Download De Rham cohomology of algebraic varieties

ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES
W. D. GILLAM
Abstract. This is an exposition on Grothendieck’s 1963 letter to Atiyah, published in
1966 as [Gro] with some additional footnotes. This is meant to accompany a talk given
at Brown in the Spring of 2012.
1. Introduction
The intention of this brief note is to discuss a “well-known” result of Grothendieck concerning the relationship between (smooth) complex varieties, (smooth) complex analytic
spaces, and, to a lesser extent, smooth manifolds. I will begin then by saying something
about these sorts of spaces, though we have very little need for any systematic theory of
complex analytic spaces or differentiable manifolds, except in the “smooth” case (i.e. the
theory of differentiable and complex analytic manifolds). Every “space” encountered in
this paper will be a certain kind of locally ringed space over C. The reader who has some
knowledge of complex analytic spaces can skip the next several paragraphs.
I assume the reader has a reasonable understanding of schemes (we will use the word
scheme to mean “scheme of locally finite type over C” unless otherwise specified), so let me
begin by recalling a thing or two about (complex) analytic spaces. If V ⊆ Cn is an open
subset (in the usual metric topology) and f1 , . . . , fk : U → C are holomorphic functions,
then we can consider the ideal I of the sheaf OV of holomorphic functions on V generated
by the fi . Note that (V, OV ) is a locally ringed space over C and that the unique maximal
ideal mv ⊆ OV,v is given by the germs of holomorphic functions which vanish at v ∈ V .
Let i : Z ,→ V denote the inclusion of the closed subset
Z := {v ∈ V : Iv ⊆ mv }
= {v ∈ V : f1 (v) = · · · = fk (v) = 0}.
We regard Z as a locally ringed space over C by endowing Z with the sheaf i−1 (OV /I) of
C-algebras. A (complex) analytic space is a locally ringed space over C locally isomorphic
to such a Z.
For every scheme X, there is an associated analytic space X an (the analytification of X)
and a map τ : X an → X of locally ringed spaces over C terminal among LRS/C-morphisms
from an analytic space to X. The points of X an are in bijective correspondence with the
C points of X. Locally, X ∼
= Spec C[x1 , . . . , xn ]/(f1 , . . . , fk ), and X an is constructed as
n
above with V = C (the polynomials fi are, in particular, holomorphic functions). If F
is a sheaf on X (i.e. a sheaf of OX modules), we write F an for the sheaf
F an := τ ∗ F
= τ −1 F ⊗τ −1 OX OX an
Date: February 14, 2012.
1
2
W. D. GILLAM
on X an . The map τ is “nice” in many senses: the maps OX,τ (x) → OX an ,x are (faithfully)
flat maps of noetherian local C-algebras, and induce isomorphisms on completions, and the
map (ΩX )an → ΩX an is an isomorphism. All of this is functorial in X, and many properties
of X and of maps f : X → Y of schemes are “inherited” by the corresponding analytic
space X an and map f an : X an → Y an : X is separated/proper iff X an is Hausdorff/compact,
open/closed embeddings are preserved, étale maps become local isomorphisms, and so
forth.
In the second paragraph above, we can replace “Cn ” with “Rn ” and “holomorphic”
everywhere with “smooth, complex valued” to define a (complex) differentiable space.1 In
analogy with analytification, one can form the differentialization Y sm of an analytic space
Y . (A holomorphic function is, in particular, smooth.) There is again an LRS/C map
σ : Y sm → Y with a similar universal property. This time σ is an isomorphism on the level
of topological spaces, but it is not true that σ ∗ ΩY → ΩY sm is an isomorphism—the differentials of holomorphic functions don’t generate the differentials of all smooth, complex
functions (...but if we also throw in the differentials of anti-holomorphic functions, then
we get everything). For our present purposes, it enough to know about this construction
in the case where Y is a complex manifold, in which case Y sm is just the same topological
manifold, endowed with the sheaf of smooth complex functions. For a scheme X, we write
X sm for (X an )sm .
A lot of the complex analytic and algebraic geometry of the 1950s and 1960s was
motivated by various problems concerning the relationship between X, X an , and X sm .
For instance, one of Serre’s GAGA theorems says:
Theorem (GAGA). Let X be a projective variety over C, F a coherent sheaf on X.
Then the natural map
Hp (X, F ) → Hp (X an , F an )
is an isomorphism for all p.
There are dozens of results along these lines in [GAGA], and various generalizations: for
example, Grothendieck showed that you can replace “projective” with “proper” and the
above theorem remains true [GAGA2]. In the present paper, the only remotely non-trivial
GAGA statement that we will need is the above theorem in the case where X is smooth
and projective.
Let us now get to the statement of the result we plan to discuss. For a scheme X, we can
consider the sheaf of Kähler differentials ΩX (relative to C of course), and the (algebraic)
de Rham complex
Ω•X := [OX → ΩX → ∧2 ΩX → · · · ].
Keep in mind that, although each sheaf ∧p ΩX is a (coherent) OX module, the differential
for Ω•X is not OX -linear—it is only CX linear.2 One can make some subtle errors by acting
as if Ω•X is a complex of OX -modules (see Example 1). The de Rham complex carries the
structure of a differential graded CX algebra, but for our purposes we will just think of it
as a complex of C vector spaces, whose terms happen to be coherent OX modules.
1In reality, this definition is too restrictive, but it is sufficient for the purposes of the present discussion.
2The constant sheaf associated to a set S on a topological space X will be denoted S .
