
flat), and we see that an integrable connection on M and a left DX -module structure on M amount
BASIC THEORY OF ALGEBRAIC D-MODULES
15
to the same thing. Given a connection ∇ on M , the maps ∇i : ΩiX ⊗OX M → Ωi+1
X ⊗OX M defined
by ∇i (ω ⊗ m) = di ω ⊗ m − ω ∧ ∇(m) form a complex
∇
DRX (M ) = (0 → M −
→ Ω1X ⊗OX M → · · · → ΩnX ⊗OX M → 0)
if and only if ∇ is integrable. Therefore, for any left DX -module M , we can build the complex
DRX (M ), which is called the de Rham complex of M . Likewise, we can define the relative version
DRX/Y (M ). The complex DRX (DX )[−n] is a locally free resolution of ωX as a right DX -module,
and the complex DRX/Y (DX )[−r] is a locally free resolution of DY ←X as a left f −1 DY -module.
R
Therefore, if M ∈ Mod(DX ), we have f M = Rf∗ (DRX/Y (M )[−r]) in Db (DY ). From this, we
R
see at once that f preserves quasi-coherence (the objects in the complex DRX/Y (M ) are quasiR
b (D ) → D b (D ) for
coherent if M is), so by our factoring argument, f descends to a functor Dqc
X
Y
qc
any morphism f .
Week 8: Kashiwara’s theorem
R0
Recall that if i : X ,→ Y is a closed immersion, we have the exact functor i : Mod(DX ) →
R0
Mod(DY ) defined by i M = i∗ (DY ←X ⊗DX M ), and the left exact functor i\ : Mod(DY ) →
R0
Mod(DX ) defined by i\ N = Homi−1 DY (DY ←X , i−1 N ). We saw last time that ( i , i\ ) form an
adjoint pair. Today, we will prove Kashiwara’s theorem, stated last time: if we restrict these functors
to
Z
0
: Modqc (DX ) → ModX
qc (DY ),
i\ : ModX
qc (DY ) → Modqc (DX )
i
(where the superscript X means those DY -modules supported on X), we obtain an (adjoint) equivR0
alence of categories. The same is true with qc replaced by c; one must simply check that i and i\
preserve coherence (we already know they preserve quasi-coherence).
R0
Proof. Since the functors form an adjoint pair, it suffices to show that the unit M → i\ i M
R0
and counit i i\ N → N are isomorphisms for all M ∈ Modqc (DX ) and N ∈ ModX
qc (DY ). By
the composition rule, we may factor i into a sequence of codimension-one closed immersions and
thus assume that i itself is of codimension one. To prove that two sheaves are isomorphic is a local
question, so we may assume that we have coordinates {yi , ∂i }ni=1 on Y such that yn = 0 is a defining
equation for X. We write y for yn and ∂ for ∂n . In these coordinates, the transfer bimodule DY ←X
takes the form k[∂] ⊗k DX . The canonical bundles of X and Y have been simultaneously trivialized
by these coordinates, so DY ←X = OX ⊗i−1 OY i−1 DY , with left i−1 DY -action given by the transpose
of the obvious right action.
We first consider i\ N for arbitrary N ∈ Mod(DY ). Since OX is just i−1 OY /(y), we see that
= Homi−1 DY (OX ⊗i−1 OY i−1 DY , i−1 N ) is just the kernel of y ∈ i−1 DY acting on i−1 N . Let
M ∈ Modqc (DX ) be given. Then
Z 0
M = i∗ (DY ←X ⊗DX M ) ' i∗ ((k[∂] ⊗k DX ) ⊗DX M ) ' i∗ (k[∂] ⊗k M ) = k[∂] ⊗k i∗ M,
i\ N
i
R0
−1
so
i M isRthe kernel of y acting on i (k[∂] ⊗k i∗ M ) = k[∂] ⊗k M . It is easy to see that the
0
unit M → i\ i M sends m to 1 ⊗ m under this identification. We claim that the kernel is precisely
1 ⊗k M , from which it will follow that the unit is an isomorphism. Write the left i−1 DY -module
k[∂] ⊗k M as ⊕j≥0 ∂ j · M , where the powers of ∂ appear on the left because the action is transposed.
For any j ≥ 0 and any m ∈ M , we have
i\
y∂ j m = ∂ j ym − j∂ j−1 m = −j∂ j−1 m;
16
NICHOLAS SWITALA
that is, ym = 0, but powers of ∂ can “absorb” powers of y before they reach m. It follows that
y∂ j m = 0 if and only if j = 0, that is, the kernel of y is 1 ⊗k M ⊂ k[∂] ⊗k M . (Caveat lector : in
Week 6, I stated incorrectly in the seminar that y annihilates all of k[∂] ⊗k M ! This has now been
fixed in the posted notes.)
We now consider the counit. Let N ∈ ModX
qc (DY ). Let θ be the linear operator y∂ : N → N , and
j
let N = {n ∈ N |θn = jn}, for j ∈ Z, be the j-eigenspace of this operator. Note that θ : N j → N j
is an isomorphism (multiplication by j) if j 6= 0, and ∂y = θ + 1 : N j → N j is an isomorphism
(multiplication by j + 1) if j 6= −1. Let n ∈ N j , for any j, be given. The calculation
θ(yn) = y∂yn = y(y∂ + 1)n = yθn + yn = y(jn) + yn = (j + 1)yn
shows that yn ∈ N j+1 , that is, yN j ⊂ N j+1 . A similar calculation shows that ∂N j ⊂ N j−1 .
