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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 • 4 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 6 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). 8 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, Â 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]