X
ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES
3
Now we can consider the (hyper)cohomology groups Hp (X, Ω•X ) of the de Rham complex. We will call these the algebraic de Rham cohomology of X. Let us just recall a few
facts about hypercohomology before proceeding. First of all, to a (bounded below, say)
complex K • of sheaves of abelian groups on a space X, there are two different spectral
sequences [God, II.4.5]
′
(1)
(2)
′′
E1p,q = Hq (X, K p ) =⇒ Hp+q (X, K • )
E2p,q = Hq (X, Hp (K • )) =⇒ Hp+q (X, K • )
abutting to the hypercomology of K • .3 From the second spectral sequence (or other
concerns...) it is clear that a quasi-isomorphism K • → L• induces isomorphisms on
hypercomology. From the first spectral sequence, we see that the hypercohomology of K •
can be computed by finding a quasi-isomorphism K • → L• to a complex L• with each Lp
a Γ-acyclic sheaf, then noting that
Hp (X, K • ) = Hp (Γ(X, L• )).
Keep in mind that the algebraic de Rham cohomology is calculated on the schemetheoretic topological space of X. Of course one can also consider the corresponding objects
for an analytic space. The map X an → X induces maps
(3)
Hp (X, Ω•X ) → Hp (X an , Ω•X an ).
Now, if X is smooth, then X an is a complex manifold, hence the natural map
(4)
CX an → Ω•X an
is a quasi-isomorphism by the Poincaré Lemma, and hence so is
(5)
Hp (X an , C) → Hp (X an , Ω•X an ),
so we see that the analytic de Rham cohomology agrees with the usual cohomology of
X an .
Grothendieck’s Theorem. Let X be a smooth scheme over C (not necessarily proper,
separated, or quasi-compact), X an the associated complex analytic space. Then the natural
map (3) is an isomorphism for all p, hence the algebraic de Rham cohomology also agrees
with the usual cohomology of X an .
In the case of de Rham cohomology, the spectral sequence (1) takes the form
(6)
E1p,q = Hq (X, ΩpX ) =⇒ Hp+q (X, Ω•X ).
This spectral sequence is called the Hodge-to-de Rham spectral sequence. One can use it to
prove Grothendieck’s theorem for a projective (or proper) X as follows: This Hodge-to-de
Rham spectral sequence maps to the analogous one for the associated complex analytic
manifold X an , and the maps on E1 are isomorphisms by GAGA, so the map on the
abutments is also an isomorphism.
Now, if X an is Kähler (e.g. if X is projective), then one knows by Hodge Theory [GH,
Pages 111-116] that the Hodge-to-de Rham spectral sequence for X an degenerates at E1 ,
hence so does the one for X by GAGA. Furthermore, the resulting filtration of cohomology
3Both spectral sequences (with the indicated “starting points”!) are natural in K • . “The E term” of
1
the second spectral sequence, however, requires a choice of Cartan-Eilenberg resolution, and is hence not
natural in K •
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W. D. GILLAM
splits naturally (using the R structure on complex cohomology and complex conjugation)
to yield the Hodge isomorphism
⊕
Hn (X an , C) =
Hq (X an , ΩpX an ).
p+q=n
The statement that the Hodge-to-de Rham spectral sequence degenerates at E1 is an entirely algebraic statement, and one may ask whether it holds over all fields, or whether one
can prove it by entirely algebraic methods. Indeed, Faltings [F] proved that this spectral
sequence degenerates at E1 for all smooth, proper schemes over a field of characteristic
zero. The method goes through characteristic p > 0, where the Hodge-to-de Rham spectral
sequence is known not to degenerate at E1 . I recommend Illusie’s article [I] for an account
of this and an explanation of his “elementary” proof (with Deligne) of the aforementioned
result of Faltings.
On an affine scheme (or its associated complex analytic space), a coherent sheaf has no
higher cohomology, so the spectral sequence (6) degenerates and Grothendieck’s Theorem
says:
Corollary 1. Let X be a smooth, affine scheme over C, X an the associated complex
analytic space. Then the natural map
Hp (Γ(X, Ω•X )) → Hp (Γ(X an , Ω•X an ))
is an isomorphism for all p.
In the remaining parts of this note, I will describe the proof of Grothendieck’s Theorem
in some detail, trying to keep everything as self-contained as possible. I’ve made no
attempt to make the proof as short and direct as possible, but the reader who wants the
most direct route to the proof, and who is willing to except various technical sheaf-theory
facts should probably just read 1) the definition of F (∗Y ) in §2, 2) the Atiyah-Hodge
Lemmas in §3, which are the essential local computation, and 3) the actual proof in §4
which is then a matter of general nonsense.
One can also ask about the de Rham cohomology for singular varieties. I worked it
out for a node in Example 2 at the end of the paper—it comes out the same as the
usual cohomology. One could probably see this by general nonsense, but, anyway, it is a
simple example intended to demonstrate that it is eminently feasible to explicitly compute
algebraic de Rham cohomology. I couldn’t think of an easy example where the de Rham
cohomology of a singular variety doesn’t compute the usual cohomology (but if you’re
willing to look at non-reduced schemes, then the de Rham cohomology of Spec C[x]/(x2 ),
for example, if not the usual cohomology of a point).
After writing this, I found a nice little survey by J. Pottharst [P] on similar topics, with
some interesting information on Hartshorne’s general setup for de Rham cohomology. The
elliptic curve example in [P] is also fun. The present survey is intended for a more general
audience.
2. Some sheaf theory
2.1. The ∗Y construction. For a locally ringed space X and an ideal I ⊆ OX , the
subset
Z(I) := {x ∈ X : Ix ⊆ mx }
ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES
5
is closed in X because a section that is a unit in the stalk is a unit on a neighborhood.