Therefore, if j < −1, the isomorphisms θ : N j+1 → N j+1 and θ + 1 : N j → N j can be factored
y
y
∂
∂
N j+1 −
→ Nj −
→ N j+1 and N j −
→ N j+1 −
→ N j , and so ∂ : N j+1 → N j and y : N j → N j+1 are both
isomorphisms for such j.
We claim that N = ⊕j>0 N −j as k-spaces. Before proving this claim, we show how the remaining
part of Kashiwara’s theorem follows from it. Assuming that N = ⊕j>0 N −j , we have N = k[∂] ⊗k
N −1 (since ∂ : N −j → N −j−1 is an isomorphism for all j > 0) and i\ N , the kernel of y acting
on i−1 N = ⊕j>0 i−1 N −j , is i−1 N −1 (since y : i−1 N −j → i−1 N −j+1 is an isomorphism for j > 1).
Therefore
Z 0
Z 0
i\ N '
i−1 N −1 ' i∗ ((k[∂] ⊗k DX ) ⊗DX i−1 N −1 ) ' i∗ (k[∂] ⊗k i−1 N −1 ) ' k[∂] ⊗k N −1 ' N,
i
i
which completes the proof of Kashiwara’s theorem modulo the (omitted) verification that the
composite isomorphism above coincides with the counit.
Finally, we prove the claimed direct sum decomposition. This step is where the quasi-coherence
of N is essential: since N is supported on X, this quasi-coherence implies that every n ∈ N is
annihilated by some power of the defining ideal (y) of X. Therefore it will be enough to show that
yk
ker(N −→ N ) ⊂ ⊕kj=1 N −j
for all k ≥ 1 (then just take the ascending union of both sides). We prove this last statement by
induction on k. For the base case, if n ∈ N is 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. Since y k−1 (yn) = 0, the induction
−j . Applying ∂ (which drops the eigenvalue by 1), we find
hypothesis implies that yn ∈ ⊕k−1
j=1 N
θn + n = ∂yn ∈ ⊕kj=2 N −j . On the other hand, the element θn + kn is also annihilated by y k−1 :
we have y k−1 (θn + kn) = y k ∂n + ky k−1 n = ∂(y k n) = ∂(0). So the induction hypothesis also gives
−j , from which it follows that the difference (k − 1)n = (θn + kn) − (θn + n)
θn + kn ∈ ⊕k−1
j=1 N
−j − ⊕k N −j ⊂ ⊕k N −j . Since k > 1, k − 1 is invertible, and the proof is
belongs to ⊕k−1
j=2
j=1
j=1 N
complete.
The version of Kashiwara’s theorem just proved can be viewed as the base case of a proof by
induction (on cohomological length) for a more general derived category version of the theorem:
R
b (D ) → D b,X (D ) is an equivalence of triangulated categories with quasi-inverse
namely, i : Dqc
qc
X
Y
Ri\ ' Li∗ [dim X − dim Y ], where the target category is the bounded derived category of DY modules whose cohomology sheaves are both quasi-coherent and supported on X. (Again, the same
is true with qc replaced by c.) We also remark that if Z ,→ X is a closed immersion where X
is smooth but Z is not, we can define (inspired by Kashiwara’s theorem) Modqc (DZ ) to be the
subcategory ModZ
qc (DX ). In Dennis Gaitsgory’s notes on geometric representation theory, there is
a proof that the resulting category is independent of the choice of X and embedding Z ,→ X.
BASIC THEORY OF ALGEBRAIC D-MODULES
17
Week 9: characteristic varieties
R0
Last week we proved Kashiwara’s theorem: if i : X ,→ Y is a closed immersion, then i :
Modqc (DX ) → ModX
qc (DY ) is an equivalence of categories. Suppose that X is a single closed point
P ∈ Y . The category Modqc (DP ) is simply the category of k-spaces. Therefore, by Kashiwara’s
theorem, every object in the equivalent category ModPqc (DY ) is isomorphic to a direct sum of copies
R0
of i k, and this last object is isomorphic to i∗ (k[∂1 , . . . , ∂n ]) where {yi , ∂i } are local coordinates
on an open affine U containing P . The left action of DY on the skyscraper sheaf i∗ (k[∂1 , . . . , ∂n ])
was described last week: the ∂i act in the obvious way, and the yi annihilate any element of
i∗ (k[∂1 , . . . , ∂n ]) that lacks a ∂i term to “absorb” them. If R = k[y1 , . . . , yn ] and Y = Spec R, then
k[∂1 , . . . , ∂n ] with this action is isomorphic as a D(R, k)-module to the top local cohomology module
Hmn (R) supported at the irrelevant maximal ideal m = (y1 , . . . , yn ). The module Hmn (R) is often
denoted E (it is an injective hull of k over R). Concretely, E is usually described as the module of
“inverse polynomials” ⊕k ·y1−α1 · · · yn−αn where (α1 , . . . , αn ) runs over all n-tuples of strictly positive
integers, and the action of R on E sends any non-negative power of a variable to zero. The map
DY /DY · m = k[∂1 , . . . , ∂n ] → E that sends ∂iαi to yi−αi −1 becomes an isomorphism once the images
yi−αi −1 are multiplied by appropriate constants determined by the quotient rule action of D(R, k)
on E.
We briefly describe one application of the preceding description of quasi-coherent D-modules
supported at a single point. Recall that if X is an affine scheme, the global section functor Γ(X, −) :
Modqc (OX ) → Mod(Γ(X, OX )) is exact. We define a smooth scheme X to be D-affine if Γ(X, −) :
Modqc (DX ) → Mod(Γ(X, DX )) is exact, and if whenever Γ(X, M ) = 0 for a quasi-coherent DX module M , then M = 0. (From these two conditions, it follows that every M ∈ Modqc (DX ) is
generated over DX by its global sections, and that the functor Γ(X, −) as above is an equivalence
of categories.) Obviously, if X is smooth and affine, then X is D-affine. However, there are other
examples of D-affine schemes:
Theorem. (Beilinson-Bernstein) Projective space Pnk is D-affine.
