D-module lecture note 16
Dongkwan Kim
April 29, 2015
Remark. Unless otherwise stated, Mod(DX ) denotes the category of ”left” DX -modules.
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Introduction: Inverse Image Preserves Quasi-coherence
Let f : X → Y be a morphism of smooth varieties. Then we already know that the (higher)
pull-back of a quasicoherent sheaf is quasicoherent. Since a pull-back as a D-module and
as an O-module are the same (with some shift,) it is still true for a D-module case. (One
might doubt it since in each case the category on which Li f ∗ is defined is different, but we
can compose a forgetful functor Db (Mod(DX )) → Db (Mod(OX )) to identify them. Here
we use the fact that DX is flat over OX .)
How about the (total) derived functor (with some shift,) i.e. f ! ? By the argument right
b (Mod(O )), and
before, we see that f ! sends objects in Db (Modqc (OY )) to at least Dqc
Y
we use the following
Theorem 1 (Bernstein). Let A be a quasicoherent sheaf of associative algebras on X.
Then the natural morphism
b
Db (Modqc (A)) → Dqc
(Mod(A))
is an equivalence of categories.
Proof. [1, Thm VI.2.10].
2
Direct Image Preserves Quasi-coherence
How about push-forward? If we consider sheaf-theoretic push-forward, it is still true by [2,
III.8]. In general, Ri f• preserves quasi-coherence if f is quasi-compact and quasi-separated
(where f• will denote the sheaf-theoretic push-forward from now on.) Also, it can be
shown that Ri f• preserves O-coherence when f is proper. But in D-module case, it needs
a (nontrivial) amount of work, since at first push-forward is not the same in either cases.
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We recall the definition; let M ∈ Modr (DX ). Then we have
f∗ M = Rf• (M ⊗L
DX DX→Y )
Now we have a general strategy; we can decompose f with a locally closed embedding (i.e.
Γf : X → X × Y ) and a projection. But before that, we need to check the following crucial
(which can be easily overlooked) fact.
Proposition 2. If g : Y → Z and Z is also smooth, then (g ◦ f )∗ = g∗ ◦ f∗ .
Proof. [3, Prop 1.5.21].
Now we break f down to several pieces. First, if f is an open embedding, then M ⊗L
DX
DX→Y = M thus f∗ = Rf• , and the result is clear. If f is a closed embedding, then we use
the general argument to assume that X is of codim 1 in Y defined by a single equation,
and in this case DX→Y is a free DX -module. Thus M ⊗L
DX DX→Y = M ⊗DX DX→Y , and
this case is similarly clear. (Note that Rf• = f• , since f is affine.)
Before we proceed for projections, we stop here(!) and use a different approach. First
we prove that M ⊗L
DX DX→Y can be represented by a complex of quasi-coherent sheaves.
We recall the definition of a de Rham complex
dR(DX ) = 0 → DX → DX ⊗OX Ω1X → DX ⊗OX Ω2X → · · · → DX ⊗OX ΩnX → 0
which is a resolution of ΩnX . Now we define the Koszul complex of M , say Kos(M ), to be
M ⊗OX dR(DX ) ⊗OX (ΩnX )∨ . Then we see that Kos(M ) is quasi-isomorphic to M . Also
note that the right DX -module structure on this complex comes from that on dR(DX ).
Thus even though M might not be locally free as a DX -module, hence so is Kos(M ), it is
still true that Kos(M ) is locally free as a k[∂i ]-module. (Here ∂i denotes the vector field
which corresponds to some local étale coordinate system.) But since DX→Y is locally free
as a OX -module, we have Kos(M ) ⊗L
DX DX→Y ' Kos(M ) ⊗DX DX→Y . Thus we see that
L
M ⊗DX DX→Y ' Kos(M ) ⊗DX DX→Y which is a complex of quasi-coherent sheaves.
(That is to say, for W = khx, ∂i/(∂x − x∂ − 1) and an inclusion 0 → k[∂]I → k[∂]J , we
still have an inclusion 0 → k[∂]I ⊗W k[x]K → k[∂]J ⊗W k[x]K ).
Now we wish to prove that Ri f• K • is quasi-coherent. (Indeed, this is the most nontrivial
part, even though it would be (relatively) obvious if K • is quasi-isomorphic to a quasicoherent sheaf.) For this, we take an affine open cover X = ∪i Ui and consider the Čech
complex Č(K • ) which corresponds to K • , or the total complex of the bi-complex
M
M
ji∗ K • |Ui →
jij∗ K • |Uij → · · ·
i
i,j
which is quasi-isomorphic to K • and which consists of quasi-coherent sheaves. The upshot
of this complex is that we have Rf• Č(K • ) ' f• Č(K • ) since we take the affine open cover.