If Y ⊆ X is a closed subset such that Y = Z(I), then we will say that I is an ideal of
definition for Y . An ideal I ⊆ OX is invertible iff it is locally isomorphic to OX , or,
equivalently, it is locally generated by a single section f ∈ I(X) ⊆ OX (X) which is not a
zero divisor in any stalk. We will say that a closed subset Y ⊆ X is defined locally by a
single equation iff Y = Z(I) for some invertible ideal sheaf I. If this is the case, then we
choose some such invertible ideal sheaf I and we define, for any OX -module F ,
F (∗Y ) := lim H omX (I n , F ).
(7)
−→
Now, to be precise, the notation F (∗Y ) doesn’t indicate that we made the aforementioned
choice of invertible ideal sheaf I.
This abuse of notation is justified by the fact that in any “reasonable” locally ringed
space, the choice won’t matter. For example:
Lemma 1. Suppose X is a scheme and I, J are two invertible ideal sheaves defining the
same closed subspace Z(I) = Z(J) =: Y . Let i : U ,→ X denote the inclusion of the open
complement U := X \ Y . Then there is a natural isomorphism
lim H omX (J n , F ) = lim H omX (I n , F )
−→
−→
for any OX -module F (not necessarily quasi-coherent). (This also holds without the “invertible” assumption if we assume X locally noetherian.) If F is quasi-coherent, then in
fact, both sheaves are identified with i∗ (F |U ). The morphism f is affine.
Proof. Locally, X = Spec A and I = (f ), J = (g) for some f, g ∈ A. The fact that
Z(I) = Z(J) means that the set of prime ideals of A containing I is the same as the set of
prime ideals of A containing
the radical of an ideal is the intersection of the
√ J.√In any ring,
a
primes containing it, so I = J, hence I ⊆ J and J b ⊆ I for some non-negative integers
a, b (this holds with no assumptions about I and J if A is noetherian). This implies that
the ideals {I n : n ∈ N} are cofinal in the ideals {J n : n ∈ N} and vice-versa, so the direct
limits in question are equal. The identification of these direct limits is clearly natural,
hence the local identifications glue and we get the result in the global case. For the final
statement, in our local situation, F ∼
= M ∼ for some A module M . Furthermore, the
complement U of Z(I) = Z(f ) is just Spec Af (a prime either does or does not contain f !)
and i is Spec of the localization map A → Af = Ag , so F |U ∼
= Mf∼ and i∗ (F |U ) ∼
= Mf∼ ,
where, here, Mf is regarded as an A module via restriction of scalars along A → Af . The
isomorphism we want is given by (∼ of)
lim HomA ((f n ), M ) → Mf
−→
[ϕ : (f n ) → M ] 7→ [ϕ(f n )/f n ].
It is easy to check that this map is well-defined with inverse taking [m/f n ] ∈ Mf to the
class of the unique A module map (f n ) → M taking f n to m. It is also easy to see
that this isomorphism would be “the same” had we used g instead of f everywhere. These
local isomorphisms hence glue to the desired global isomorphism, and the global f is affine
because the question of whether a map is affine is local on the codomain.
In fact, for the proof of Grothendieck’s Theorem, one only needs the F (∗Y ) construction
in the case where X is a smooth projective variety and Y ⊆ X is a normal crossings
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W. D. GILLAM
divisor, and in the analytification of this situation. In either case, we can equivalently
define F (∗Y ) by
F (∗Y ) = lim F (nY ).
−→
The first statement of the above lemma is also true for complex analytic spaces, but
the final statement is not true, and this fact is perhaps the crux of what Atiyah and
Hodge were studying in [AH] and what inspired Grothendieck to prove the result we are
discussing and write his letter to Atiyah. For example, suppose X = C and I = (z) is the
ideal defining the origin Y , so U = C∗ . Then there is a containment
OX (∗Y ) ⊆ i∗ (OU )
which is certainly not an equality: Functions like exp(1/z) are in the right-hand side but
not the left hand side. I think what Atiyah, Hodge, and Grothendieck realized here is that
the sheaf F (∗Y ) is often a more well-behaved substitute for i∗ F (|X \ Y ).
2.2. Cohomology of filtered direct limits. In this section, we give a more-or-less selfcontained proof that sheaf cohomology commutes with filtered direct limits on a compact
or noetherian space. The noetherian case is more well-known and can be found in [Har].
The compact case is [AH, Lemma 5], but they use some esoteric language and constructions
in their proof; the proof given here is a little more standard and parallels the noetherian
case closely.
Theorem 2. Let X be a topological space, c 7→ Fc a direct limit system of sheaves on X
indexed by a filtered category C . If X is quasi-compact, then the natural map
(8)
lim Γ(X, Fc ) → Γ(X, lim Fc )
−→
−→
is injective. If, furthermore, X is Hausdorff (i.e. compact), or noetherian, then it is an
isomorphism.
Proof. Recall that the direct limit sheaf F := lim Fc is given by the sheafification of the
−→
presheaf direct limit F pre , whose sections are given by
F pre (U ) = lim Fc (U ).
−→
The map (8) is the map on global sections induced by the sheafification map F pre → F .
The injectivity when X is quasi-compact is not hard: Recall from the usual construction
of filtered direct limits that an element [s] ∈ lim Fc (X) is represented by an index c ∈ C
−→
and an element s ∈ Fc (X). Two such pairs [s] = (c, s), [t] = (d, t) are equivalent iff there
are C -morphisms f : c → e and g : d → e so that
F (f )(s) = F (g)(t) ∈ Fe (X).
