Proc. Nati. Acad. Sci. USA Vol. 75, No. 10, pp. 4666-4667, October 1978 Second Leray of relative hypercohomology (generalized Mayer-Vietoris sequence) SAUL LUBKIN AND GORO C. KATO Department of Mathematics, The University of Rochester, Rochester, New York 14627 Communicated by Jacob Bigeleisen, June 19, 1978

ABSTRACT A second of relative E~,q = HP(X,Hq(X,U,5f*)) and the abutment is Hn(X,U,J*). hypercohomology is constructed. (This is skew in generality to Cn = an earlier one constructed by S. Lubkin [(1968) Ann. Math. 87, Note. Let X be an open subset D in CX...XC and let 105-255].) The Mayer-Vietoris sequence of relative hyperco- n times homology [Lubkin, S. (1968) Ann. Math. 87, 105-255] is also U = D - Rn D and let J* be the single 0, sheaf of generalized. germs of holomorphic functions on D, then Hn(D,D - Rn D,(0) is called the sheaf of on Rnn D (ref. This note provides a generalization of the Leray spectral se- 2). quence. This material was originally presented in the seminar COROLLARY 4. Let f be a continuous map from X to Y; let "Zeta Matrices of an Algebraic Family," given by S.L. at U = V = 0; let 5r* be a single sheaf 5 F S(X); let (Rqf*)pre(5) Harvard University during the fall of 1969. Our second Leray be the presheaf defined by W Hq(f-'(W),'I); and let spectral sequence of relative hypercohomology has important (Rqf*)(5r) be the sheafassociated to this presheaf. Then in this applications in , complex analytic geometry, special case we obtain the most familiar, ordinary Leray and the theory of hyperfunctions of several variables. spectral sequence with EEq = HP(Y,(Rqf*)(5r)) and with Also we generalize the Mayer-Vietoris sequence in ref. 1, I.6, abutment Hn(X,'7). Corollary 4.3. (This generalized Mayer-Vietoris sequence was Note. In this case, when U = V = X and 7* = 57 in the the- also presented in the same seminar, "Zeta Matrices of an Al- orem, we have that Hyq(X, 7) = (R f*)(5). gebraic Family," in 1969.) COROLLARY 5. If F is a continuous map from X to Y, F* 1. The second Leray spectral sequence of F Co+(e(X)), and U = V = X, then we have a spectral se- hypercohomology modulo an open subset quence of ref. 3, chapter III, section 2. THEOREM. Let X and Y be topological spaces; let f be a con- Proof of the theorem. We use the theorem of the spectral tinuous map from X to Y; let U be an open subset in X; and sequence of a composite . First note that the abelian let V be an open subset in Y. Let us denote by Co+(Y(X)) the Co + (@(X)) has enough injectives (ref. 1, I.1, Theorem category of positive cochain complexes ofsheaves ofabelian 2) and that the relative hypercohomology is a system of derived groups on X. For each open subset W in Y and F* E on Co+ (@(X)) (ref. 1, I.1, pp. 117-118). We must first to be the associated to show that HO(Y,V,Hy(X,U,J(*)) = HO(Xf-'(V) U USA) for Co+(e(X)), define Hyq(X,U,5*) sheaf V* F Co+(eV(X)). This is true because W H°(f-I(W), the presheaf (W Hq(f-l(W),f-1(W) n u,5I*)). Then there f-'(W) n U, V) is obviously a sheaf. It remains to show that is induced a first-quadrant cohomological spectral sequence Ho(X,U,Y*) is flask for an 5* in Co+ (&(X)) with Eq = HP(Y,V,HI(X,U,5(*)) abutting to Hn(X,f-l(V) X 50 S .. - 5f+ 1 UU,5*). i.e., a cochain complex such that Note. We call this spectral sequence "the second Leray ... is exact, such that 5i is injective, all j > 0 and such that spectral sequence of relative hypercohomology" because there ker(°0 X, 51) is injective (ref. 1, 1.1, Theorem 2). But is another well-known Leray spectral sequence of relative hy- Ho(X,U,*) = Hy(X,U,J) in which 9 = ]fO(5*), so that J is percohomology (ref. 1, I.6; pp. 151-152), which we now call injective (ref. 1, I.1, Theorem 2). Therefore it remains to show the first Leray spectral sequence of relative hypercohomology. that Ho(X,U,5) is flask when .Y is an . If W is an This second Leray spectral sequence of relative hypercoho- open subset in Y, then it suffices to show that r(Y,H°(X,U,5)) mology is very general (but is skew in generality with respect F(W,H°(X,U,J)) is an epimorphism. Let s . F to the first Leray spectral sequence of relative hypercohomol- Ho0f-'(W),f-1(W) n UJ) = F(WHO(XUJ)) and let S2 F ogy as defined in ref. 1, 1.6, pp. 151-152). F(U,5) be the zero section of U. Then there exists s3 F As special cases of the theorem we have the following co- r(U U f-'(W),5) such that s3Jf'(W) = s, and s31 U = S2 rollaries. [If f = idx, we write Hq(X,U,j7*) for Hence we have s eF (X,J) such that s I (U U f '(W)) = S3 HI (X,U,J ). ] because 5 is flask. Then s F H0(X,U,J) maps into s . COROLLARY 1. If f is an identity map, then ESq = 2. The generalized Mayer-Vietoris sequence HP(X,V,Hq(X,U,5f*)) with the abutment Hn(X,V U US*). THEOREM. Let X be a ; let U, U', V, and V' COROLLARY 2. If f is an identity map and if U = 4, then be open subsets in X such that U' C U and V' C V; and let J* E~,q = HP(X,V,yiq(j7*)) with the abutment Hn(X,V,J*), in G Co+(&P(X)). Then there is induced a long which ]jq(5f*) = ker(5fq 5rq+l)/Im(jq-l q). Note that 2 of Corollary is the second spectral sequence relative hyper- dn-1 pn Hn(U,U , r) as defined in ref. 1, I.6, p. 141, equation 2. Thus, .... -> HHn(U U VU' U that spectral sequence is a special case of the theorem. V',"7*) H>(U COROLLARY 3. If f is an identity map and if V = X, then Hn(V,V',l*) ain The publication costs of this article were defrayed in part by page 30 Hn(u (a v,u, (n v/,5*) charge payment. This article must therefore be hereby marked "ad- vertisement" in accordance with 18 U. S. C. §1734 solely to indicate dn this fact. -o Hn+l(U U V,U' U V',*) * * *. 4666 Downloaded by guest on September 24, 2021 Mathematics: Uibkin and Kato Proc. Natl. Acad. Sci. USA 75 (1978) 4667

