D-module lecture note 9
Dongkwan Kim
March 12, 2015
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Proof of Kashiwara’s Theorem
Theorem 1.1 (Kashiwara). Let i : X → Y be a closed embedding. Then i∗0 is an equivalence between Mr (DX ) and MrX (DY ), where MrX (DY ) is the category of right DY -modules
set-theoretically supported on X, i.e. for any f ∈ JX and m ∈ M , mf N = 0 for some
N ∈ N.
To prove this, we first show the necessary condition.
Theorem 1.2. The image of i∗0 : Mr (DX ) → Mr (DY ) is contained in MrX (DY ).
Proof. Indeed, i∗0 (M ) = M ⊗DX DX→Y , and DX→Y = OX ⊗OY DY = DY /J DY for a
closed immersion. So we only need to prove this for DX→Y . But note that the DY action
is defined on the right, that is we need to prove for any m ∈ DX→Y , mJ N = 0 for some
N ∈ N. But it is still true by using commutator relation, i.e. for any f ∈ J and for
any ∂ N , ∂ N f N +1 = ∂ N −1 f N +1 ∂ + ∂ N −1 (N + 1)f 0 f N = · · · = f (∂ N + · · · ), whence the
assertion.
Now we define a functor i!0 : Mr (DY ) → Mr (DX ) as
i!0 M = HomOY (OX , M ) = {m ∈ M | mJ = 0}
To give a right DX -module structure, for any vector field v ∈ V ec(X) you need to define
something like mv such that mvJ = 0. For this, we need the following lemma.
Lemma 1.3. For any v ∈ V ec(X), we can locally find ṽ ∈ V ec(Y ) such that ṽ|X = v and
ṽ preserves J .
Proof. Since X is smooth, it is a local complete intersection [1, Theorem 8.17, Example 8.22.1]. Thus for any x ∈ X, there exists U ⊂ X and an étale coordinate system
x1 , · · · , xk , y1 , · · · , yl such that X ∩ U = V (y1 , · · · , yl ) and x1 , · · · , xk gives an étale coordinate system on X. Now it is clear that any vector field on X can be lifted locally, and
they indeed preserve J .
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Now we define m · v = mṽ. It is invariant under the choice of the lifting, since if ṽ 0
is another lifting we have m(ṽ − ṽ 0 ) = 0 since ṽ − ṽ 0 ∈ J V ec(Y ). (The restriction on X
is zero. Use the local chart around the point as before!) Also mvJ = 0 is zero since v
preserves J ; indeed, for any f ∈ J , mvf = m[v, f ] + mf v = 0 since [v, f ] = v(f ) ∈ J .
Why do we define this functor?
Theorem 1.4. (1) i!0 is the right adjoint to i∗0 . i.e. i∗0 : Mr (DX ) ↔ Mr (DY ) : i!0 .
(2) If we restrict i!0 to MrX (DY ), then they are mutually inverses.
Proof. HomDY (i∗0 M, N ) = HomDX (M, i!0 N )? f ∈ HomDY (i∗0 M, N ), define f˜ : M →
i∗0 M → N by M → i∗0 M : m 7→ m ⊗ 1 ∈ M ⊗DX DX→Y . Since for a ∈ J and m ∈ M ,
f˜(m)a = f (m ⊗ 1)a = f (ma ⊗ 1) = f (m ⊗ a) = 0, f˜(M ) ⊂ i!0 N . Also for any v ∈ V ec(X),
f˜(mv) = f (m ⊗ ṽ) = f (m ⊗ 1)ṽ where ṽ is the lifting of v on Y (locally.) (Note the
definition of DX -action on i!0 N ! Also v · (1 ⊗ 1) = 0 + 1 ⊗ v by the DX -module structure
of DX→Y .)
Conversely, for g ∈ HomDX (M, i!0 N ), we define g̃ = i∗0 M = M ⊗DX DX→Y → N :
m ⊗ L 7→ g(m)L. Since DX→Y = DY /J DY , it is well-defined by the definition of i!0 N . Is
it DY -linear? if v ∈ V ec(Y ), then g̃(m ⊗ Lv) = g(m)Lv. Clear! Now they are inverse to
each other, by just following the definition.