Sketch of proof. We use the following notation: V = k n+1 with coordinates x0 , . . . , xn ; V ◦ = V \{0};
X = Pnk ; π : V ◦ → X is the projection; and j : V ◦ ,→ V is the inclusion. Let 0 → M 0 → M →
M 00 → 0 be a short exact sequence of quasi-coherent DX -modules. The functor π ∗ is exact (since
π is smooth) and j∗ is left exact, so we obtain a long exact sequence
0 → j∗ π ∗ M 0 → j∗ π ∗ M → j∗ π ∗ M 00 → R1 j∗ π ∗ M 0 → · · · ,
where the term R1 j∗ π ∗ M 0 and all later terms are quasi-coherent DV -modules supported at V \V ◦ =
{0} ⊂ V . By Kashiwara’s theorem, all of these sheaves P
are direct sums of copies of the skyscraper
n
k[∂0 , . . . , ∂n ] at 0. Consider the Euler operator θ =
i=0 xi ∂i . By an explicit calculation, the
eigenvalues of θ acting on k[∂0 , . . . , ∂n ] are strictly negative integers, while the 0-eigenspace of θ
acting on Γ(V, j∗ π ∗ N ) = Γ(V ◦ , π ∗ N ) is exactly Γ(X, N ) for any DX -module N . By first taking
global sections (an exact functor on the affine space V ) and then passing to the 0-eigenspaces of θ,
we obtain the desired short exact sequence 0 → Γ(X, M 0 ) → Γ(X, M ) → Γ(X, M 00 ) → 0, since the
global sections of the later terms in the long exact sequence have trivial 0-eigenspaces.
We turn next to the problem of assigning a dimension to every coherent DX -module. Recall
that the associated graded sheaf gr DX (with respect to the order filtration) is isomorphic to the
symmetric algebra Sym ΘX and locally takes the form OU [ξ1 , . . . , ξn ], where {xi , ∂i } are coordinates
on an open affine U ⊂ X and ξi , the (principal) symbol of ∂i , is the class of ∂i in F1 DX /F0 DX ⊂
gr DX . We can glue the spectra of the rings OU [ξ1 , . . . , ξn ] as U varies, obtaining a smooth scheme
SpecX gr DX of dimension 2n together with an affine projection map π : SpecX gr DX → X such
18
NICHOLAS SWITALA
that π∗ OSpecX gr DX = gr DX . Since gr DX ' Sym ΘX , the scheme SpecX gr DX ' SpecX Sym ΘX
together with the affine morphism π is the geometric vector bundle over X whose sheaf of sections
is the OX -dual of ΘX , namely the cotangent sheaf Ω1X (see exercise II.5.18 in Hartshorne). We
therefore refer to SpecX gr DX as the cotangent bundle over X, denoted T ∗ X. Locally, π takes the
form of a projection π|U : T ∗ U = U × k n → U , where the ξi are coordinates for k n .
Let M be a coherent DX -module. We saw in Week 6 that there exists a coherent OX -submodule
M0 of M that generates M over DX . If we set Fi M = (Fi DX ) · M0 ⊂ M for all i ≥ 0, we
obtain a global good filtration on M : the associated graded sheaf of modules grF M is coherent
over gr DX = π∗ OT ∗ X . (Note that the existence of M0 means that we do not need to worry about
patching good filtrations on M |U as U varies.) By pulling this coherent sheaf back to the cotangent
bundle, we obtain a coherent OT ∗ X -module
−1
F
^
F M = O ∗ ⊗ −1
gr
T X
π π∗ OT ∗ X π (gr M )
^
F M = grF M .
such that π∗ gr
^
^
F M of gr
F M , a closed subset of T ∗ X which we endow with the
Definition. The support Supp gr
reduced subscheme structure, is called the characteristic variety of M and denoted Ch(M ).
p
Anngr DX grF M . Since
The closed subscheme Ch(M ) is defined by the characteristic ideal
grF M is a gradedpmodule over gr DX , and the grading on the latter comes (locally) from the ξi ,
the graded ideal Anngr DX grF M is homogeneous with respect to the ξi , from which it follows
that Ch(M ) is conic (closed under scalar multiplication on the fibers of π). In other words, if
~ ∈ Ch(M ), then (~x, λξ)
~ ∈ Ch(M ) for all λ ∈ k.
(~x, ξ)
In order for the characteristic variety to be a well-defined construction, we must show that
^
F M is independent of the choice of good filtration F on M . For this, we may assume X is
Supp gr
affine and M is a finitely generated left module over the ring DX = DX (X).
Proposition. Let M be a finitely generated left module over DX , and let F and G be good filtrations
on M . Then Supp grF M = Supp grG M as subsets of gr DX .
Proof. We may assume that Fp M ⊂ Gp M ⊂ Fp+1 M for all p, since by a result from Week 5,
any two good filtrations can be linked by a finite chain of such pairs. Fix p and consider the
natural map ϕp : Fp M/Fp−1 M → Gp M/Gp−1 M induced by the inclusions. The kernel of ϕp
is Gp−1 M/Fp−1 M , and using the fact that Gp−1 M ⊂ Fp M , we see that the cokernel of ϕp is
(Gp M/Gp−1 M )/(Fp M/Gp−1 M ) ' Gp M/Fp M . That is, we have ker ϕp ' coker ϕp−1 for all p.
