Inventiones math. 7, 275-296 (1969)

De Rham Cohomology of an Analytic Space

THOMAS BLOOM* and MIOUEL HERRERA (Paris and Seattle)

Introduction

It is well known that the sheaf f2~: of germs of holomorphic differential forms on a complex analytic manifold X is a resolution of if7 on X. As a consequence, the complex cohomology H*(X,~) of X is isomorphic, when X is Stein, to the cohomology of the De Rham complex F(X, f2*) of global sections.

Recent examples show that if X is an analytic space, then /2], as defined by Grauert and Grothendieck, need not be a resolution ([7] and [16]), so that no such isomorphism exists.

We show in this paper that, nevertheless, the complex cohomology of the analytic space X can still be obtained from the De Rham complex: there exists a canonical splitting

H*(X; 12~)'-~ H* (X, ~) O A*

of the hypercohomology H* (X; (2") of f2* into the classical cohomology and a second factor.

The result also holds in the real analytic and semianalytic cases, and for the complex of smooth (c~) differential forms. It is proved construct- ing a right inverse, by means of integration, to the canonical edge homo- morphism H* (X, ~) --~ H* (X, I2~) (Theorem 3.11).

As a corollary, one deduces the following complement to Grothen- dieck's results on the algebraic De Rham cohomology of regular alge- braic schemes [9]. Let X be a complete variety (not necessarily regular) or a prescheme locally of finite type over r with only isolated singularities. Then the hypercohomology of the complex f2a* x of rational regular dif- ferential forms on X splits canonically (cf. 3.14):

H*(X, 12*x)"~ H* (X, r O A *

Particular cases of these results have been announced in [12]. To our knowledge, Norguet [15] posed the question about the relationship between (smooth) De Rham and classical cohomologies of an analytic variety.

�9 Supported by a National Research Council of Canada fellowship.

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276 T. Bloom and M. Herrera:

1. Semianalytic Sets We refer to [13] for proofs of the properties stated in this section.

1.1. Let (X, Ox) be a real analytic space, not necessarily reduced, and let (gxr be the sheaf of analytic functions of X, that is, functions associated to sections of (9 x on the open subsets of X. For each xEX, let S(x) be the smallest family of germs at x of subsets of X such that:

(1) a,b~S(x) implies aubeS(x) and a-b~S(x); (2) if f is an analytic function on a neighborhood of x, then the

germ at x of the set ( f > 0 ) belongs to S(x). A subset M of X is called semianalytic if its germ at xeX belongs

to S(x), for all xeX. The semianalytic sets of a complex analytic space are defined by considering the associated real analytic structure.

Let U be open in IR", J a coherent sheaf of ideals of the sheaf (9 v of analytic functions on U and Y the set of zeros of ~r Then the semi- analytic sets of the real analytic space (Y, (gv/j[ Y) are the semianalytic sets of (U,(gv) included in Y.

Locally finite unions and intersections, complements, closures, interiors and boundaries of semianalytic sets are semianalytic.

1.2. Suppose M is semianalytic in (X, (gx). A point xeM is q-simple (or q-regular), for q an integer >=0, if there is a neighborhood U of x in M such that (U,(gxrlU), is isomorphic (as a ringed space of local N-algebras) to an open subspace (V, 6v) of F,q; in particular the 0-simple points of M are the isolated points of M. The set of simple points of M (i.e., q-simple points for some q) is dense in M. The dimension, d imM, of M is < p if there are no q-simple points of M with q>p; d i m M = p if dim M < p but not dim M < p - 1•

Suppose dim M=p. Then dim M=p and dim(M-M)<p, where M is the closure of M in X. The semianalytic set b M = M - M is called the border of M; bM is closed if and only if M is locally closed. If M* is the set of p-simple points of M, then (M*, (gxrIM*) is a real analytic submanifold of X of dimension p, and s M = M - M * is semianalytic in X, dimsM<p.

2. Semianalytic Chains

Throughout this section, K is a principal ideal domain and M is a closed semianalytic subset in the real analytic space (X, (~x)- We allow the possibility that M=X.

If �9 is a family of supports in the locally compact space X and ~ is a sheaf of K-modules on X, then H,(X;~,~)(H,~(X;~-))denotes Borel- Moore homology with coefficients in ~ and closed supports (supports in 4) [2]. We will denote by K the constant sheaf with value K and we will sometimes use the notation H,(X) instead of H , ( X ; K). If F c X is

De Rham Cohomology of an Analytic Space 277

closed and U = X - F, there is a natural exact sequence ([2], 1.6).

...Hq(F;K) 'FX,Hq(X;K) sXV,Hq(U;K) ~ (1)

Suppose Y is also closed in the (locally compact) space X, and let V= X - Y. Then there are homomorphisms i,j, ~ connecting (1) with the similar sequences for (Fc~ V, V) and (Fc~ Y, Y); and the corresponding diagram commutes ([3], p. 252 (30)). When Y c F, that diagram reduces to'

�9 ..--,Hq(F) ' ,Hq(X) J , H q ( X - F ) - * . . .

T, I ...--~ Hq(Y) ' , Hq(X) ~ , Hq(X- Y)-~. . .

We shall often use these properties without reference.

2.1. For each integer q>=0, denote by ~5~q(M; K) the family of the pairs (N, c) such that N c M is a locally closed semianalytic set in X with dimN__<q and c~Hq(N;K); if d i m N < q , then necessarily c = 0 ([2J, 1.10). The elements of ~3~q(M;K) are called q-semianalytic pre- chains of M with coefficients in K.

We define a commutative and associative operation on ~3~q(M)= ~ q (M; K) as follows. If (N 1, q ) and (N2, c2) are prechains, then

(NI, cI)+ (N2, c2)= (N, cl + c2),

where N = N 1 u N 2 - L, L = b N 1 ~ b N2 (cf. 1.2) and c s is the image of c s by the maps

Hq(Ns) / 's '~-L,Hq(N~-L) i~_, . . . . ,Hq(N) ( s= l , 2 ) .