Unwinding the sheafification construction, we see that [s] and [t] map to the same element
of Γ(X, F ) iff there is a cover {Ui : i ∈ I} of X, a function e : I → C , and C -morphisms
fi : c → e(i), gi : d → e(i) for each each i ∈ I so that
(9)
F (fi )(s|Ui ) = F (g)(t|Ui ) ∈ Fe(i) (Ui ).
If X is quasi-compact, we can assume that I is finite, hence, since C is filtered, we can
find a single index e ∈ C and maps hi : e(i) → e so that hi fi = hj fj and hi gi = hj gj
ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES
7
for every i, j ∈ I. Call the respective common maps f : c → e and g : d → e. From the
equality (9) we see that
F (f )(s)|Ui =
=
=
=
=
=
=
F (f )(s|Ui )
F (hi fi )(s|Ui )
F (hi )(F (fi )(s|Ui ))
F (hi )(F (gi )(t|Ui ))
F (hi gi )(t|Ui )
F (g)(t|Ui )
F (g)(t)|Ui .
Since F is a sheaf, we conclude F (f )(s) = F (g)(t), hence [s] = [t].
We now prove surjectivity of (8) when X is compact. In general, a section [s] ∈ Γ(X, F )
is represented by an open cover {Ui : i ∈ I}, a function c : I → C , and sections si ∈
Fc(i) (Ui ) which satisfy the following compatibility condition on each pairwise intersection
Uij = Ui ∩ Uj : For every x ∈ Uij , there are: an index c(i, j, x) ∈ C , C morphisms
f = f (i, j, x) : c(i) → c(i, j, x), g = g(i, j, x) : c(j) → c(i, j, x), and a neighborhood
W = W (i, j, x) of x in Uij such that
F (f )(si |W ) = F (g)(sj |W )
(10)
∈ Fc(i,j,x) (W ).
The reader can spell out the details of when two such collections of data represent the
same section—we will implicitly replace one such data set with an equivalent one in what
follows.
First, since X is compact, we can assume I is finite, hence, since C is filtered, we can
assume that c := c(i) is independent of i ∈ I. (Choose an index c so that there are
C morphisms fi : c(i) → C for each i ∈ I and replace si by F (fi )(si ).) Now, by a
standard topological argument, we can find a cover {Vi : i ∈ I} (same indexing set I!) of
X such that V i ⊆ Ui for all i ∈ I. Now, using compactness of V ij and the fact that C
is filtered, we can find an index c(i, j) ∈ C , a neighborhood W (i, j) of V ij in Uij , and a
map f (i, j) : c → c(i, j) so that c(i, j) (resp. f (i, j), f (i, j), W (i, j)) can serve as c(i, j, x)
(resp. f (i, j, x), g(i, j, x), W (i, j, x)) in the compatibility condition for each x ∈ V ij . In
fact, since there are only finitely many pairs (i, j), we can (using C filtered) even arrange
that d := c(i, j) and the map f := f (i, j) : c → d are independent of the pair (i, j). By the
same kind of calculation we made in the injectivity argument, the compatibility condition
(10) now implies that
F (f )(si )|Vij = F (f )(sj )|Vij .
Hence the local sections F (f )(si )|Vi ∈ Fd (Vi ) glue to a global section s ∈ Fd (X). The
image of s in lim Fc (X) clearly maps to our [s] under (8).
−→
The argument in the case where X is noetherian is similar, but easier. In this case,
since an open subset of a noetherian space is quasi-compact, the intersections Uij are
quasi-compact. Using this quasi-compactness and the fact that C is filtered, we find an
index c(i, j) and a map f (i, j) : c → c(i, j) so that c(i, j) (resp. f (i, j), f (i, j), Uij ) can
serve as c(i, j, x) (resp. f (i, j, x), g(i, j, x), W (i, j, x)) in the compatibility condition for
each x ∈ Uij . The rest of the argument is identical (replace U with V everywhere).
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W. D. GILLAM
Corollary 3. Let X be a noetherian topological space, c 7→ Fc a filtered direct limit
system of sheaves on X. The direct limit sheaf F := lim Fc coincides with the direct limit
−→
presheaf. That is,
F (U ) = lim Fc (U )
−→
for every open subset U ⊆ X.
Proof. An open subset of a noetherian topological space is noetherian, so this follows from
the previous Theorem.
Recall that a sheaf F on a space X is called flasque iff the restriction map F (X) →
F (U ) is surjective for every open subset U ⊆ X. Similarly, F is called soft iff the natural
map
Γ(X, F ) → Γ(Z, i∗ F )
is surjective for any inclusion i : Z ,→ X of a closed subset.
Corollary 4. A filtered direct limit of flasque sheaves on a noetherian topological space is
flasque. A filtered direct limit of soft sheaves on a compact topological space is soft.
Proof. Since a filtered direct limit of surjective maps (of sets) is again surjective, the first
statement is immediate from Corollary 3, and the second is immediate from Theorem 2
(note Z is compact when X is compact).
Theorem 5. Sheaf cohomology commutes with filtered direct limits on a compact or noetherian space X. That is: Let c 7→ Fc be a filtered direct limit system of sheaves of abelian
groups on a compact or noetherian space X. Then the natural map
(11)
lim Hp (X, Fc ) → Hp (X, lim Fc )
−→
−→
is an isomorphism for all p ≥ 0.