By adjointness, we have a canonical morphism M → i!0 i∗0 M for M ∈ Mr (DX ) and
i∗0 i!0 N → N for N ∈ MrX (DY ). We will show that they are isomorphisms. WLOG
we may look at this locally, and by induction on codimension we may assume that X
is a smooth codim 1 hypersurface defined by a single f , thus locally we have an étale
coordinate y1 , · · · , ym and corresponding vector fields ∂1 , · · · , ∂m = ∂ on Y such that
ym = f . We will show that DX→Y = ⊕n∈N DX ∂ n , thus in particular free over DX . But
nm−1
indeed DX→Y = DY /f DY , and DX = ⊕OX ∂1n1 · · · ∂m−1
.
Now, thus, i∗0 M = ⊗n M ∂ n . What is killed by J , i.e. f ? Since m∂ n f = mf ∂ n +
nm∂ n−1 = nm∂ n−1 , (f m∂ n = 0) only M ∂ 0 part is killed, thus i!0 i∗0 M = M .
×f
Conversely, for N ∈ MrX (DY ), i!0 N = ker(N −−→ N ). Then i∗0 i!0 N = ⊕n∈N (i!0 N )∂ n .
First note that i∗0 i!0 N ⊂ N is an injection; if n∂ l = 0 ∈ i∗0 i!0 N , then by applying f l
times on the right we get n = 0. Also it is the direct sum since each part has different
eigenvalue w.r.t. f ∂. Meanwhile, we know that f acts locally nilpotently on N since N is
set-theoretically supported on X. Thus it suffices to show that on N/i∗0 i!0 N f has zero
kernel. But if nf ∈ i∗0 i!0 N , then nf = ñf for some ñ in i∗0 i!0 N since f : i∗0 i!0 N → i∗0 i!0 N
is a surjection. But it means n − ñ ∈ i!0 N ⊂ i∗0 i!0 N . Thus i∗0 i!0 N = N as desired.
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Definition of Mr (DX ) for a Singular Variety X
Now for any affine X not necessarily nonsingular, we first choose a closed embedding
f : X → An , or more generally f : X → Y where Y is a smooth affine variety and
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define Mr (DX ) = MrX (DY ). If X is smooth, then it is the same as our original definition
by Kashiwara’s theorem. To show that this is well-defined, you need to check that it is
invariant under the choice of embedding.
Remark. We only need to consider the closed embedding to affine spaces, since for general
X → Y , then we embed Y into an affine space and use Kashiwara’s theorem to deduce
that MrX (DY ) = Mr (DX ).
Suppose we have two embeddings i1 : X → An1 = A1 and i2 : X → An2 = A2 . Then
we will construct the canonical equivalence φ21 : Mi1 (DX ) → Mi2 (DX ) where Mir (DX )
i
j
1
are defined in the obvious sense. First, we assume X −
→
A1 →
− A2 where j is also a closed
r
embedding and j ◦ i1 = i2 . Then we set φ21 = j∗0 : MX (DA1 ) → MrX (DA2 ). We claim this
is an equivalence; to prove this, we mimic the proof of Kashiwara’s theorem. i.e.
(1) j∗0 MrX (DA1 ) ⊂ MrX (DA2 )?
(2) define j !0 : MrX (DA2 ) → MrX (DA1 )? i.e. HomOA2 (OA1 , M ) is supported on X?
(3) how to define DA1 -action?
(4) are they adjoint to each other?
(5) are they mutually inverses?
The only difference is that the ideal of X becomes smaller, but we can still use an étale
coordinate system since A1 and A2 are smooth regardless of X.
In general, we will use the following theorem.
Theorem 2.1 (Schwede, [2], 3.4). Suppose A and B are rings. Further suppose I is an
ideal of A and there exists a map γ from B to A/I. We will denote the quotient map
from A to A/I by π. Let X = SpecA, Y = SpecB, and Z = SpecA/I, so that Z is a
closed subscheme of X. Then X ∪Z Y is an affine scheme with Y a closed subscheme,
(X ∪Z Y ) − Y ∼
= X − Z, and the maps α : X → X ∪Z Y and β : Y → X ∪Z Y are
morphisms of schemes.
Also following the proof, we see that if X, Y, Z are affine k-varieties the the pushout is
at least an affine scheme of finite type over k. In other words, in general we can always
have an affine space A12 and the following commutative diagram;
X
i2
i1
A1
j1
/ A2
j2
/ A12
r
r
Now we define φ21 = (j2 )−1
∗0 (j1 )∗0 : MX (DA1 ) → MX (DA2 ).