(i.e. push-forward is exact.)
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Now we come back to the case of projections. In this case, the first part of previous
argument is relatively easy, since for a projection X × Y → Y , we have DX×Y →Y =
OX DY , and we have
n ∨
M ⊗L
DX DX→Y ' M ⊗DX×Y (dR(DX ) ⊗OX (ΩX ) ) DY
(It is just the reformulation of Kos(M ).) However, there is no way to avoid the second
part, even though we may reduce to the case dim X = 1, where dR(DX ) consists of only
two nontrivial terms.
Remark. From now on, we mainly deal with quasi-coherent D-modules.
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Properties of Direct Image
We already observed some properties of direct images. But to use the mixture of direct
images, inverse images, and duality functor, we need to correct it so that it is a functor
of left D-modules using the equivalence. Following the notation of [3], we define f∗ =
−1 K ∨ .
Rf• (DY ←X ⊗L
Y
DX −), where DY ←X = KX ⊗OX DX→Y ⊗f −1 OY f
If f is an open embedding, then DY ←X = DX→Y = DX , thus f∗ = Rf• by the
argument above. If f is a closed immersion, then by similar argument we see that f∗ =
f• (DY ←X ⊗DX −).
Now if f is a projection, we have an explicit description of DY ←X ⊗L
DX M as before.
DY ←X×Y ⊗DX×Y M = f −1 (DY ⊗OY KY∨ ) ⊗f −1 OY KX×Y ⊗DX×Y M
= (KX DY ) ⊗DX×Y M
since KX×Y ' KX KY . Now we use the resolution dR(DX ) of KX to obtain
DY ←X×Y ⊗DX×Y M ' (dR(DX ) DY ) ⊗DX×Y M.
where
dR(DX ) = 0 → DX → DX ⊗OX Ω1X → · · · → DX ⊗OX ΩnX → 0.
Now assume f is smooth. Then we can directly convert the argument above to the
local picture of this case, and we see that dR(DX ) DY corresponds to the relative de
Rham complex. In other words, DY ←X ⊗L
DX M is quasi-isomorphic to
X−dim Y
dRX/Y M := 0 → M → M ⊗OX Ω1X/Y → · · · → M ⊗OX Ωdim
→0
X/Y
Now I copy the statement on the lecture note: ”If M = O, it is a bundle with a flat
connection whose fibers are the de Rham cohomology of the fibers of π, and the connection
is the Gauss-Manin connection.”
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4
Base Change
Suppose we have a following commutative diagram
P
f˜
S
g̃
g
/X
f
/Y
where all varieties are smooth. Then for any F ∈ Db (Modqc (DX )), we may define
g ! f∗ F, f˜∗ g̃ ! F, where the shriek pullback is defined on the previous talk;
−1 •
f ! K • = DX→Y ⊗L
K [dim X − dim Y ] = Rf ∗ K • [dim X − dim Y ].
f −1 DY f
In some good situation, indeed push-forward and shriek pull-back are adjoint, i.e. f∗ a f ! ,
thus there exists a natural morphism F → f ! f∗ F ⇒ g̃ ! F → g̃ ! f ! f∗ F = f˜! g ! f∗ F ⇒ f˜∗ g̃ ! F →
g ! f∗ F.
Now we assume furthermore that this is a pull-back diagram, i.e. P = X ×Y S. Then
we have the following base change property.
Theorem 3. There is a natural isomorphism of functors g ! f∗ (−) → f˜∗ g̃ ! (−). (NOTE: It
is always well-defined without using adjunction if the diagram is cartesian.)
To prove this, we use the similar trick, i.e. break g into easier pieces. First, if g
is an open embedding, then this is clear since g ! and g̃ ! are just restrictions and direct
image is compatible with it. Now we switch the order and assume g is a projection. Thus
g : S = Y × Z → Y , and P = X × Z. Then we see that g̃ ! F = F OZ [dim Z] by definition,
thus f˜∗ g̃ ! F = f∗ F OZ [dim Z] = g ! f∗ F.
Before we delve into the case of closed embedding, we state some properties without
detailed proof.
Proposition 4. Suppose we have a closed embedding i : Z ,→ X of smooth varieties and
let j : U ,→ X be its complement.
(1) We have a functorial isomorphism (where Hom is derived internal Hom, i.e. HomX =
R(HomMod(DX ) ) = HomDb (Mod(DX )) )
HomX (i∗ M • , N • ) ' i• HomZ (M • , i! N • )
for M • ∈ D− (Mod(DZ )) and N • ∈ D+ (Mod(DX )).