These maps are defined because N S - L is closed in X - L, and so closed in N; they are injective because of the dimension of N~ and exactness in (1). Associativity follows because, if el = (N~, c) are prechains (i = 1, 2, 3), one can check (using [3t, p. 252 (30)) that ~1+(~2+a3) and (cq+e2)+a 3 are both equal to (N, c), where N= N1u N2 u N3 - L, L=b Ni u b N2 • b N 3,

s " , o " N s , N s - L c = c l + c 2 + c 3 a n d c =tNs_L N J (Cs) for s=1 ,2 ,3 . A product by elements in K, linear with respect to this addition, is

also defined by k(N, c)= (N, k c), for each prechain (N, c) and keK. Consider now the relation on ~ ( M )

(N1, Cl),~(N2, c2) <:*. (N1, cl)+(N2, - c 2 ) = ( N , 0),

where 0 is the zero in Hq(N; K). It is easy to see that this relation is an equivalence compatible with the addition and product on N~q(M), and that these operations induce a K-module structure on the quotient

(M; K) = ~ (M; K) /~ . The elements of ~ (M; K) are called q-semi- analytic chains of M with coefficients in K, and the class in ~ ( M ; K)

20*

278 T. Bloom and M. Herrera:

of a prechain (N, c) is denoted by [N, c]. In particular, [N, 0] is the zero element in ~(M). Observe that ~ ( M ) = 0 if q < 0 or q > d i m M.

By means of naturality properties of Borel-Moore homology ([3], p. 252 (30)), one can show that the maps

�9 ~ G (M; K)--* ~5~q_,(M; K), (U, c)-* (b N, C?N,bN (C)) (q~Z)

are linear, and compatible with the relations just defined on prechains. They induce then a boundary homomorphism 0 on

5P, (M)= ~(5~q(M): q~Z);

that c? o c3 = 0 is clear.

Let now M' be an open subset of M and U any open set in X such that U r~ M =M' . The module 5:, (M'; K) is defined, if we consider M' as a closed semianalytic set in U, and does not depend on the particular chosen set U. Moreover, if M" c M' is also open in M, a restriction homo- morphism rra,,,•, : 5P, (M'; K) --~ 5:, (M"; K), compatible with boundaries, is induced by the maps

~5~q(M';K)-~S~q(M";K) , (N,c)-- , (Nc~M";jN'N~"(c)) .

It is immediate that 5:,: M'--* 5r K) (M' open in M), together with the defined restriction and boundary homomorphisms, is a differ- ential presheaf. We are going to prove, after some lemmas, that it is a sheaf.

2.2. The homology sheaf o~,(Y; K) of a locally compact space Y is the sheaf associated to the presheaf U --~ H, (U; K) (U open in Y). If (the cohomological dimension of Y)=q, then U-~Hq(U;K) is already a sheaf ([2], 1.10), and the support ofceHq(Y; K) is defined as the support of the corresponding section of ~ ( Y ; K) on Y. If M is semianalytic, the cohomological dimension of M is equal to dim M.

2.3. Lemma. For each semianal ytic pr echain ( N, c) in ~ 5t~q ( M ; K ), there exists another (N', c') ~ ~5~q (M ; K) such that:

a) N' is a closed subset of N. b) The support of c' is N'.

C) iN, ' N (C') = C.

Proof. We can suppose dim N = q > 0 , c4=0. Let N* be the manifold of the q-simple points of N, and N * = U (N*: t e Y ) the partition into its connected components; define

~ 0 = ( t e Y : jN'N'(C)#:O), N0* = [,.)(N~* : te~00), N' =/~0" ,

S = N ' - N * c s N and Co=ff'U~(c). Now, is,sN is injective in dimension q - l , since H~(sN-S)=O, and iS,sNoON~.s(Co)=ON, sNOff'N*(c)=O, SO

De Rham Cohomology of an Analytic Space 279

iN"N~ (r"~-- P More- that C~No, s(Co)=0. Thus c'~Hq(N') exists such that : ~ / - ~ o . over, ff'N*(c)=iN~,N.(Co)=ff'N*(iN,,n(C')), SO that c=iN, n(c' ) since i f , n. is injective in d imension q. It is clear that the suppor t of c' is N' .

2.4. Lemma. For each prechain (N, c)s~S~q(M; K), there exists an- other (L, d ) ~ S ~ q ( M ; K) such that a) the support of d is L; b) the support of ~L, bL(d) is b L," c) (L, d)~(N, c).

Proof. Let t=t~N,bN(C ). By 2.3, a prechain (S,t ') exists such that S~b N is closed, the suppor t of t' is S and is,bN(t')=t. Define N o = N - S and consider the commuta t ive d iagram

O----~ HqlN ) J > Hq(No)[j ~ , Hqi:(S ) ~ ~ Hq_~(N)I

0- ->H~(~ r) ~ ,Hq(N) 0 ,H,_~(bN)~Hq_,(N)

t' ,~ /c'~ for some c'eHq(No), since i~,s(t')=ibN, s(t)=O _. By exactness, =~s0 ,s t J N o w t~N,bSff~ by commutat iv i ty , so that ~eHq(N) exists such that jS'N(~)=C--ff~ Consequent ly c=jS~ where c o = c' + i f ,no (~)~ Hq (No). This implies that (N o, Co) ~ (N, c), and the sup- por t of ~So,bno(Co) = t' is b N o = S.

By 2.3 again, we now find a prechain (L, d) such that L ~ N o is closed in No, so that b L ~ b N o, iL, No(d)=co and the suppor t of d is L. Then (L,d)~(No,co)~(n,c) a n d 63L, bno(d)=t3No,bNo(iL, no(d))=Ono,bNo(eO)=t ' has suppor t b N o. We conclude that b L=bNo, because otherwise OL, bno(d) would be zero on bNo-bL. The suppor t of OL, bL(d) is then b L, as it remained to show, and the l emma is proved.

2.5. Definition. A prechain (L, d)e~5~q (M; K) is faithful if the supports of d~H~(L; K) and OL, bL(d)EHq_I(b L; K) are L and b L, respectively.

Observe that if (L,d) is faithful, its restrictions (Lc~ M', jL'L~M'(d)) to open subsets, and its boundary ~(L,d), are also faithful.

-2.6. Corollary. Each semianalytic chain aeS~q(M; K) has one and only one faithful representative.

Proof By 2.4, faithful representat ives of at always exist. We have then to prove that, if (NI, q ) ~ ( N 2, c2) and both prechains are faithful it follows N~ = N 2 and c 1 = e 2 .

(a) Suppose N~ and N 2 are closed. In this case, equivalence between prechains reduces to in, n (ci) = iN~ n (c2), where N = N 1 w N 2 . If U = N - N 2 is not empty, ff'v(inb~(c,))=jn"O(Cl)+O, since the suppor t of c, is N~; on the other hand, the exact homology sequence for N2~N gives i f , v (in: ' n (c2)) = 0. Tha t contradicts in, , n (q ) = in~, n (c2). We conclude that U = 0 , N I = N 2 and c l = c 2.