Proof. Let C be the filtered category indexing the direct limit system c 7→ Fc . Let
A := Hom(C , Ab(X)) be the abelian category of C -indexed (hence filtered) direct limit
systems of sheaves of abelian groups on X. For a fixed p, both sides of (11) are natural
in (c 7→ Fc ) and may viewed as functors A → Ab. These functors agree when p = 0 by
Theorem 2, so, by general nonsense [T, 2.2], they will agree for all p once we prove that
both sides are effaceable δ-functors A → Ab. To see that both sides are δ-functors, first
note that the formation of filtered direct limits of abelian groups
lim : Hom(C , Ab) → Ab
−→
is an exact functor (this is an easy exercise and is well-known). It follows easily from this
fact and the fact that sheafification is exact that
lim : Hom(C , Ab(X)) → Ab(X)
−→
is also exact. Now the fact that both sides are δ-functors follows from the fact that
cohomology H• is a δ-functor.
It remains only to prove that both sides are effaceable. For any F ∈ Ab(X), we can
find an injection F ,→ F ′ from F into a flasque sheaf, functorial in F (use the first step
of the Godement resolution [God, II.4.3], for example), so we can find an injection from
ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES
9
any limit system (c 7→ Fc ) into a limit system (c 7→ Fc′ ) where each Fc′ is flasque. It now
remains only to prove that, for p > 0, both sides of (11) vanish when the Fc are flasque.
The left side vanishes (with no hypotheses on X) because a flasque sheaf has no higher
cohomology [God, II.4.4.3(a)]. In the noetherian case, the right side vanishes because a
filtered direct limit of flasques is flasque (Corollary 4). In the compact case, the right side
vanishes because a flasque sheaf on a (para)compact space is soft [God, II.3.3-4], a direct
limit of soft sheaves is soft (Corollary 4), and a soft sheaf on a (para)compact space has
no higher cohomology [God, II.4.4.3(b)].
If you don’t like general nonsense, you can argue a bit more directly as follows. By
using, say, the Godement resolution, find a flasque resolution Fc → Ic• of each Fc , natural
in c. Then, if we let I p := lim Icp , then F → I • is a resolution of F (check on stalks, use
−→
exactness of filtered direct limits). By applying the same results mentioned in the above
paragraph, one sees that each I p is Γ-acyclic, so, by alternative general nonsense, and the
results of this section we compute
Hp (X, F ) = Hp (Γ(X, I • ))
= Hp (lim Γ(X, Ic• ))
−→
= lim Hp (Γ(X, Ic• ))
−→
= lim Hp (X, Fc ).
−→
3. Poincaré lemmas
I refer to any explicit computation of de Rham cohomology as a “Poincaré Lemma.”
Algebraic Poincaré Lemma. Let A = C[z1 , . . . , zn ] or its completion C[[x1 , . . . , xn ]].
Then the natural map C → Ω•A/C is a quasi-isomorphism of complexes of C vector spaces.
Poincaré Lemma. Let M be a complex analytic manifold, or any smooth manifold
endowed with the sheaf of smooth, complex-valued functions. Then the natural map CM →
Ω•M is a quasi-isomorphism of complexes of sheaves on M .
Algebraic Atiyah-Hodge Lemma. For integers 0 ≤ k ≤ n, let
A := C[z1−1 , . . . , zk−1 , z1 , . . . , zn ]]
denote the subring of the ring of (formal) Laurent series in z1 , . . . , zn consisting of Laurent
series not involving negative powers of zk+1 , . . . , zn . Then the map
(12)
∧• Ck → Ω•A
taking the standard basis vector ei ∈ Ck to dlog zi := zi−1 dzi is a quasi-isomorphism of
differential graded Q algebras (the domain has the trivial differential, while the codomain
has the de Rham differential). In other words, H0 (Ω•A ) = C and for p > 0, Hp (Ω•A ) is
freely generated as a C vector space by the elements
dlog zi1 ∧ · · · ∧ dlog zip ,
for 1 ≤ i1 < i2 < · · · < ip ≤ k.
10
W. D. GILLAM
Proof. We first note that the forms dlog zi are clearly closed, so the map (12) is welldefined (a wedge of closed forms is closed). We next claim that (12) is injective. If not,
then we could write some nontrivial linear combination of the forms
dlog zi1 ∧ · · · ∧ dlog zip+1
(1 ≤ i1 < i2 < · · · < ip+1 ≤ k) as dϕ for a p-form ϕ ∈ ΩpA . A p-form ϕ ∈ ΩpA can be
written uniquely
∑
∑
(13)
Ai1 ,...,ip ;j1 ,...,jn z1j1 · · · znjn dzi1 · · · dzip ,
ϕ=
1≤i1 <···<ip ≤n j1 ,...,jn
where (j1 , . . . , jn ) ∈ [−r, ∞)k × Nn−k (for some interval [−r, ∞) ⊆ Z). But
d(z1j1 . . . znjn dzi1 · · · dzip ) =
n
∑
jl z1j1 · · · zljl −1 · · · znjn dzl dzi1 · · · dzip ,
l=1
so it is clear that the coefficient of dzi1 · · · dzip+1 in dϕ cannot be zi−1
· · · zi−1
.
1
p+1
The main issue is to prove that (12) is surjective. We proceed by induction on k. The
case k = 0 is the Poincaré Lemma, so we can now consider the case where k > 0. By
collecting the terms in (13) where i1 = 1 into one sum, and the other terms in another sum,
p−1
we see that ϕ can be uniquely written as ϕ = αdz1 + β, where α ∈ ΩA
is a (p − 1)-form
p
not involving dz1 and β ∈ ΩA is a p-form not involving dz1 . Set
B := C[z2−1 , . . . , zk−1 , z1 , . . . , zn ]]
C := C[z2−1 , . . . , zk−1 , z2 , . . . , zn ]].