0 v 0

H0(U', a7) I 1~~~p c 0. ,H(U' V',y) ° 0 H0(V',Y) 4

P2 Ho(u, X) ci2 0 iH HOWUV, Y) -O H0(UflV,) -O H°(V, X)

o-I H°(U,U',.Y) ,o I 0 -o Ho(UUVU'UV',Y) HO(UflVU'nV',Y) -. 0 I 1HO(V,V, JI) I 0 0 0 FIG. 1. Commutative diagram.

where the map p is induced by restrictions, and the map a" prove that the map TO in Eq. 2 is an epimorphism, we are re- is induced by the restriction Hn(U,U',5I*) - Hn(U n vu' n duced to the case in which U' = V= , 5* = 5 a sin- V',F*) and the negative of the restriction Hn(V,Vf,"7*) gle injective sheaf. But then the restriction map HO(U,5) Hn(u n vxr n v,,5r*). .HO(U n V,5) is an epimorphism because J is Proof. In general if we are given an exact sequence of flask. functors and natural transformations 0 - F - G - H (from q.e.d. one , having enough injectives, into another Remark 1. In ref. 1, 1.6, Corollary 4.3, Mayer-Vietoris se- abelian category) such that G(I) H(I) is an epimorphism quences and the first Leray spectral sequence of relative' hy- whenever I is injective, and if H is left exact and G is half exact, percohomology are established for punctual cohomology and then there is induced a longexact sequence 0 F G H punctual sheaves. Of course those results [and also the first and RIF D-*RIG RIHdz ... a -1*RnF RnG second spectral sequences of' relative hypercohomology (ref. RnHdn -Rn+ F -.*. (ref. 4,III, Exercise 5, p. 52). There- 1, I.6, pp. 140-141) in that paper] work equally well for ordi- fore it suffices to show that, for J E Co +(4(X)), the se- nary by the methods analogous to this quence note. Conversely, the second Leray spectral sequence of relative HO (U,Uf,5*) hypercohomology and the generalized Mayer-Vietoris se- 0 HO(U U V,U.' U Vf'J*) do quence established in this paper hold equally well for punctual H V , 5* cohomology and punctual sheaves by the analogue of the °(V, technique in ref. 1, Chapter 1. a0 HO(u n vup n v ) Remark 2. The first and second' Leray spectral sequence of [ii relative hypercohomology; the generalized Mayer-Vietoris is exact, in which p0 and a0 are defined as in the statement of sequence, and the two spectral sequences of relative hyperco- the theorem, and also to show that for an injective object J* in homology go through equally well to combinatorial punctual Co+ (S(X)) the sequence hypercohomology of positive cochain complexes of etale sheaves on a HO(U,Uf',*) proscheme, as defined in ref. 5, Chapter 1. 0O HOWUU VU'1 U VI',Y*) 30 01 Note Added in Proof. In the theorem (resp.: Corollary 1; in Corollary HO(V,V",a*) 3) in Section 1, the sheaf HI(X,U,9*) is concentrated on the closed subset D = f(X - U) (resp.: D= X- U; D = X- U) of Y (resp.: X; X). a0 Therefore, = HP(D, D n V, HO(U n v,u' n v',5*) - o [2] Eg'q Hj(X,U,57*)). is exact. Equation 1 is plainly true from the definitions of p0 and 1. Lubkin, S. (1968) Ann. Math. 87,105-255. a0. To prove Eq. 2, we can replace 5* by 5 = which 2. Harvey, R. (1969) Am. J. Math. 91,853-873. 110(5*), 3. Grothendieck, A. & Dieudonne, J. (1961) Elements de Geometrie is injective (ref. 1, I. 1, Theorem 2). Consider the commutative Algebrique (Publ. Math. I.H.E.S. 11). diagram with exact columns shown in Fig. 1. We must show 4. Cartan, H. & Eilenberg, S. (1956) (Princeton that the bottom row is exact. But by the nine lemma, it suffices Univ. Press, Princeton, NJ). to show that the top two rows are exact. Therefore in order to 5. Lubkin, S. (1967) Am. J. Math. 89,443-548. Downloaded by guest on September 24, 2021