280 T. Bloom and M. Herrera:

(b) Suppose N~ and N 2 are locally closed. Consider S= b N 1 ~ b N 2 and C't=jN~'N'-S(c,) ( t = 1, 2). Both N 1 - S and N z - S are closed in M - S , and the equivalence implies (N 1 - S, c ] ) ~ ( N 2 - S , c~), so that by (a) we have N 1 - S = N 2 - S and c' 1 = c~. Moreover ,

~Nt_S,s(C't)=ibNt,S(~Nt,bN,(Ct)) for t = l , 2, (2)

and the two left hand homology classes are equal. It follows then that (b N 1 , 0N,, bU, (C,))~ (b N z , OU2,b N2 (C2)), and (a) assures b N I = b N z = S. This implies N 1 = N 1 - S = N 2 - S = N 2 and c 1 = c' 1 = c 2 = c 2 , as wanted.

2.7. Remark. In the p r o o f of next theorem, we shall make use of the following e lementary relat ions between simplicial and semianalyt ic chains.

Let ~V" be a Jocally finite simplicial complex in affine space IR", and choose a definite simpticial or ienta t ion of each (open) simplex in ~ . The or ienta t ion of a q-simplex a determines a fundamenta l class ~ , s Hq(v(a); K) of the affine variety v(a) generated by a (cf. [2], 2.9 and [11], I.B. 6). Deno te by e~=ff(~)"(6~) its restriction to a.

T o a q-chain Z k ~ a ( k ~ K )

of the or iented complex ~U with coefficients in K, we associate the semi- analyt ic chain

Z [a, k~ %3 ~9~q(lRm; K),

where the sum is defined since ~F" is locally finite. This cor respondence is a m o n o m o r p h i s m , and one can check that is compat ib le with the simpliciat boundar ies in ~f and the boundar ies in NORm;K), as it follows immedia te ly f rom the definitions (cf. Ell], I.B. 6.1).

2.8. Theorem. Let M be a closed semianalytic set of dimension n in the real analytic space (x , (gx). The presheaf 5e, : V ~ Se. (V; K) ( V open in M ) of semianalytic chains of M with coefficients in K is a ~-soft differen- tial sheaf ([31, II, 9.1), for any paracompactifying family of supports ([31, I, 6.1) on M.

l f o~ is a sheaf of K-modules, there is a natural isomorphism

Proof 1. Let v=Uv

$

be an open cover ing of V and take ~ = [ N , c ] ~ ( V ; K) such that the restrictions rvs,v(~)E~(V~; K) are zero for all s. This meansff'N~Vs(c)=O

De Rham Cohomology of an Analytic Space 281

for all s, so that c = 0 (cf. 2.2) and ~ = 0. The first axiom of sheaves ([31, 1.5) is then proved.

To check the second axiom, let

v=Uvs $

be an open covering of V. Suppose that chains ~,=[Ns, cs]eS~q(V~; K) are given such that rvs,,vs(~s)=rvs,,v,(~t) for all s,t such that V~,= V~c~ V~+0. In terms of prechains, we have

(N s c~ V~,, jus, ~v~ ~vs, (c~))~ (N t c~ V~,, ju,, u, nvs, (ct)) (3)

for such s, t. By 2.6, we can suppose the prechains [Ns, cs] to be faithful so that the prechains in (3) are also faithful. By 2.6 again, (3) implies

Ns~V~t=N~nVs, and jus'u~nvs'(c~)=ju"u'~v~'(c,)

for each (s, t) with V~ c~ Vt+ 0. Therefore

u=UNs $

is a locally closed semianalytic set in K and a class C~Hq(N; K) exists such that jmNs(c)=c~ for all s (cf. 2.2). The chain o~=[N,c]E~(V;K) then satisfies rv~,v(~)=~ for all s, as wanted.

We show now that the sheaf ~ is ~-sofl if the family of supports is paracompactifying. Let ~ be a section of ~ on F ~ . Extend ~ to a section IN, c] E~(V; K), where V is a paracompact neighborhood of F in M ([31, II.9.4). Choose an open neighborhood U of F in M, such that U is semianalytic in X and 0 ~ ~ Then N ~ U is semianalytic in V and, since N c~ U ~ V, is also locally closed and semianalytic in X. The chain [N ~ U, jN, N ~ v (c)] e ~ (M) is therefore an extension of a to M, so that ~ is ~b-soft.

2. To prove the second part of the theorem, consider the differential precosheaf S. = F c 5~. of the sections with compact support of 5P., that is, the covariant functor V~S.(V)=F~(V,Sr (V open in M), with boundaries Oq: Sq~ Sq_ 1 deduced from those of 5P. ([3], V. 1.1). Define on M the precosheaves Bq=Im ~q+l, Zq=ker0q and Hq(S.)=ZJBq (q ~ Z). We are going to construct a natural homomorphism q: H 0 (S.) ~ K and show that the couple (S.,r/) is a torsion free quasi-coresolution of K on M by flabby cosheaves, according to Bredon's terminology ([31, V. II.4). Our assertion will follow then from Theorem V. II.15 of [31.

For any open V in M, the faithful representative (N,c) of a chain [N,c]ESo(V ) is a finite set of points in V with coefficients in K. Let r?': S o ( V ) ~ K be the map that sends (N, c) into the sum of its coefficients

282 T. Bloom and M. Herrera:

equivalently, r/'(N, c) is that element in K image ofc by the homomorphism v(N): Ho(N)-*K induced by mapping N into a point. Suppose that IN, c] is the boundary of a chain [N',c']~SI(V), and choose (N',c') faithful, so that necessarily N' is compact, N=bN' and C=ON~N(C'). By naturality, and exactness in the homology sequence for (N, N'), we get v(N)(c)= v(N')o i:r s~(c)= 0, so that t/' is zero on B o (V) and induces a homomorphism

tlv: Ho (S. (V)) = S o (V)/B o (V) -* K.

It is clear that t/=(t/v ) is natural. The precosheaves Sq = F~ ~ are flabby cosheaves because the sheaves (qeZ) are soft ([3], V.1.5); they are torsion free because, if O+-keK

and (N,c) is a q-prechain with c+0, then k(N,c)=(N, kc)+O, since Hq(N;K) being a submodule of Hq(N*,K), is torsion free.

By definition, (S,,r/) is a quasi-coresolution if the precosheaves H~(S.) (q>0), kerr/ and K/Im tl are locally zero. Let T denote any of these precosheaves. We prove now the apparently stronger assertion. (A). - For any open set U in M and xe U, there exists an open neigh- borhood Vc U of x such that T(V)=0.