In the sums defining α and β, we collect the terms with j1 ≥ 0, the terms with j1 = −1,
. . . , and the terms with j1 = −r to uniquely write
α = α0 + α1 z1−1 + · · · + αr z1−r
β = β0 + β1 z1−1 + · · · + βr−r z1−r ,
p
p−1
p
where: α0 ∈ Ωp−1
B ; β0 ∈ ΩB doesn’t involve dz1 ; α1 , . . . , αr ∈ ΩC ; and β1 , . . . , βr ∈ ΩC .
Now we make a little computation4 to put dϕ back into this “standard form”:
dϕ = dαdz1 + dβ
(
)
= dα0 + dα1 z1−1 + (−1)p−1 α1 d(z1−1 ) + · · · + dαr z1−r + (−1)p−1 dz1−r dz1
+dβ0 + dβ1 z1−1 + (−1)p β1 d(z1−1 ) + · · · + dβr z1−r + (−1)p βr d(z1−r )
(
)
= dα0 + dα1 z1−1 + · · · + dαr z1−r dz1
+dβ0 + dβ1 z1−1 + (−1)p−1 β1 z1−2 dz1 + · · · + dβr z1−r + (−1)p−1 rβd z1−r−1 dz1
[
= dα0 + dα1 z1−1 + (dα2 + (−1)p−1 β1 )z1−2 + (dα3 + 2(−1)p−1 β2 )z1−3 + · · ·
]
+(dαr + (r − 1)(−1)p−1 βr−1 )z1−r + r(−1)p−1 βr z1−r−1 dz1
+dβ0 + dβ1 z1−1 + dβ2 z1−2 + · · · + dβr z1−r .
4It seems to me that Atiyah and Hodge are sloppy with signs in their proof of this [AH, Lemma 17]. In
the de Rham complex (or any differential graded algebra), one has d(ab) = (da)b + (−1)|a| adb.
ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES
11
Suppose now that ϕ is closed (dϕ = 0). From the above formula for the standard form
of dϕ, we see that dα0 , dα1 , and dβ vanish and we have the following equalities:
(−1)p dα2 = β1
(−1)p
dα3 = β2
2
..
.
(−1)p
dαr = βr−1 .
r−1
Set
(
)
1
1
−2
−r
θ := (−1)
+ α3 z1 + · · · + αr z1
.
2
r
Then we compute using the equalities listed above that
p
α2 z1−1
dθ = β1 z1−1 + β2 z2−1 + · · · + βr−1 z1r−1
+α2 z1−2 dz1 + α3 + z1−3 dz1 + · · · + αr z1−r dz1 .
This implies that
(14)
θ = dθ + α1 dlog z1 + α0 dz1 + β0 .
Now, by the induction hypothesis applied to B and the fact that dα0 = 0 and dβ0 = 0,
we can write
α0 = dθ1 + α0′
β0 = dθ2 + β0′ ,
p−1
′
′
where θ1 ∈ Ωp−2
B , θ2 ∈ ΩB , and α0 and α1 are in the exterior algebra on dlog z2 , . . . , dlog zk .
By the induction hypothesis applied to C and the fact that dα1 = 0, we can write
α1 = dθ3 + α1′ ,
p−2
where θ3 ∈ ΩC
and α1′ is in the exterior algebra on dlog z2 , . . . , dlog zk . Plugging this
all back in to (14) shows that ϕ is equal, modulo exact forms, to a form in the exterior
algebra on dlog z1 , . . . , dlog zk . This proves that the forms in the exterior algebra on
dlog z1 , . . . , dlog zk generate the cohomology of Ω•A .
Atiyah-Hodge Lemma. Let M be a smooth complex manifold, f : V ,→ M an open
subspace with normal crossings complement W ⊆ M . Then the natural map
Ω•M (∗W ) → f∗ Ω•V
is a quasi-isomorphism of complexes of sheaves on M .
Proof. This can be checked on stalks. In particular, we can assume M = Cn , W =
{z1 · · · zk = 0} (for some k ∈ {0, . . . , n}) and we can work in the stalk at the origin
0 ∈ Cn . In this case, we have V = (C∗ )k × Cn−k . Furthermore, we can explicitly compute
the cohomology of (f∗ Ω•V )0 . For a positive integer N , let
UN := {z ∈ Cn : ||zi || < 1/N for i = 1, . . . , n}.
The neighborhoods UN are cofinal in the set of all neighborhoods of the origin. Note that
VN := UN \ W sits inside V and all the inclusions
· · · ⊆ VN +1 ⊆ VN ⊆ · · · ⊆ V1 ⊆ V
12
W. D. GILLAM
are homotopy equivalences (they are all homotopy equivalent to (S 1 )k ). The complex
(f ∗ Ω•V )(UN ) = (Ω•V )(VN )
= Γ(VN , Ω•VN )
is just the usual complex analytic de Rham complex whose cohomology computes the usual
complex cohomology of VN (in light of the Poincaré Lemma and the fact that the coherent
sheaves ΩpVN on VN have no higher cohomology).5 This cohomology is the exterior algebra
on
dlog z1 , . . . , dlog zk
(or, more precisely, the restrictions of these global differential forms on V to VN ). One
way to see this is to take an explicit basis for the homology of VN (small loops winding
counterclockwise around the origins) and integrate wedges of the dlog zi over these loops
via the Cauchy Integral formula. Since this discussion is valid for all N , we have, in
particular, that the cohomology (f∗ Ω•V )0 is the exterior algebra on
dlog z1 , . . . , dlog zk
(or, more precisely, their images in the stalk).