Passing to the direct sum over all p ≥ 0, the map ϕ = ⊕ϕp : grF M → grG M has ker ϕ ' coker ϕ
(disregarding the gradings). By considering the short exact sequences
0 → ker ϕ → grF M → im ϕ → 0, 0 → im ϕ → grG M → coker ϕ → 0
we have equalities
Supp grF M = Supp ker ϕ ∪ Supp im ϕ = Supp coker ϕ ∪ Supp im ϕ = Supp grG M,
where the second equality uses the isomorphism ker ϕ ' coker ϕ. The proof is complete.
If M ∈ Modc (DX ), the preceding proposition shows that Ch(M ) is a well-defined invariant of
M , and therefore so is its dimension, d(M ) = dim Ch(M ), which we refer to as the dimension of
M . Given a short exact sequence 0 → M 0 → M → M 00 → 0 in Modc (DX ), we have Ch(M ) =
Ch(M 0 )∪Ch(M 00 ) (a good filtration on M induces good filtrations on M 0 and M 00 , then we consider
the supports of the terms of the short exact sequence obtained upon passing to associated graded
objects); and therefore d(M ) = max{d(M 0 ), d(M 00 )}. A priori, the dimension of M is an integer
BASIC THEORY OF ALGEBRAIC D-MODULES
19
d(M ) such that 0 ≤ d(M ) ≤ 2n (since Ch(M ) ⊂ T ∗ X). Next week, we will prove Bernstein’s
inequality, a much stronger lower bound for d(M ).
Week 10: Bernstein’s inequality; holonomic modules
Recall from last week how we assign a dimension d(M ) to every coherent DX -module M : we
choose a global good filtration F on M , form the associated graded gr DX -module grF M , and
then pull this module back along the cotangent bundle π : SpecX gr DX = T ∗ X → X, obtaining a
^
^
F M . The support Supp gr
F M , a closed subset of T ∗ X which we endow
coherent OT ∗ X -module gr
with the reduced subscheme structure, does not depend on the choice of good filtration F on M : it
is called the characteristic variety Ch(M ) of M , and we take for d(M ) its dimension dim(Ch(M )).
Because Ch(M ) ⊂ T ∗ X, we have 0 ≤ d(M ) ≤ 2n.
Theorem. (Bernstein’s inequality) Let M 6= 0 be a coherent DX -module. Then we have d(M ) ≥ n.
Proof. We will use induction on dim(X), the base case (where 0 = n = 2n) being obvious. We
observe first that the support of M as a sheaf on X is precisely π(Ch(M )). To see this, note that
if M0 is a coherent OX -submodule of M that generates M over DX , then M and M0 have the
same support on X: clearly Supp M0 ⊂ Supp M , and if x ∈
/ Supp M0 , then there is an open affine
neighborhood U of x such that (M0 )|U , and consequently M (U ) = DX (U ) · M0 (U ), vanishes. If we
consider the good filtration F on M defined by Fi M = (Fi DX ) · M0 , then grF M is generated over
gr DX by M0 , from which it follows that π(Ch(M )) = Supp M0 = Supp M . Note that this subset
is closed, since M0 is coherent over OX .
Suppose for contradiction that dim(Ch(M )) < n = dim(X). Then S = π(Ch(M )) is a proper
closed subscheme of X. By the generic smoothness theorem (Corollary III.10.7 in Hartshorne), after
replacing X with a dense open subset (which must have the same dimension), we may assume that
S is smooth.
Let i : S ,→ X be the inclusion. By the coherent version ofRKashiwara’s theorem,
R0
0
M ' i N for some coherent DS -module N , so we simply replace M with i N . By induction on
dimension, we may assume that d(N ) = dim(Ch(N )) ≥ dim(S). We claim that dim(Ch(N )) + r =
dim(Ch(M )), where r is the codimension of S in X (this claim will imply that dim(Ch(M )) ≥
dim(S) + r = dim(X), a contradiction that completes the proof).
By the composition rule for direct images, it suffices to assume that r = 1. We also assume that
we have coordinates {xi , ∂i } on X such that xn = 0 is a defining equation for S. We write x for xn , ∂
for ∂n , and ξ for the principal symbol ξn . We are simply going to prove here that dim(Ch(N )) + 1 =
dim(Ch(M )). A stronger statement is true: if ρ is the natural map S ×X T ∗ X → T ∗ S, which is
smooth of relative dimension 1, then ρ−1 (Ch(N )) = Ch(M ) ⊂ S ×X T ∗ X ⊂ T ∗ X. (This statement
is local on S and would be needed for our passage to local coordinates to be rigorous.)
In coordinates, we know that M ' k[∂] ⊗k i∗ N ' ⊕j≥0 ∂ j · N . We will show that if we choose our
good filtrations carefully, then gr M is obtained from gr N by adjoining a coordinate (namely ξ).
Let G be a good filtration on N ; after shifting, we may assume G−1 N = 0. Define a good filtration
F on M by setting
Fj M = ∂ j · G0 N + ∂ j−1 · G1 N + · · · + Gj N ;
then we have
∂ j · G0 N + ∂ j−1 · G1 N + · · · + ∂ · Gj−1 N + Gj N
Fj M
=
Fj−1 M
∂ j−1 · G0 N + ∂ j−2 · G1 N + · · · + Gj−1 N
Gj N
Gj−1 N
'
⊕ξ·
⊕ · · · ⊕ ξ j · G0 N,
Gj−1 N
Gj−2 N
20
NICHOLAS SWITALA
so that
grF M = ⊕j Fj M/Fj−1 M
M Gj N
Gj−1 N
j
'
⊕ξ·
⊕ · · · ⊕ ξ · G0 N
Gj−1 N
Gj−2 N
j
' ⊕j ξj · grG N
' k[ξ] ⊗k grG N,
from which it is clear that dim Supp grG N + 1 = dim Supp grF M , as desired.