Since semianalytic subsets of countable real analytic manifolds are triangulable (cf. [6] and [14]), they are locally contractible. Given the open set U in M and xE U, we can always suppose U countable and find a neighborhood Vc U of x that is contractible and semianalytic in U.

Consider a q-cycle [N,c]~Sq(V), with q>0, and suppose (N,c) is faithful, so that N is compact. By Giesecke-Lojasiewicz's theorem ([6] and [14]), we can choose a semianalytic triangulation of V compatible with N. This is a locally finite simplicial complex ~ in an affine space ~ " , together with an homeomorphism z: V ~ V such that: (a) for any open simplex g e U , z(a) is semianalytic in V and has only simple points; (b) N is the image of a (finite) subcomplex JV of U.

Let ~4~q=(a~Jff: dim a=q) and c~,=jN'~(~)(c)~Hq(z(a); K), for a ~ . Then (N, c) and the prechain a = ~ ((~ (a), c~): a ~ ) are equivalent, since N ' = N - U (z(a): a 6 ~ ) has dimension <q. Therefore a represents the cycle [X, c], and 0 (a) = (N', 0). Let z -~ (~) = ~ ((a, z , ' (c~)): a 6 ~ ) , with z,l(c,,)~Hq(a;K), be the corresponding prechain in JR", and ~ the associated simplicial cycle (2.7).

Suppose q>0. The simplicial q-homology with compact supports of ~U is zero, because I~//'t is contractible, so that a finite (q + 1)-chain/~ exists such that ~/~ =~. It is clear that/~ determines a prechain (N o, Co)6 5~q+,(V; K) such that O(No,Co),,~(N,c). This implies that Hq(S.(V))=O for q>0. In the same way one checks that ker q and K/Im ~ are zero on V, so that (S., r/) is a quasi-coresolution, and the theorem is proved.

De Rham Cohomology of an Analytic Space 283

2.9. The Kiinneth Isomorphism. Let (X, Ox) and (Y, (9 0 be real analytic spaces, and let 5P, x, 5P, r and ~,x• r be the corresponding sheaves of semianalytic chains with coefficients in K. Consider on the total tensor product 5~, x ~ 5P, r on X x Y the total differential and grading. There is a differential homomorphism

defined as follows. If U and V are open in X and Y, and if [M, e]e@x(u) and [N,t]e~r(v), then

( [M, c] | [U, t] ---> [M x N, c Q) t],

where M x N is semianalytic in X x Y with dimension <=p+q, and c Q)t is the image of r | t by the canonical homomorphism

Q): Hv(M;K)| Hp+q(MxN;K)

(cf. [11], I. B. 5).

Taking sections with compact support in (3), one gets a chain map (cf. 2.8.2)

S , (X; K) | S,(Y; K)-~ S,(X x Y; K).

It is simple to show that the algebraic KiJnneth formula corresponding to this map reduces to the classical one (cf. [3], V. 13):

0 --* ~ H;(X; K)| H~,_v(r; K)---~ H~(X x Y; K) P

- , • H;_,(X; K)| H~,_p(Y; K)-=,O. P

2.10. Semianalytic Cochains. For each q~Z, consider on M the pre- sheaves 5fq: V--)Hom(Sq(V; K),K) (V open in M), with differentials 5q: ~ q ( W ) - - ) . ~ , ~ q + l ( V ) defined by 5qf(a)=f(3~), for fe~q(V) and aESq+l(V ) (cf. 2.8.2). The elements in 5Pq(V) are called semianalytic cochains on V with coefficients in K. There is an injective sheaf map ~: K - , 5 p~ defined by associating to k~K the constant function k on the open subsets of M. One checks that 6 ~ reasoning like in the construction of r/, in 2.8.2.

2.tl. Proposition. Suppose K is afield. Then 5 r qEZ), to- gether with e, is a resolution of K on M by flabby sheaves.

The presheaves 5 pq are sheaves because the Sq are cosheaves, and are flabby because the Sq are flabby and K is a field (cf. [3], V. 1.9). Given an open set U in M and x e U, choose a contractable neighborhood V c U of x. By the last theorem, Hq(S,(V; K))=Hq(F~(V, SP,))~-H~(V; K)

284 T. Bloom and M. Herrera:

is zero if q > O, and isomorphic to K if q = O. The corresponding assertion for H q ( 5 : * ( V ) ) ~ - H o m ( t t q ( S , ( V ) ) , K ) i m p l i e s that 5 e*, together with e, is a resolution.

3. De Rham Cohomology

Throughout this section, K denotes the field of complex or real numbers. Let U be an open set in the affine space Km. We shall make use of the following notations:

d~176 q~Z) is the sheaf of germs of smooth (i.e., infinitely differentiable) differential forms on U, with coefficients in K.

s = ~ (O~: q~ Z) is the sheaf of germs of analytic differential forms on U.

d is the exterior differential operator on either of d~ or O~.

@*(U)=F~(U, eq*) is the space of smooth forms on U with compact support, with its usual locally convex inductive topology.

@, (U) is the space of currents on U, or continuous linear functionals on ~*(U). A boundary b: ~ ( U ) - - - , ~ q _ I ( U ) (qeZ) is defined on @,(U) by b T(cO=T(dc O, for T e ~ ( U ) and e e ~ q - l ( U ) .

Suppose K =lR, and let x' ( i= 1 . . . . , m) be the coordinate functions of ~ " . Then F(U, gf:) (F(U, f2~)) is representable as finite sums of forms like a (x) dx s~/x dx s2 ^ . . . / x dx sp, 1 < s I < . . . < sp < m, where a (x) is a real valued smooth (analytic) function on U.

If K=I~, let zS=xS+ i x m+S (s = 1 . . . . . m) be the complex coordinates of I12 m. Then F(U, gg) and F ( U , f ~ ) are representable, respectively, as finite sums of forms like

a ( x ) d x ~ A . . . A d x ~, l < s l < . . . < s p < 2 m , o r

b(z) d z ~ A . . . A d z ~, l ~ S l < . . . < s p ~ m ,

where a(x) is a complex valued function on U, smooth with respect to the variables x ~ (s = 1 . . . . . 2 m), and b (z) is a holomorphic function on U.

3.1. Summary on Integration. Let U be an open set in Km. Consider a prechain ( N , c ) ~ ( U ; K ) , and the manifold N* of the q-simple points of N. The class c* =jN'N*(c)eHq(N*;K) defines orientations and multiplicities on the connected components of N*, and we denote by

~ e ~ ( N * ) r

the corresponding integration current.