The holomorphic differential forms dlog zi on V are of course the images of the corresponding elements of (ΩM (∗W ))0 with the same name, so what we want to show is that
the cohomology of (Ω•M (∗W ))0 is also the exterior algebra on these guys. Now, by the
definition of OM (∗W ) as a direct limit, and the fact that this direct limit commutes with
stalks, we have
(OM (∗W ))0 = lim (z1 · · · zk )−m OCn ,0 .
−→
We can view an element of (OM (∗W ))0 as a formal Laurent series in z1 , . . . , zn which
becomes a convergent power series in z1 , . . . , zn (i.e. an element of OM,0 = OCn ,0 ) after
multiplying by (z1 · · · zk )m for some large enough m, so we can view (OM (∗W ))0 as a
subring of the ring A = C[z1−1 , . . . , zk−1 , z1 , . . . , zn ]] of the Algebraic Atiyah-Hodge Lemma.
Similarly, we can view p-forms of (OM (∗W ))0 as p-forms
∑
ω =
fi1 ,...,ip
1≤i1 <···ip ≤n
of A whose coefficients fi1 ,...,ip ∈ A actually lie in (OM (∗W ))0 (i.e. have the aforementioned
convergence property). Now the computation of the cohomology of (Ω•M (∗W ))0 is identical
to the calculation made in the proof of the Algebraic Atiyah-Hodge Lemma... one just
has to look through that proof and note that none of the steps is going to destroy the
convergence property.
4. Proof of Grothendieck’s Theorem
We first prove the Theorem in the affine case (Corollary 1). Let U be a smooth, affine
scheme over C. By Hironaka’s results on resolution of singularities [Hir], we can find an
open embedding f : U ,→ X, where X is a smooth, projective scheme over C and the
complementary closed subset Y := X \ U is a normal crossings divisor in X. The result
5The same theorem would hold if we used smooth differential forms on V instead of holomorphic forms.
ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES
13
we want follows from the following sequence of numbered isomorphisms, to be explicated
momentarily:
(15)
(16)
(17)
(18)
(19)
Hp Γ(U, Ω•U ) =
=
=
=
=
Hp (U, Ω•U )
Hp (X, R f∗ Ω•U )
Hp (X, f∗ Ω•U )
Hp (X, Ω•X (∗Y ))
lim Hp (X, H omX (I n , ΩX ))
−→
(20)
= lim Hp (X an , H omX an ((I an )n , ΩX an ))
(21)
(22)
(23)
(24)
=
=
=
=
−→
p
H (X an , ΩX an (∗Y an ))
Hp (X an , f∗an Ω•U an )
Hp (X an , R f∗an Ω•U an )
Hp (U an , Ω•U an ).
Isomorphism (15) is the degeneration of Hodge-to-de Rham in the affine case, as discussed
before Corollary 1, (16) is the Leray spectral sequence for f , (17) is because f is affine
(Lemma 1), (18) is Lemma 1, (19) is a definition (§2.1), (20) is the GAGA Theorem, (21)
is a definition, (22) is the Atiyah-Hodge Lemma, (23) is because f an is the analytification
of an affine morphism and the ΩpU an are coherent, and (24) is the Leray spectral sequence
for f an .
Hp Γ(U, Ω•U ) =
=
=
=
=
Hp (U, Ω•U )
Hp (X, R f∗ Ω•U )
Hp (X, f∗ Ω•U )
Hp (X, Ω•X (∗Y ))
lim Hp (X, H omX (I n , ΩX ))
−→
(1)
(2)
(3)
(4)
(5)
= lim Hp (X an , H omX an ((I an )n , ΩX an )) (6)
−→
=
=
=
=
=
Hp (X an , ΩX an (∗Y an ))
Hp (X an , f∗ Ω•U sm )
Hp (X an , R f∗ Ω•U sm )
Hp (X an , R f∗ CU an )
Hp (U an , C)
(7)
(8)
(9)
(10)
(11).
Isomorphism
Next we prove that the affine case (Corollary 1) of the Theorem implies the Theorem
in general. In fact, we first prove that Corollary 1 implies the Theorem for separated X.
To do this, take an affine cover U = {Ui : i ∈ I} of X. By basic Čech cohomology, there
is a spectral sequence
E2p,q = Ȟ (U, H q (Ω•X )) =⇒ Hp+q (X, Ω•X ),
p
where H q (Ω•X ) denotes the presheaf on X given by V 7→ Hq (V, Ω•V ). Explicitly, E2p,q is
the pth cohomology of the Čech complex:
∏
∏
q
q
(25)
i∈I H (Ui , ΩUi ) →
(i,j)∈I 2 H (Uij , ΩUij ) → · · ·
14
W. D. GILLAM
Note that each finite intersection6
Ui0 ,...,in := Ui0 ∩ · · · ∩ Uin
is affine because we assume X is separated. We also have a similar spectral sequence for
the corresponding open cover Uan = {Uian : i ∈ I} of the analytic space X an , with E2p,q
given by the pth cohomology of
∏
∏
q
q
an
an
an
an
(26)
i∈I H (Ui , ΩUi ) →
(i,j)∈I 2 H (Uij , ΩUij ) → · · · .