(2) i! : Db (DX ) → Db (DZ ) is right adjoint to i∗ : Db (DZ ) → Db (DX ), i.e. i∗ a i! .
b (D ) → D b
!
(3) We have an equivalence i∗ : Dqc
Z
qc,Z (DX ) whose quasi-inverse is i . It is also
true for D-coherent case.
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b (D ), we have i! j M • = 0.
(4) For M • ∈ Dqc
∗
U
b (Mod(D )), we have a distinguished triangle
(5) For K • ∈ Dqc
X
i∗ i! K • → K • → j∗ j ! K • → i∗ i! K • [1].
Proof. The proof of (1) and (2) is given in [3, Prop 1.5.25]. Note that (2) follows from (1)
by taking H 0 (X, −) on both sides. (3) is [3, Cor 1.6.2]. For (4), the result follows from
−1
•
L
−1
•
i• (OZ ⊗L
i−1 OX i j∗ K ) = i• (OZ ⊗i−1 OX i Rj• K )
•
= i• OZ ⊗L
OX Rj• K
= Rj• (j
−1
i• OZ ⊗L
OU
(∵ projection formula )
•
K ) = 0.
For (5), note that we have a sheaf-theoretic distinguished triangle
RΓZ (K • ) → K • → Rj• j −1 K • → RΓZ (K • )[1].
Indeed, by [2, Ex. II.1.20] we have an exact sequence 0 → ΓP (F) → F → j• j −1 F, and the
last map is surjective if F is flabby. Thus the result follows by deriving this. Then since j
is an open embedding j −1 = j ! and Rj• = j∗ . Also, one can show that RΓP (K • ) = i∗ i! K • ,
even though it is not trivial. (I think (3) and (4) are essential to prove this.) We refer
readers to [3, Prop 1.7.1].
Now we assume that g is a closed embedding. Set i = g. Let U = X \ P and V = Y \ S
which are open on X and Y respectively, and consider the following diagram.
ĩ
P
f |P
S
i
/ X o j̃
U
f
/Y o
j
f |U
V
As before we have a distinguished triangle ĩ∗ ĩ! F → F → j̃∗ j̃ ! F → ĩ∗ ĩ! F[1]. Thus by
applying f∗ we have a following diagram
f∗ ĩ∗ ĩ! F
i∗ i! f∗ F
/ f∗ F
/ f∗ F
/ f∗ j̃∗ j̃ ! F
/ f∗ ĩ∗ ĩ! F[1]
'
/ j∗ j ! f∗ F
/ i∗ i! f∗ F[1]
where the second isomorphism is from the argument for open embeddings. We wish to
show that we have an isomorphism f∗ ĩ∗ ĩ! F = i∗ f∗ ĩ! F → i∗ i! f∗ F, since then we can use
Kashiwara’s theorem. Also it suffices to construct some morphism which makes the diagram
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commute by five lemma. For this, we apply Hom(f∗ ĩ∗ ĩ! F, f∗ −) to the distinguished triangle
ĩ∗ ĩ! F → F → j̃∗ j̃ ! F → ĩ∗ ĩ! F[1] to get
· · · → Hom(f∗ ĩ∗ ĩ! F, f∗ ĩ∗ ĩ! F) → Hom(f∗ ĩ∗ ĩ! F, f∗ F) → Hom(f∗ ĩ∗ ĩ! F, f∗ j̃∗ j̃ ! F) → · · · .
But
Hom(f∗ ĩ∗ ĩ! F, f∗ j̃∗ j̃ ! F) = Hom(i∗ f∗ ĩ! F, j∗ f∗ j̃ ! F)
(or j∗ j ! f∗ F)
= Hom(f∗ ĩ! F, i! j∗ f∗ j̃ ! F) = 0
by the proposition (3) and (4) above. Thus Hom(f∗ ĩ∗ ĩ! F, f∗ ĩ∗ ĩ! F) → Hom(f∗ ĩ∗ ĩ! F, f∗ F)
is an isomorphism, and we can take the image of natural morphism f∗ ĩ∗ ĩ! F → f∗ F of
the pull-back of ĩ∗ ĩ! F → F defined by adjunction. Now it is straightforward that this
morphism makes the diagram commute (by algebraic nonsense.)
Remark. One can show that this isomorphism is independent of the choice of the decomposition of g.
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More Properties
Lemma 5. If f : X → Y is smooth, then f ! preserves D-coherence.
Proof. First note that f is smooth, thus flat, which means OX is flat over f −1 OY , thus f ∗
is exact and we have f ! = Lf ∗ [dim X − dim Y ] = f ∗ [dim X − dim Y ]. Now it suffices to
show that DX → DX→Y : L 7→ L(1 ⊗ 1) is surjective, and we use local coordinate system
(note that f locally looks like a projection.)