The embedding i: N*---~ U defines a homomorphism i*: g~-~g~, . We remark that, if ~e~q(U), its image i*(~)eF(N*,g**) does not in general have compact support on N*. We refer to [ i l l for the proof,

De Rham Cohomology of an Analytic Space 285

in case K=IR, of the following lemma; its extension to the case K = C is immediate (for an alternate proof, see also [4] and [5]).

Lemma. For all ctE~q(u), the integral

i* c*

exists, and the map ~a ~ (U, K) --~ ~ (U), (N, e) --~ I(N, c), (q ~ Z), where

xIm, 5 c*

is an injection that satisfies the following property (Stokes):

b X(N, c)= I(a(X, e)). (1)

Moreover, I (N l ,q )=1(N2 ,c2) i f and only if (Nl,CO~(N2,c2). In fact, with the notation of 2.1, we have that I(N~, cs)=I(N,d ) by Pro- position II. A.3.2 of [11] (s=1,2), and I(N, c l )=I(N,e 2) if and only if c~= c 2, which is the definition of the equivalence.

The induced map I: ~ (U; IR) ~ ~ (U), IN, c] ~ I IN, c] is a homo- morphism by [11] II.A.3.3, and is compatible with boundaries by the Stokes formula (I).

3.2. Analytic Differential Forms. Let (X, (fix) be an analytic space over K. We denote by O ~ = ~ (f2}: qeZ) the sheaf of Grauert-Grothendieck analytic differential forms on X (cf. [7, 8, 10, 15]). g2* is an (fix-module, and has a K-linear differential d: ~ x ~ 2 q+l (qeZ) together with a canonical monomorphism e: K ~ f2 ~ = (fix such that d o e = 0. For each morph i smf : X---, Y of analytic spaces, there exists a naturalf-cohomo- morphism f * : f 2 * ~ O*, which is compatible with differentials.

Let U be open in K ~, (fly the sheaf of analytic functions on U, j ~ (fly a coherent sheaf of ideals with set of zeros Y, and (Y,,(fir) the closed analytic subspace of U with (fir = (~x/J] Y.. Denote by ~ff* ~ f2* the sub- sheaf of germs wyef2* y (ye U) that can be written

~=1 ),=1

where % and fl~ belong to O'v. r and f~ and g~ belong to A~,. Then

f2~ - O v / W d l r , (2)

and the differential on f2~ is induced by the one of f2* thanks the in- clusion d(oU*) m oU~ (cf. [16], w 1).

3.3. Smooth Differential Forms. Let U be open in K m, and let M be a closed semianalytic subset of U. Consider the subsheaf Jffv*M c gt~ of germs of smooth forms cr such that: for any smooth manifold W and

286 T. Bloom and M, Herrera:

any smooth map g: W---~ U with g ( W ) c M , one has g*(a)=0, where g*: 8 t ~ g* is the induced map. This is equivalent to ask that i*(a)=0, where i is the embedding in U of the set of simple points of M.

3.4. Definition. g~=g~l/J~v*M[m is the sheaf of germs of smooth differential forms on M, with coefficients in K, with differential d induced by the inclusion d(Jffv*M)C Jffv*~t-

Observe that JffV*M is fine, as a sheaf of g~ In particular,

F~(M, g*) ~_ F~ (U, g~)/F~(U, JVv*M). (3)

Let now U' be open in K", and M' be closed and semianalytic in U'. Then, for any smooth map f : U'-~ U such that f (M ' )cM, the induced map f * : g ~ g * , satisfies f*(Y~N)~JVV*,M,, and defines a cohomo- morphism f* : g~t~g~, which is compatible with differentials. Con- sequently, g~t can be defined for any locally closed semianalytic set M in an analytic space (X, t~x) over K, thanks to the local characterization of morphisms of analytic spaces, g~t is ~b-fine, for any paracompactifying family of supports q~ on M. Moreover, it follows that, if M i is locally closed and semianalytic in the analytic space (Xi,Ox,) (i--1,2), then each morphism f : XI---, X z such that f ( M O ~ M 2 induces a differential f-cohomomorphism f * : g~2 ~ g*~-

3.4. Integration on Analytic Spaces. Let M be closed and semianalytic in the open set U in K", and let IN, c ] e ~ ( M ; K). For all eeF~(U, ~q. ~t), one has I[N,e] (~)=0, since i*(e)~F(N*,N~.) is zero. We deduce by (3) a homomorphism,

I [N, e] : F~ (M, g~t) ---' K that satisfies (1).

Suppose that [N~, ci] are chains in the open sets U/of K"' (i = 1, 2), and that a smooth morphism f : U~ ~ U 2 is given such that g = f l N 1 is a homomorphism onto N 2 and c 2 is the image of c~ by g. Then I[N2, c2] (~)=I[N1, c1] (f* ~) for any form 0~E~q(u2), as is obvious from the definitions.

Let now M be a locally closed semianalytic set in the analytic space (X,(gx) over K. Consider a form ~ F ( V , g q) on the open set V in M and a chain [N,c]~S~(V;K)=F~(V, SC~,~_(K)). We define I[N,c-I(~) by decomposing a, on a neighborhood of N, in a finite sum of forms ~ti with compact supports contained in the domains of analytic charts of X. The integrals ! [N, c] (,~) are defined, and do not depend on the charts, by the previous remarks. One shows as usual that

I IN, c] (~) = ~ I [U, c] (~ti) i

does not depend on the decomposition.

De Rham Cohomology of an Analytic Space 287

Each form eEF(V,8~) defines then a semianalytic cochain I(e)e 6~ K) = Uom(Sq(V); K) by I(~): [N, c] ~ I[N, c] (c~), for all IN, c] e Sq(V). By 3.1, I: F(V,8*)--.6~*(V;K) is a monomorphism compatible with differentials. I clearly commutes with restrictions to open subsets, so that we have proved the following.

3.5. Proposition. Let M be a locally closed semianalytic set in the analytic space (X, (gx) over K. Let 8" be the sheaf of smooth differential forms on M, and 6e~ be the sheaf of semianalytic cochains on M, both with coefficients in K. Then integration defines a differential monomorphism I such that the following diagram commutes:

M M

~ / (4)

K

where e and ~' are the canonical injections.

3.6. Remark. Suppose that M = X . Then there is a canonical map q: f2*--, 8* from the analytic into the smooth forms on M, which is locally induced, with the notation of 3.2 and 3.3 by the inclusion K v ~ v * ~. This map is not, in general, an injection. In fact 8 1 = 0 for q >d im R X, whereas f~x may not be zero. By composition with I we get an integration homomorphism

am: ~ ~ ~ * (5)

compatible with augmentations but not necessarily injective.