Furthermore, the algebraic spectral sequence maps to the analytic one and the abutment
of this map of spectral sequences is the map
H• (X, Ω•X ) → H• (X an , Ω•X an )
that we want to show is an isomorphism. By a standard fact about spectral sequences, it
is enough to show that the map on E2 terms is an isomorphism, i.e. that the map from
the pth cohomology of (25) to the pth cohomology of (26) is an isomorphism for all p...
but by Corollary 1, the map from (25) to (26) is even an isomorphism of complexes (not
just a quasi-isomorphism). This proves the theorem in the separated case. In the general
case, we now repeat exactly the same argument, noting that the finite intersections
Ui0 ,...,in := Ui0 ∩ · · · ∩ Uin ,
while not necessarily affine for n > 0, are at least separated, as they are open subschemes
of affine schemes.
This completes the proof of Grothendieck’s Theorem.
Example 1. Let A be the local ring of a smooth complex variety at a C point. Let A → Â
denote the completion. Then the de Rham complex of A maps to the de Rham complex
of Â:
/ ΩA
/ ∧2 ΩA
/ ···
A
Â
/Ω
Â
/ Ω2
Â
/ ···
The pth term in the bottom row is obtained from the pth term in the top row by extending
scalars along A → Â, but the complex in the bottom row is not obtained in this way, and
it does not even make sense to “extend scalars in the top row.” The problem is that the
differential d in the top row is not A-linear so the usual formula ω ⊗ a 7→ ω ⊗ â one would
right down for the extended differential d ⊗A Â does not make sense (i.e. is not A bilinear).
The upshot is that, even though A → Â is faithfully flat, there is no easy relationship
between the cohomology of Ω•A and the cohomology of Ω•Â . By Cohen structure theory,
 is just a formal power series ring, so the cohomology of the former is computed by
the Algebraic Poincaré Lemma, but the cohomology of Ω•A , on the other hand, is a mess.
Even when A = C[x](x) is the local ring of A1 at the origin, C → Ω•A can’t be a quasiisomorphism—if it were, then we’d see that CP1 → Ω•P1 is a quasi-isomorphism by checking
on stalks, where, here CP1 is the constant sheaf C on the Zariski space of P1 (as opposed
an
to (P1 ) ). But this is absurd because a constant sheaf on an irreducible space is flasque,
6In this argument in [Gro], when he discusses this E p,q term, Grothendieck uses “q” to denote the
2
number of finite opens in an intersection, which is rather confusing as the E2p,q term has more to do with
the intersections of p opens than q opens. At least I found this confusing.
ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES
15
hence has no higher cohomology, so we’d conclude that H2 (P1 , Ω• ) = 0, contradicting
Grothendieck’s Theorem. Of course you don’t need this roundabout argument to see that
C → Ω•A isn’t a quasi-isomorphism, but it illustrates in general that one does not even
hope for this.
Example 2. Let A = C[x, y]/(xy), so A is the coordinate ring of the axes in A2 . Topologically, (Spec A)an is a wedge of two copies of C, so it is contractible. Let us compute
the de Rham cohomology of A. The module ΩA is generated by dx and dy subject to the
relation xdy + ydx = 0. Using this relation and the fact that xy = 0 in A, we see that
x2 dy = 0, y 2 dx = 0, xdx ∧ dy = 0, and ydx ∧ dy = 0. From this, we see that any element
of A can be uniquely written as
α = a + b1 x + b2 x2 + · · · + bn xn + c1 y + · · · + cn xn ,
for a, bi , ci ∈ C. Any 1-form on A can be written uniquely as
ω = (a + b1 x + · · · + bn xn + cy)dx + (e + f1 y + · · · + fn y n )dy,
for a, bi , c, e, fi ∈ C. Any 2-form on A can be uniquely written tdx ∧ dy, t ∈ C. We have
dα = (b1 + 2b2 x + · · · + nbn xn−1 )dx + (f1 + 2f2 y + · · · + nfn y n−1 )dy
dω = −cdx ∧ dy.
It is clear from these formulas that Ω•A has cohomology C supported in degree zero, in
agreement with the usual cohomology of (Spec A)an .
References
M. Atiyah and W. Hodge, Integrals of the second kind on an algebraic variety, Ann. Math.
(Series 2) 62(1) (1955) 56-91.
[EGA]
A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique. Pub. Math. I.H.E.S.,
1960.
[F]
G. Faltings, p-adic Hodge theory, J.A.M.S. 1 (1998) 255-299.
[GAGA] J. P. Serre, Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier 6 (1956) 1-42.
[GAGA2] A. Grothendieck, Sur les faisceaux algébriques et les faisceaux analytiques cohérents, Séminaire
H. Cartan, Exposé 2, 1956-1957.
[GH]
P. Griffiths and J. Harris, Principles of algebraic geometry. John Wiley and Sons, Inc. 1978.
[God]
R. Godement, Théorie des faisceaux. Actualités scientifiques et industrielles 1252, Hermann,
Paris, 1958.
[Gro]
A. Grothendieck, On the de Rham cohomology of algebraic varieties, Pub. Math. I.H.E.S. 29
(1966) 95-103. (This is freely available through www.ihes.fr/IHES/Publications.)
[T]
A. Grothendieck. Sur quelques points d’algèbra homologique, Tôhoku Math. J. 9 (1957) 119-221.
[Har]
R. Hartshorne. Residues and duality. Lec. Notes Math. 20. Springer-Verlag, 1966.
[Hir]
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero,
Ann. Math. 79 (1964) 109-326.
[I]
L. Illusie, Frobenius and Hodge degeneration, in Introduction to Hodge Theory. Amer. Math.
Soc., Soc. Math. France, SMF/AMS Texts and Monographs, Volume 8, 1996.
[P]
J. Pottharst, Basics of de Rham cohomology, available at: math.harvard.edu/∼jay/derham.pdf
[AH]
Department of Mathematics, Brown University, Providence, RI 02912
E-mail address: [email protected]