If M is any coherent DX -module, there exists (see Appendix D in HTT) a filtration
0 = C 2n+1 M ⊂ C 2n M ⊂ · · · ⊂ C 1 M ⊂ C 0 M = M
by coherent DX -submodules such that, for all j such that C j M/C j+1 M 6= 0, every irreducible
component of Ch(C j M/C j+1 M ) has dimension 2n − j. Note that C j M/C j+1 M is a coherent DX module. It follows from Bernstein’s inequality that we must have C n+1 M = C n+2 M = · · · =
C 2n+1 M = 0, and therefore
Corollary. Let M 6= 0 be a coherent DX -module. Then every irreducible component of Ch(M ) has
dimension at least n = dim(X).
The coherent DX -modules that are “as small as possible” (either zero, or of dimension exactly
n) form a privileged class with especially nice properties.
Definition. Let M be a coherent DX -module. We say that M is holonomic if M = 0 or d(M ) = n.
It follows from the previous corollary that if M 6= 0 is holonomic, every irreducible component
of Ch(M ) has dimension exactly n (that is, Ch(M ) is equidimensional). Recall from last week
that if 0 → M 0 → M → M 00 → 0 is a short exact sequence in Modc (DX ), then Ch(M ) is the
union of Ch(M 0 ) and Ch(M 00 ), and consequently d(M ) is the maximum of d(M 0 ) and d(M 00 ). It
follows that M is holonomic if and only if M 0 and M 00 are holonomic. Therefore, the full subcategory
Modh (DX ) ⊂ Modc (DX ) consisting of holonomic modules is a Serre subcategory; since the category
Modc (DX ) is Abelian, so also is Modh (DX ).
If M ∈ Modc (DX ) is actually coherent as an OX -module, we saw in Week 2 that M must be
locally free over OX . Given such an M , we can define a filtration F on M by setting Fi M = 0 if
i < 0 and Fi M = M if i ≥ 0. Since every Fi M is coherent over OX , F is a good filtration, with
grF M ' M . Note that, for all i, ξi ∈ gr DX annihilates grF M (because it increases degree by 1).
Therefore Ch(M ) is only the zero section TX∗ X ⊂ T ∗ X (locally, TU∗ U = U × {0} ⊂ U × k n = T ∗ U ).
Since dim TX∗ X = dim X = n, M is holonomic. In fact, if M 6= 0, Ch(M ) = TX∗ X if and only if M
is coherent over OX (if and only if M is locally free over OX ). From this characterization, we can
see that every holonomic DX -module is generically of this form:
Proposition. Let M be a holonomic DX -module. There exists an open dense subset U ⊂ X such
that M |U is coherent as an OU -module.
Proof. Let S = Ch(M ) \ TX∗ X. If S is empty, then either M = 0 or Ch(M ) = TX∗ X; in the latter
case, M is already coherent over OX and so we can take X for U . Otherwise, since Ch(M ) is conic,
the fibers of π|S : S → π(S) are at least one-dimensional, so dim π(S) < dim S = dim X, and there
is an open dense subset U of X that lies within the complement X \ π(S). But then Ch(M |U ) is
contained in TU∗ U , so M |U is coherent over OU .
BASIC THEORY OF ALGEBRAIC D-MODULES
21
We remark that in fact π(S) in the preceding proof is already closed, so we can simply take
X \ π(S) for U . To see this, note that the restriction of π to T ∗ X \ TX∗ X factors through the
projectivized cotangent bundle ProjX Sym ΘX = P(ΘX ) → X, and this last map is proper; since
S is conic, its image in X must be closed. We also remark that if M is merely coherent over DX
(not necessarily holonomic), there exists an open dense subset V (not the same as the U above)
such that M |V is projective over OV .
Week 11: more on holonomic modules; sins of omission
Last week we defined the category Modh (DX ) of holonomic DX -modules: the Serre subcategory
of Modc (DX ) consisting of 0 together with all coherent DX -modules M such that d(M ) = n (by
Bernstein’s inequality, this is the smallest possible dimension). In order to prove our next structural
result for Modh (DX ), we need to introduce multiplicities. If M is any coherent DX -module with
a chosen good filtration F , and C ⊂ T ∗ X is an irreducible component of Ch(M ), the multiplicity
^
^
F M along C is the length of the stalk of gr
F M at the generic point η of C, viewed as a
of gr
module (which must be of finite length) over the local ring OT ∗ X,η . Let md(M ) (M ) be the sum of
^
F M along the d(M )-dimensional irreducible components of Ch(M ). This
the multiplicities of gr
sum is independent of the choice of good filtration F (the proof of this fact is similar to the proof
that Ch(M ) is independent: prove it first for neighboring filtrations), and since length is additive
in short exact sequences, we have md(M ) (M ) = md(M ) (M 0 ) + md(M ) (M 00 ) for every short exact
sequence 0 → M 0 → M → M 00 → 0 in Modc (DX ).
Theorem. Every holonomic DX -module M is of finite length.
Proof. Every irreducible component of Ch(M ), as well as of Ch(M 0 ) for any DX -submodule M 0 ⊂
M or of Ch(M 00 ) for any DX -module quotient M M 00 , has dimension exactly n. Consider the
multiplicity m(M ) = mn (M ). If m(M ) = 0, then Ch(M ) is empty, and therefore M = 0. Therefore,
if M 0 ⊂ M is a proper DX -submodule (so that M 00 = M/M 0 6= 0 and thus m(M 00 ) > 0), we have
m(M ) = m(M 0 ) + m(M 00 ) > m(M 0 ). The finite length of M follows by induction on m(M ).