3.7. The Poincar~ Lemma. In general, the sheaves f2] and 8~ are not resolutions of K on X. Grauert and Kerner [7] have given an example of a non-reduced complex analytic space (X, (gx), with X a point, where the sequence~ ~ , (9 x d , f2~ is not exact. Reiffen ([16, 17]) has given a family of examples of reduced analytic spaces over C where the Poincar6 lemma does not hold. In [12], it is shown that for M the algebraic set y ( y - x 3 ) ( y - x ) = O in N 2, 8~ is not a resolution oflR. We will now give another example with M irreducible.

Let f : N ~ N z be the map y~__~yS, y6+ yr. Let U be a neighborhood of 0 in IR, so that f (U) is analytic and irreducible. Set M =f (U) . Since M is one-dimensional d(g~t)=0. We will show however that N ~ n , 8~ is not surjective.

Let we(8~)o, that is, let w be the germ at 0, of a 1-form on M. Now f * we(8~o and d( f* w)=0 so there exists an element he(8~ such that d h = f * w. Suppose there existed an element ge(8~ such that dg=w. Then, we would have f * dg=d( f* g)= f * w and so h=f*g+cons t .

288 T. Bloom and M. Herrera:

Evidently a necessary condition for h to be of such a form is that the formal power series of h (at 0) can be expressed as a power series in y5 and (y6+y7). (This condition is also sufficient, although, this is not so evident.) For w=x 1 dx 2, the resulting h does not have that form and so the Poincar6 lemma does not hold.

3.8. De Rham Cohomology. Let X be a topological space, 5~ (50q: q~Z) a differential sheaf on X such that 50q=0 if q<0 , and @ a family of supports on X. Consider the double complex cg*(X;50*), where cg*(X;50q) is the canonical flabby resolution of 50q (qEZ) (cf. E3], II.2). We recall that the hypercohomology H~(X, 50*) of 50* on X with supports in q~ is the cohomology of the complex of sections F~ cg*(X, 50*) with its usual total differential and graduation.

3.9. Definition. Let (X, Ox) be an analytic space over K. Then

(a) For any sheaf of K-modules ~ on X, and any family of supports cb on X, the analytic De Rham cohomology H*R,+(X;~ ) is the hyper- cohomology

H~R.+(X, ~ ) = H+(X, •x |

(b) Let M be a locally closed semianalytic set in X, ~ a sheaf of K-modules and cb a family of supports on M. The smooth De Rham co- homology H~R,~(M;~ ) is the hypercohomology

H~R,+ (M; ~ ) = H~,(M; g* |

3.10. Remark. When 4) is the family of all closed subsets, we drop it from the notations. If M = X, we distinguish, with the notations aH~R and slieR, between the analytic and smooth De Rham cohomologies.

If H* (H~(X; Q* | ~ ) ) = 0 for q > 0, then the canonical edge homo-

morphisms np(Fo(X, (2~c @~))_~ H~R,o(X,~) (p6Z)

are isomorphisms (cf. [-3], IV.2). This happens, for instance, if ~ = K = ~, X is a Stein space and @ is the family of all closed subsets of X, or if ~ = K = I R and @ is the family of closed subsets. In the smooth case, the corresponding edge homomorphisms are isomorphisms when @ is paracompactifying, since B* | is @-fine.

3.11. Theorem. Let (X, (gx) be an analytic space over K, not necessarily reduced, and M a locally closed semianalytic set in X. Let ~* (resp. E~) be the sheaf of germs of analytic (resp. smooth) differential forms on X (resp. on M) and let ~ and q~ be a sheaf of K-modules and a family of sup- ports on X (resp. M). Then there exist canonical splittings natural in ~:

H~R,+(X , ~)~- H~(X, ~ ) + A~(~) resp.

H~R,+(M , o~),,, H~(M, ~ ) @ B~,(~),

De Rham Cohomology of an Analytic Space 289

where H~( ;if) denotes classical cohomology. In the case ~ = K and oh=all closed subsets, the splittings are compatible with morphisms of paracompact analytic spaces.

Proof. We find the splitting only in the smooth case. The analytic case may be handled in the same may, using the map t/ (3.6). Denote by Y * = ( f f q : q~Z) the complex of sheaves such that f f q = 0 if q+0, and f r o = ~-. We have a commutative diagram of complexes of sheaves:

0~-*

where I: g~t ~ ~r~ has been defined in 3.5 and v and/~ are, in dimension 0, the injections

K | ~ | 1 7 6 1 7 4 and K | ~'|176176174

(cf. (4)). Apply the functor F ~ * to this diagram, and consider total differen-

tials and graduations in the corresponding diagram of double complexes:

F ~ c ~ * i M ' ~ * ) ~ . ~

After identifying the cohomology of the upper complex with H~ (M; if), we get in the cohomology level the commutative diagram

Hg(M;~).

HgR,~(M;ff ) ~ , H P ( F ~ * ( M , ~ a * |

where fi and ~ are edge homomorphisms. But 5~ f f | is a resolution of f f on M (cf. 2.10), so that 7 is an isomorphism (cf. [3], IV.2), and I is a right inverse of ft. This gives the splitting.

We proceed now to prove that the splitting is compatible with morphisms. Let M i be a locally closed semianalytic set in the paracompact analytic space (X i, (gx) (i= 1, 2), and f : X 1 --* X 2 a morphism such that f (MOcM 2. The f-cohomomorphism f* : 8"~---~ g~, (cf. 3.3) induces a chain map Fog * (M z , g*2)--~ FOg* (M1, g'l), and also the homomorphism f'R: H~R(M2, K)-+ H*R(M 1, K) in cohomology. We have to show that

290 T. BloomandM. Herrera:

the following diagram commutes:

HP(M1, K) - ~ , HgR (M 1 , K)

HP(M2,K) ~ , H~R(M2,K )

r, , H P ( M l , S p . ) 4"

t2, HP(M2,5,~

(7)

where fP is the usual map in classical cohomology, and the right arrow is deduced from fP by the isomorphisms I~ o fi~ and I o fiz-

Commutativity of the left diagram is clear, by naturality of the edge homomorphisms. Commutativity of (7) is equivalent to f~R(keri2)c ker il, but this cannot be deduced from a diagram in the chain level. In fact, the image of a semianalytic set by a (even proper) morphism is not in general semianalytic (cf. [13], p. 135), so that no differential homomorphism 5e~2 ~ 5a~ can be constructed from f