Lemma 6. If f : X → Y is projective, then f∗ preserves D-coherence.
Proof. If f is a closed embedding, then f∗ = f• (DY ←X ⊗DX −) and it is trivial since
f∗ DX = f• DY →X is coherent. (Locally it is isomorphic to DY /DY IX .) Note that any
complex in Dcb (Mod(DX )) can be resolved by direct summands of locally free DX -modules
of finite rank. Thus we may assume X = PnY . By the similar reason, we only need to prove
that f∗ DX is DY -coherent. But since DY ←X = KPn DY , we see that
f∗ DX = Rf• DX→Y = DY [−n]
by computing the cohomology of the canonical bundle of a projective space. The result
follows.
From now on, we only deal with D-coherent cases.
Proposition 7. If f : X → Y is projective, then f∗ is left adjoint to f ! and f∗ commute
with D.
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Proof. If f is a closed embedding, then the first part is true by the proposition above. For
the second part, we have
DY f∗ M • = HomY (f∗ M • , DY )[dim Y ] ⊗OY KY∨
= f• HomX (M • , f ! DY )[dim Y ] ⊗OY KY∨
= f• HomX (M • , DX→Y [dim X − dim Y ])[dim Y ] ⊗OY KY∨
= f• (DX→Y ⊗DX HomX (M • , DX )[dim X]) ⊗OY KY∨
= f• (KX ⊗OX DX→Y ⊗X DX M • ) ⊗OY KY∨
= f• (KX ⊗OX DX→Y ⊗f −1 OY f −1 KY∨ ⊗DX DX M • )
= f∗ DX M • .
Now we assume X = PnY . Since DX is a projective generator of Db (Mod(DX )), it suffices
to show for DX . First, we wish to show HomY (f∗ DX , M • ) = Rf• HomX (DX , f ! M • ). But
we already know that f∗ DX = DY [−n], thus HomY (f∗ DX , M • ) = M • [n]. On the other
hand, f ! M • = OPn M • [n], thus Rf• HomX (DX , f ! M • ) = Rf• (OPn M • [n]) = M • [n] by
computing the cohomology of a projective space. The second part can be also similarly
proved as above.
Proposition 8. If f : X → Y is smooth, then Df ! [dim Y − dim X] = f ! D[dim X − dim Y ]
and f ! [2(dim Y − dim X)] is left adjoint to f∗ .
Proof. By similar reason, we wish to prove
Rf• HomX (f ! [2(dim Y − dim X)]DY , M • ) = HomY (DY , f∗ M • ).
•
and we have HomY (DY , f∗ M • ) = f∗ M • = Rf• (DY ←X ⊗L
DX M ) where DY ←X = KX ⊗OX
DX→Y ⊗f −1 OY f −1 KY∨ . Thus it suffices to check
•
HomX (f ! [2(dim Y − dim X)]DY , M • ) = DY ←X ⊗L
DX M .
Also, f ! DY = DX→Y [dim X − dim Y ], thus it reduces to
•
HomX (DX→Y , M • )[dim X − dim Y ] = DY ←X ⊗L
DX M .
Meanwhile, for f smooth we have a nice relation DY ←X = KX ⊗OX DX→Y ⊗f −1 OY f −1 KY∨ =
DX→Y ⊗OX KX/Y , and also DY ←X = dRX/Y DX . Thus
∨
∨
DX→Y ' 0 → DX ⊗OX KX/Y
→ DX ⊗OX Ω1X/Y ⊗OX KX/Y
→ · · · → DX → 0.
Using this resolution, we see that
∨
HomX (DX→Y , M • )[dim X − dim Y ] ' M • ⊗OX (Ω−•
X/Y ) ⊗OX KX/Y
•+dim X−dim Y
' M • ⊗OX ΩX/Y
•
•
which is nothing but DY ←X ⊗L
DX M ' dRX/Y (M ). Now the first assertion follows by
the similar argument above.
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References
[1] Borel, Armand, et al. Algebraic D-modules. Vol. 14. Boston: Academic press, 1987.
[2] Hartshorne, Robin. Algebraic geometry. Vol. 52. Springer Science & Business Media,
1977.
[3] Hotta, Ryoshi, and Toshiyuki Tanisaki. D-modules, perverse sheaves, and representation theory. Vol. 236. Springer Science & Business Media, 2008.
[4] Etingof, Pavel. Lecture 16 (D-module lecture note). http://www-math.mit.edu/
~etingof/769lect16.pdf
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