R
Recall that if f : X → Y is a morphism, the direct image f and (shifted) inverse image
f † = Lf ∗ [dim
R X − dim Y ] preserve quasi-coherence, but
R do not preserve coherence in general (if f
is proper, f preserves coherence). Remarkably, both f and f † preserve holonomy for general f :
Theorem.
(Preservation of holonomy) Let f : X → Y be a morphism. If M • ∈ Dhb (DX ), then
R
•
b
•
b
†
•
b
f M ∈ Dh (DY ), and if M ∈ Dh (DY ), then f M ∈ Dh (DX ), where the subscript h indicates that
we are considering the derived category of complexes whose cohomology objects are all holonomic.
Next week (the final seminar), we
R will give a proof of the first part of this theorem. In fact the
most difficult step is proving that f preserves holonomy when f is the projection k n → k n−1 ; the
general statement reduces
without too much trouble to this one, and the statement for f † reduces
R
to the statement for f using some standard distinguished triangles in Db (DX ). We consider an
application. Suppose that Y = P is a point. Since X is smooth,
the projection p : X → P is a
R
smooth morphism. Therefore, we can compute the complex p M ∈ Db (DP ) using the de Rham
R
complex: p M is represented by Rp∗ (DRX ), since the de Rham complex of X relative to the
point P is just DRX . The functor pR∗ is simply the global section functor on X, and so Rp∗ is the
hypercohomology functor. That is, p M is represented by RΓ(X, DRX ). The cohomology objects
∗ (M ) of M . If
(k-spaces) of the complex RΓ(X, DRX ) are called the de Rham cohomology spaces HdR
∗ (M ) are holonomic D -modules, that is, finite
M is holonomic, then by the theorem above, all HdR
P
22
NICHOLAS SWITALA
dimensional k-spaces (a DP -module is coherent if and only if it is holonomic). We have therefore
proved
Corollary. If M is a holonomic DX -module, the de Rham cohomology spaces of M are finitedimensional.
Finally, we are going to summarize a couple of the important pieces of the algebraic theory
(the first three chapters of HTT) that we did not cover in detail: the “sins of omission” of this
mini-course. One such omission is the behavior of direct and inverse images for smooth morphisms,
which was summarized in Week 7. We discuss two more here: duality and minimal (or intermediate)
extensions.
Duality. We define a contravariant dual functor D : Db (DX ) → Db (DX ) by
⊗−1
DM • = RHomDX (M • , DX ) ⊗OX ωX
[n],
a derived category version of the usual dual module construction over a non-commutative ring.
Here RHomDX (M • , DX ) becomes a complex of right DX -modules by the right action of DX on
itself, and so the side-changing operation is necessary to recover a complex of left DX -modules.
∼
The functor D preserves coherence, and if M • ∈ Dcb (DX ), then M • −
→ DDM • (the indicated map
always exists; the coherence is necessary for the map to be an isomorphism).
Now we consider the case where M • is concentrated in degree zero, that is, the case of a
single coherent DX -module M . Consider the cohomology objects ExtiDX (M, DX ) of the complex
RHomDX (M, DX ). It is true in general that ExtiDX (M, DX ) = 0 for i < 2n − d(M ), and since DX
has global homological dimension ≤ n, we also know that ExtiDX (M, DX ) = 0 for i > n. Therefore,
if M is holonomic (d(M ) = n), there is exactly one nonvanishing Ext, namely ExtnDX (M, DX ). In
this case, DM is isomorphic in Db (DX ) to a single DX -module concentrated in degree zero (because
of the degree shift in the definition of D) and so we identify DM with this module, namely the left
⊗−1
. (We abuse notation by writing DM instead of D0 M .) If M is
DX -module ExtnDX (M, DX )⊗OX ωX
holonomic, so again is DM ; in fact, their characteristic varieties coincide. As a special case, if M is
locally free as an OX -module, DM ' HomOX (M, OX ) as left DX -modules (recall from Week 2 that
we can define a left DX -module structure on the right-hand side by specifying how the derivations
act). Taking M = OX , we see that ExtnDX (OX , DX ) ' ωX , which is nonzero when restricted to any
open affine U ⊂ X; from this it follows that the ring DX (U ) has global dimension exactly n.
Returning
R to the general framework of derived categories with coherent cohomology, D commutes with f when f : X → Y is proper; the commutativity of D with inverse images is a more
complicated story.
Minimal extensions. Let M be a holonomic DX -module. Since M has finite length, there exists
a composition series 0 = Ml+1 ⊂ Ml ⊂ · · · ⊂ M1 ⊂ M0 = M of (holonomic) DX -submodules such
that Mi /Mi+1 is simple (contains no nontrivial DX -submodule) for all i. The theory of minimal
extensions provides a classification of all simple holonomic DX -modules. We remark that there exist
simple DX -modules that are not holonomic. If X = A2 = Spec k[x, y] and we let
δ = x + y + ∂x + ∂y + y∂x ∂y ∈ DX (X) = A2
(the second Weyl algebra), then δ generates a maximal left ideal in A2 . The quotient A2 /A2 · δ is
a simple left A2 -module that is not holonomic (its dimension is 3). This example is due to Toby
Stafford.
R
R
If f : X → Y is a morphism, we define a functor f ! = DY ◦ f ◦DX : Dhb (DX ) → Dhb (DY ) (we are
R
R
using the theorem on preservation of holonomy). There is a natural transformation f ! → f which
BASIC THEORY OF ALGEBRAIC D-MODULES
23
R
is an isomorphism when f is proper (because in that case f commutes with the dual functors).