We observe, instead, that the edge homomorphisms

HP(F(MI, 8"))-+ HP(MI, 8*) and HP(F(M,, 5e~,))-* HP(MI, 5g~)

(i=1, 2) are isomorphisms, since 8 * is fine and 5P~ is flabby, so that we can replace diagram (7) of hypercohomologies by

Hp(r(M,, 8*,)) HP(r(M ,

* (81

reducing the problem to the proof of fP(ker i2)~ker ]l- Let ~eHP(F(M 2, ~f*)) be the cohomology class of the closed form

~peF(M2,g~t~). The image ]2(~p) in

H p (F(M2, 90* )) = H p (Hom (S. (M2), K)) = H o m (Hp (S. (M 2)), K)

"-~ Hom (H~(M2, K), K)

is zero if and only if the evaluation (p (a) of q~ at all classes a ~ H~ (S, (M 2)) H~(M2;K ) (cf. 2.8) is zero. The inclusion fP(ker i2 )~ker I 1 will be a consequence, therefore, of the following.

3.12. Lemma. Let M~ be locally closed semianalytic sets in the analytic spaces (Xi, (gx,) (i= 1, 2), and let f : X 1 --, X 2 be a morphism such that f ( M x ) ~ M 2. Let g,: H~(MI;K)--*H~(Mz;K) be the induced map in homology; and q~ ~ F(M 2 , 8~h ) a closed form. Then, for all a ~ H~ (M 1 ; K), the evaluation f * qg(a) o f f * qg6F(M1,8~) at a and the evaluation of q~ at g,(a) are equal:

f * (p(a)= ~0(g. (~)) (~e H~,(M,; K)).

De Rham Cohomology of an Analytic Space 29!

Proof We can always suppose that M~ = X i (i= 1, 2) and, since the assertion is trivial for immersions, we can factor f through the graph morphism X 1 ~ X1 • X2, reducing the lemma to the case where the morphism is the projection n: X~ x X z ~ X 2.

Choose then a closed form (p~F(X2, ~fr and a class c~H~(X 1 • X2; K). By the Kfinneth isomorphism (cf. 2.9), ~ has a representative a such that

p

a = E [Mp_q • Nq, cp_q @ tp], q=0

where [Mp_q, cp_q] and [Nq, tq] ( q= l . . . . . p) are semianalytic cycles in X 1 and X 2, of dimensions p - q and q, respectively. It can be assumed, moreover, that the (0,p)-component of a is of the form [x 0 • Np, 1 @ tp], for some point xo~X1, so that [Nv, tp]ESp(X2) is a representative of n, (oOe H~(X2) , and

q~ (n. (ct)) = I [Np, t .] (~o).

On the other hand,

n* tp (ct) = l [ x 0 • Np, 1 | tp] (n* q~),

because the integrals of n* ~o on all components of a of type other that (0, p) are zero. The last integral is equal to I[N., tp] (~o), which implies ~o(n. (c0)=n* q~(~t), as wanted.

To check that the splitting of the analytic De Rham cohomology is compatible with morphisms, it suffices to see that in the diagram

.Hg R (X1, K) or,, Hp ( X, , S'~t~)

similar to (7), one still has f ~ R ( k e r j 2 ) c ke r j 1 . This follows from the factorizations ,]~ = i~ o fli, where the maps fl~: ,Hf~g (Xi; K) ~ ~Hf~ R (X~; K) are canonical (cf. 3.6).

3.13. Remark. In the conditions of 3.11, one has that

B~ (~) ~- H~-' (M; (ff'*/s | ~ ) ,

as follows from the exact sequence of hypercohomology associated to the short exact sequence

0 - , e~, --, ~ ,7 - ~ ~ / ~ , --, 0.

Thus B* is isomorphic to a hypercohomotogy with a shifted grading.

21 Inventiones math., Vol. 7

292 T. Bloom and M. Herrera:

3.14. Corollary. Let X be a complete variety (not necessarily regular) or a reduced prescheme locally of finite type over C with only isolated singularities. Then the hypercohomology of the complex s of rationat regular differential forms on X splits canonically:

H* (X, * Oa.x)~-H*(X,C)OA*.

Proof Suppose X is a complete variety, and let X h be the associated analytic space. By Gaga [18] and [8a], the 1-terms

E~ q = H q (X, O~,x) ~ H* (X, f2* x) and

E~ q = H q (X h, f2],) ~ H* (X h, Q*~)

of the "second" spectral sequences of the algebraic and analytic hyper- cohomologies are isomorphic, so that the hypercohomologies themselves are isomorphic, and the splitting follows from 3.11.

In the case X is a prescheme as stated above, one still proves that the algebraic and analytic hypercohomologies are isomorphic, and the algebraic splitting is induced by the analytic one. First, one takes an affine covering of X and, by means of the spectral sequence argument in [9] (Theorem 1'), one reduces to consider the case where X is an affine scheme U of finite type over C (with isolated singularities, reduced).

Then one applies Theorem 2 of [9], which is local, rephrased in the following form; let X be a projective closure of U, Y= X - U the hyper- plane section at infinity and f : U ~ X the natural immersion. Let f2*h(* yh) be the complex of modules on X h of "differential forms with polar singularities on Y"". Then the canonical homomorphisms

q ~-~* * y h q " *

are isomorphisms.

In fact, U h is regular on a neighborhood of y h , SO that the proof in [9] holds. Consequently, the 2-terms of the spectral sequences

EPzq=H"(Xh; d/gq(f2*~(* Yh))) ~ HP+q(Xh; f2}~(*Vh)),

E~" = H p (Xh; R q f . (Q~)) ~ H p +q (uh; f2~,)

are isomorphic, so that the limits also are. One has, moreover, that the second spectral sequences of both hypercohomologies degenerate (because U n is Stein and because of the Corollary in [9]), so that they are respectively isomorphic to

HP+q[F(X h, O~,(*Yh))] and HP+q[F(U h, (2uQ ].

De Rham Cohomology of an Analytic Space 293

But F(Xh,~h(*Yh))"~F(U, f2*~,tl), SO that

u~+q(r( c, ~*,.~)~- u~*q(rt~ h, ~vh)), as wanted.

3.15. Corollary. Let X be a Stein space. Then H q ( X , ~ ) = 0 for q > d i m c X .

Proof. Recall that, by 3.6, the diagram

s2q ol , Sq

gq

is commutative. Now, since X is a complex analytic space, the map t/ is the zero map for q > dim e X. (Though neither f~ nor gq are necessarily the zero sheaf.) For, t 1 is the zero map over the regular points of X and a section of d ~ zero at the regular points of X is the zero section.