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Is it well-defined? i.e. φ21 is canonical? for this, we use the following diagram;
X
/ A12
=
/ A1
!
/ A0
12
A2
g
f
!
/ A00
12
which exists by the theorem of Schwede. Since f∗0 : MrX (A012 ) → MrX (A0012 ) and g∗0 :
MrX (A12 ) → MrX (A0012 ) are isomorphisms, the assertion follows from the functoriality and
diagram chasing.
Now we need to show that φ21 satisfies the cocycle condition, from which the welldefinedness of Mr (DX ) follows. For this, we use the same trick;
X
> A1
/ A12
=
/ A123
<
/ A2
/ A23
=
0
<A
A3
/ A13
and again this is justified by diagram chasing.
Now note that for i : X → An , we have a functor Γ : Mr (DX ) = MrX (DAn ) →
r
M (OX ) : M 7→ {m ∈ M | mJ = 0} = i!0 M as an OX -module. It’s called the functor
of global sections. Is it well-defined? Again using diagram chasing, we see that it is welldefined regardless of the embedding and the corresponding affine space. Thus we indeed
have a functor Γ : Mr (DX ) → Mr (OX ) for an arbitrary affine variety. Note that if X is
smooth, then it is just ”forgetting” the DX -module structure.
Remark. For an embedding X → Y , we have DX→Y = DY /J DY , and by definition
Γ = HomDY (DX→Y , •). Now if we have another embedding j : Y → Z, then j∗0 DX→Y =
DX→Y ⊗DY DY →Z = DY /J DY ⊗DY DZ /J 0 DZ = OX ⊗OZ DZ = DX→Z , thus using the
same argument above we have a well-defined object DX = {DX→Y } ∈ Mr (DX ). In this
sense, Γ is representable. Note that if X is singular then DX is not an algebra and also
D(X) 6= DX . But it is still true that D(X) = End(DX ). Now it means D(X) acts on
Γ(M ) = HomDY (DX , M ) on the right, but it does not mean that we can reconstruct M
by using this information. Also note that if X is singular then DX may not be projective
and thus Γ may not be exact. On the smooth case, DX is projective and Γ is exact.
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!-crystal
If X is an affine scheme of finite type (not necessarily reduced,) then we can still use
the same argument to define Mr (DX ). Since this construction only uses the set-theoretic
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support of X under the embedding X → Y for a smooth Y , if we let Xred ⊂ X be
a corresponding reduced scheme, i.e. variety, then Mr (DX ) = Mr (DXred ). However,
Γ is √
different, because for an embedding X → Y , DX = DY /J DY , whereas DXred =
DY / J DY .
It is related to the notion of !-crystal. Let X be a variety, M ∈ Mr (DX ), and X 0 be
0
an affine scheme of finite type such that Xred
= X. Then by the previous argument, M
r
defines an object in M (DX 0 ). Thus we can define MX 0 = ΓX 0 (M ). Now if η : X 0 → X 00 is
a finite morphism such that its restriction on X gives the identity, then by functoriality we
have an isomorhpism αη : MX 0 → η ! MX 00 . (Note that Γ is as the same as i!0 if we forget
the DX -module structure.) Also this αη satisfies the obvious relation which is compatible
η
ρ
with compositions of morphisms, i.e. for Y −
→Z−
→ W we have αρ◦η = (η ! (αρ )) ◦ αη . Such
a structure is called !-crystal on X. Now we have the theorem, which is as follows.
Theorem 3.1. The category Crys(X) of !-crystals on X is equivalent by the above functor
to Mr (DX ).
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Push-forward for General Varieties
For f : X → Y where X, Y are singular, we wish to define f∗0 . Note that by embedding
Y to an affine space, we only need to deal with the case f : X → An . Now we fix an
embedding i : X → Am and consider the following diagram
X
/ Am
i
f
! θ
An
which exists since OAn is free. Now we define f∗0 = θ∗0 ◦ i∗0 . Where i∗0 is the equivalence
Mr (DX ) → MrX (DAm ).
References
[1] Hartshorne, Robin. Algebraic geometry. Vol. 52. Springer Science & Business Media,
1977.
[2] Schwede, Karl. ”Gluing schemes and a scheme without closed points.” Contemporary
Mathematics 386 (2005): 157.
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