Now suppose that Y ⊂ X is a locally closed smooth subscheme such that the inclusion map
i : Y ,→ X is affine. Then Ri∗ ' i∗ (i∗ is exact), and DY ←X is a locally free right DY -module:
j
σ
if we factor i as Y →
− W −
→ X where j is an open immersion and σ a closed immersion, then
DW ←Y = j −1 DW (= DY ) and so
DX←Y ' j −1 DX←W ⊗j −1 DW DW ←Y ' j −1 DX←W ,
and since σ is a closed immersion, DX←WR is a locally
if RM is
R free right DW -module. Therefore,
R0
0
a holonomic DY -module, we can think of i M and i! M as single DX -modules i M and i! M
(all derived functors involved in their definitions become exact), and both of these DX -modules are
R0
R0
holonomic. We have a morphism i! M → i M in Modh (DX ) as above.
Definition. Let Y ⊂ X be a locally closed smooth subscheme such that the inclusion
map
R0
R 0 i : Y ,→ X
is affine. Let M be a holonomic DY -module. The image of the morphism i! M → i M is called
the minimal (or intermediate) extension of M , and denoted L(Y, M ).
Observe that if Y is actually a closed subscheme, then i is proper, in which case L(Y, M ) is
R0
simply i M . The following is the main classification theorem for simple holonomic DX -modules
using minimal extensions:
Theorem. Let Y , X, and i be as in the definition above. If M is a simple holonomic DY -module,
then L(Y, M ) is a simple holonomic DX -module; indeed, it is the unique simple DX -submodule of
R0
i M . Conversely, given X, any simple holonomic DX -module is of the form L(Y, M ) for some
locally closed smooth subscheme Y of X with affine inclusion map and some simple DY -module
that is coherent as an OY -module.
Week 12: preservation of holonomy
We begin by revisiting the example from last week of a simple, non-holonomic module. Let
X = An = Spec k[x1 , . . . , xn ], so that DX = DX (X) is the nth Weyl algebra An . Let δ ∈ DX
be given. We are going to compute the characteristic variety of DX /(DX · δ). Suppose that δ ∈
Fl DX \ Fl−1 DX , and let ξ = σl (δ) 6= 0 be the principal symbol of δ in Fl DX /Fl−1 DX ⊂ gr DX .
The order filtration F on DX induces filtrations F 0 on the left ideal DX · δ and F 00 on the quotient
0
DX /DX · δ. We have Fp0 (DX · δ) = (Fp−l DX ) · δ for all p, and so grF (DX · δ) = (gr DX ) · ξ. The
short exact sequence 0 → DX · δ → DX → DX /(DX · δ) → 0 induces a short exact sequence of
associated graded objects (since F 0 and F 00 are induced from F ), from which we conclude that
00
grF (DX /(DX · δ)) ' gr DX /(gr DX · ξ),
a hypersurface ring. Recall that gr DX is simply the polynomial ring k[x1 , . . . , xn , ξ1 , . . . , ξn ]. The
characteristic variety Ch(DX /(DX · δ)) is the closed set V (ξ) ⊂ Spec k[x1 , . . . , xn , ξ1 , . . . , ξn ]. In
particular, its dimension (the dimension of the DX -module DX /(DX · δ)) is 2n − 1. In our example
from last week, n = 2 and δ = x + y + ∂x + ∂y + y∂x ∂y . Since δ ∈ F2 DX \ F1 DX , its principal
symbol in k[x, y, ξx , ξy ] is yξx ξy , and its characteristic variety is V (yξx ξy ) ⊂ Spec k[x, y, ξx , ξy ], a
three-dimensional closed subscheme (the union of three coordinate hyperplanes in affine 4-space).
Before turning to the proof of the theorem on preservation of holonomy, we need to discuss a
b (D ). Suppose first that X is merely a
certain distinguished triangle in the derived category Dqc
X
topological space and Z ⊂ X is a closed subset. Write i for the closed immersion Z ,→ X and j for
the complementary open immersion U = X \ Z ,→ X. If I is an injective object in the category of
Abelian sheaves on X, there is a well-known short exact sequence
0 → ΓZ (I) → I → j∗ j −1 I → 0
24
NICHOLAS SWITALA
where ΓZ denotes the subsheaf of sections supported on Z. (The kernel of F → j∗ j −1 F is ΓZ (F)
for any Abelian sheaf F on X, more or less by definition, but if F is not an injective sheaf, then
F → j∗ j −1 F may have a nontrivial cokernel.) This short exact sequence for injective sheaves induces
a distinguished triangle
+1
RΓZ (F• ) → F• → Rj∗ j −1 F• −−→
in the bounded derived category of Abelian sheaves on X. Now suppose X is a smooth scheme
R −1 (sat•
b
−1
•
isfying our current laundry list of hypotheses) and M ∈ Dqc (DX ). Then Rj∗ j M = j j M • =
R † •
†
j j M , where j is the shifted inverse image functor (there is no degree shift in this case, because
R
U is a dense open subset). If we assume furthermore that Z is smooth, then RΓZ (M • ) ' i i† M •
(this is an application of the derived version of Kashiwara’s theorem; here the quasi-coherence
is necessary). Therefore, if Z is a smooth closed subscheme of a smooth scheme X, we have a
distinguished triangle
Z
Z
+1
j † M • −−→
i† M • → M • →
j
i
in
b (D )
Dqc
X
whenever
M•
belongs to this derived category.
We now sketch a proof of the “forward” direction of the theorem on preservation of holonomy,
which we restate below. (The “backward” direction, for the inverse image functor f † , can be reduced
to the “forward” direction using the distinguished triangle above.)
Theorem.
Let f : X → Y be any morphism between smooth schemes. If M • ∈ Dhb (DX ), then
R
•
b
f M ∈ Dh (DY ).