Thus, the induced map:

H q (F(X, ~*)) ~L~ H q (F(X, S*)) is also zero.

In the commutat ive diagram

Hq(r(x ,~*) )

~ x /-/~Rfx, ~:)

by Remark 3.10, fi is an isomorphism for X a Stein space. Thus, in the splitting H~R(X,C)~-Hq(X,C)OAq(C) the projection map from HqR(X,~E) to Hq(X,~) is zero (for q > d i m e X ). That is, Hq(x ,~ ) - -O for q > dime X.

3.16. Remark. More delicate results on the topology of Stein spaces have been obtained by Kaup [21] and Narasimhan [22].

3.17. In Section 3.7 an example where the Poincar6 lemma does not hold, was given. In [12] it is shown that if M is one-dimensional then the complex

0 -~ ~ -~ ( : ) o -~ (~,~)0 - ' 0

has finite dimensional cohomology. We now prove a finiteness theorem for complex analytic spaces with isolated singularities. Specifically:

Theorem. Let X be a complex analytic space and x o e X an isolated singular point. Then the cohomology groups of the complex 0 --, ~ --~ fl~, ~o are finite dimensional.

21"

294 T. Bloom and M. Herrera:

Proof. Following Hironaka [20] we may take a neighborhood U of x o and a resolution n: Y--~ U. That is, Y is a complex manifold, n a proper, holomorphic map and n: Y - n - l ( X o ) ~ U-{x0} is a biholo- morphism. Let M=n- l (Xo) . Note that M is compact.

Consider the map n*: t2*V,~o-.F(M,t~) which takes an element weQ~,xo to the induced form n* w~F(M, 0"). We thus obtain a map of complexes

(a): 0 ~ , 0 " UO x0

(b): 0 , r , r (M, 0") .

We will show that the complex (a) has finite dimensional cohomology by showing that n* has a finite dimensional kernel and cokernel, and that the complex (b) has finite dimensional cohomology.

Lemma. n* has finite dimensional kernel and cokernel.

Proof. For each integer p > O, consider the direct image sheaf n , (~2~). t~p~ - o p for x4:x o and (n,O~,)xo~-F(M,~2~). By the result NOW, 7C,~ y j x - - ~ U , x

of Grauert [I9], n , ~ , is a coherent analytic sheaf and so, the kernel and cokernel of the map (also denoted by n*)

are coherent analytic sheaves. Since their support is the point x o, they are finite dimensional vector spaces. It remains to show that the com- plex (b) has finite dimensional cohomology.

We begin by noting that the higher direct image sheaves R q n , ((2~,) are coherent analytic sheaves [193. Their support is the single point x 0 and so, Hq(M,(2(,) which is isomorphic to (Rqn,(Y2~,))~,o, is a finite- dimensional vector space. Thus, the complex 0 - - ~ - - ~ ~ , restricted to M, is a resolution of ~ by sheaves whose higher dimensional co- homology groups are finite dimensional. That the complex (b) has finite dimensional cohomology, will now result from the following lemma and corollary.

3.18. Lemma. Let 0-* d--~ ~ * be a resolution of the sheaf d over the topological space X. Suppose that

1. d and ~ * are sheaves of vector spaces 2. dim H q (X, s < ~ for p > O, q > 1.

Then the canonical map H~(F(X, .~*))--* H~(X, ~t) has finite dimensional kernel and cokernel for q>_O.

De Rham Cohomology of an Analytic Space 295

Proof Consider the double complexes (each in a quarter plane).

APq=F(X, Cgp(X,~q)) p>=O, q>=O

where 0-~ L~ 'q ~ c~, (X, ~q'q) is the canonical resolution of s 1-3] and

BPq=F(X,o~c~'q) p=0 , q>=O

= 0 p > 0

That is, B pq has only one non-zero row. As usual, Tot(A) (resp. Tot(B)) denotes the complex A (resp. B) with its total graduation and differential.

There is a morphism 4: B-+A as follows B~ ~ is the injection of F(X, Sf q) into F(X,C~~ LPq)) while BP'q~A vq for p > 0 is just the injection of zero into A pq. ~ is compatible with differentials and will induce morphisms on the associated spectral sequences. Note that for the second spectral sequence associated with each double complex, the induced morphism

t t~pqlDX ttl'TpqlaX E, 2 [D)----~" 12, 2 tzq_J

has finite dimensional kernel and cokernel. It is the isomorphism for the "E~ ~ terms. For the "E~ q terms

for q > l , "E~q(B)=O, while, the hypotheses imply that "E~q(A)= HP(Hq(X,~c~*)) are finite dimensional. Thus "EP~q(B)~"'~Pq"'~, t,a) has finite dimensional kernel and cokernel for every r > 2.

Now, since we are dealing with quarter plane spectral sequences, we deduce that the maps "E~ q (B) ~"E~ q (A) and H q (Tot (B)) ~ H q (Tot (A)) have finite dimensional kernel and cokernel for every q>0 . But /-t*(Tot(B))~_/-/*(r(x,~e*)) and since the first spectral sequence asso- ciated to A is degenerate H*(Tot(A)) ~ -H*(X,d) . Thus, the canonical map Hq(F(X,~'*))~Hq(X,d) has finite dimensional kernel and co- kernel for q > 0.

Corollary. I f dim Hq(X, d ) < oo for q>O then dim Hq(F(X, ~ * ) ) < oo for q>O.

We note that we may apply the above corollary to prove that the complex (b) has finite dimensional cohomology, for, M is compact and so dim H q (M, r < oo for q > 0.

Remark. For a singularity which is not an isolated point, the co- homology groups may be infinite dimensional. Such an example was communicated to the authors by Bfiiske. He showed, using similar methods to those of [16], that for

X = { Z ~ _ C 5 I Z 2 Z29 Z32 j_ Z32 -10 - - - 1 0 Z 4 -'i'- Z 4 Z 52 ..~ Z32 Z5_~.Z 2 2 9 Z10 _[_ Z5 = 0} ,

4- , H (f2x, o) is infinite dimensional.

296 T. Bloom and M. Herrera: De Rham Cohomology of an Analytic Space

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Miguel Herrera Thomas Bloom University of Washington Institut Henri Poincar6 Seattle, USA Rue Pierre Curie

Paris (5e), France

(Received October 14, 1968; in revised form February 17, 1969)