Sketch of proof. By induction on the cohomological length of M • , it suffices to prove the theorem
in the case where M ∈ Modh (DX ) is a single holonomic DX -module concentrated in degree zero.
Using the graph of f , we can factor f into a closed immersion followed by a projection, so it
suffices to prove the theorem
R 0 in case f is a morphism of one of these two types. If f = i is a closed
immersion X ,→ Y , then i is exact, and a coherent DX -module M is holonomic if and only if
R0
R0
i M is a holonomic DY -module, as we saw in the proof of Bernstein’s inequality (the functor i
increases the dimension of Ch(M ) by exactly dim Y − dim X). The fact that this is an if and only
if statement in this case will be important below.
Now suppose f =
R p is a projection X = Y × Z → Y . We need only to show that if M ∈
Modh (DY ×Z ), then p M ∈ Dhb (DY ). We may assume that both X and Y are affine: the problem
is local on Y , so we may immediately reduce to the case of affine Y , and as for X, we use the Čech
complex. We can find a finite affine open covering {Ui = Y × Vi } of X. Since we have assumed X is
separated, M is isomorphic in Db (DX ) to the Čech complex of M with respect to the cover {Ui }.
If U is the intersection of some subcollection
of the Ui and j : U ,→ X is the corresponding (affine)
R
open immersion, then j∗ (M |U ) = j j ∗ M (since j is affine and hence Rj∗ = j∗ ). The Čech complex
R
of M is a direct sum of sheaves of the form j∗ (M |U ). Therefore, to show that p M ∈ Dhb (DY ),
it
(since
finite direct sums of holonomic modules are again holonomic) to show that
R Ris enough
R
∗M =
∗ M ∈ D b (D ), and therefore we may assume X = U is affine as well. Fix
j
j
Y
h
p j
p◦j
embeddings α : X ,→ k n , β : Y ,→ k m of X and Y into affine spaces, and consider the graph
morphism Γp : X → X × Y . The composite
Γp
α×β
π
X −→ X × Y −−−→ k n+m = k n × k m −
→ km ,
where π is the projection, coincides with the composite
p
β
X→
− Y −
→ km
BASIC THEORY OF ALGEBRAIC D-MODULES
25
R
R R
by definition of the graph. Since β is a closed immersion, p M ∈ Dhb (DY ) if and only if β ◦ p M =
R
R
M ∈ Dhb (Dkm ). Since β ◦ p = π ◦ (α × β) ◦ Γp , it is enough to check that π◦(α×β)◦Γp M =
β◦p
R R
R
◦ Γp M ∈ Dhb (Dkm ). But both Γp and α × β are closed immersions, so we already
π ◦ (α×β)
R
R
know (α×β) ◦ Γp M ∈ Dhb (Dkn+m ). We are therefore reduced to the case where p is the projection
π : k n+m → k m . By factoring this projection into n projections of relative dimension one, we may
assume finally that X = k n = k × k n−1 , that Y = k n−1 , and that p : X → Y is the projection onto
the final n − 1 coordinates. Beginning now, we will not distinguish between the sheaf Dkn and its
ring of global sections Dkn (the nth Weyl algebra); likewise for Dkn−1 and for modules over these
two rings.
We write i for the closed immersion {0} × k n−1 = k n−1 ,→ k n and j for the complementary open
immersion k ∗ × k n−1 ,→ k n . What follows is a summary of the proof that the projection p preserves
holonomy. We have a distinguished triangle
Z
Z
+1
†
i M → M → j † M −−→
j
i
which, upon taking cohomology, induces an exact sequence
Z
Z
Z
0 → h0 ( i† M ) → M → h0 ( j † M ) → h1 ( i† M ) → 0
i
j
i
R
since M is concentrated in degree 0. We know M is holonomic, so if h0 ( j j † M ) is holonomic,
R
R
then hl ( i i† M ) is holonomic for l = 0, 1 by the exact sequence above. As i i† M has nontrivial
R
cohomology only in degrees 0, 1 ( i is exact, and i† has cohomological dimension equal to the
R
codimension of i), we have i i† M ∈ Dhb (Dkn ). Since i is a closed immersion, this implies i† M ∈
Dhb (Dkn−1 ).
For the proof outline above to be complete, we would need to fill the following gaps:
R
R
(a) if M is holonomic, h0 ( j j † M ) is holonomic (here h0 ( j j † M ) is just the localization Mx1 , but
proving this is holonomic takes
some work);
R
(b) if i† M ∈ Dhb (Dkn−1 ), then p M ∈ Dhb (Dkn−1 ) as well.
The key to filling in the gaps above is the Fourier transform operation. Given a Dkn -module M ,
c is the Dkn -module whose underlying Abelian group is the same but whose
its Fourier transform M
D-action is defined by xi ∗ m = −∂i · m, ∂i ∗ m = xi · m. Here {xi , ∂i }ni=1 is a coordinate system
c, and the minus sign is necessary for the new action to respect
for k n , ∗ defines the D-action on M
n
if and only if
the relations in D. It is a general fact that a coherent
R Dk -module M is holonomic
†
c
M is as well. We can pass from the cohomology of p M to the cohomology of i M by a sequence
of Fourier transforms and degree shifts, which is how (b) is proved.
In order to prove (a) and the fact that Fourier transform respects holonomy, one uses the
Bernstein filtration, where both xi and ∂i have degree 1, rather than the usual order filtration. If F
is a good filtration on a coherent Dkn -module M with respect to the Bernstein filtration, then Fl M
is a finite-dimensional k-space for all l, which allows us to use Hilbert function techniques to study
such filtered modules (there is a criterion for holonomy of M based on the asymptotic growth of
the k-dimensions of Fl M ).