Second Leray spectral sequence of relative hypercohomologyhypercohomology (generalized Mayer-Vietoris Mayer-Vietoris sequence)

SAUL LUBKIN AND GORO CORO C. KATO

ABSTRACT A second Leray spectral sequence of relative E~,qEE,q = HP(X,Hq(X,U,5f*))HP(X,Hq(X,V,~*)) and the abutment is Hn(X,U,J*).Ho(X,V,':J*). hypercohomology is constructed. (This is skew in generality to Note. Let X be an open subset D in Cnen = CX...~XC and let an earlier one constructed by S. Lubkin [(1968) Ann. Math. 87, 105-255].) The Mayer-Vietoris sequence of relative hyperco­hrperco­ n times homology [Lubkin,rLubkin, S. (1968) Ann. Math. 87, 105-255] is IS also U = D - RnRn n D and let J*~* be the single sheaf 0,(Q, sheaf of generalized. generalizea. germs of holomorphic functions on D, then Hn(D,DHO(D,D - Rn n D,(0)D,(Q) is called the sheaf of hyperfunctions on Rn nn D (ref. This note provides a generalization of the LerayLeray spectral se- se­ 2). I quence. This material was originally presented in the seminar COROLLARY 4. Let f be a continuous map from X to Y; let et "Zeta Matrices of an Algebraic Family," given by S.L. at VU = VV·= = 0;algebraic geometry, complex analytic geometry, special case we obtain the most familiar, ordinary Leray and the theory of hyperfunctions of several variables. spectral sequence with EEqE~,q = HP(Y,(Rqf*)(5r)) HP(Y,(Rqf*)(~)) and with Also we generalize the Mayer-Vietoris sequence in ref. 1,1.6,1, I.6, abutment Hn(X,'7). HO(X,':J). Corollary 4.3. (This generalized Mayer-Vietoris Mayer-Vietoris sequence was Note. In this case, when U = V = Leray spectral sequence of EF Co+(8(X)),Co+(e(X)), and VU = V = functor. First note that the abelian tinuous map from X to Y; let VU be an open subset in X; and category Co+Co + (8(X))(@(X)) has enough injectives (ref. 1,1.1,1, I. 1, Theorem let V be an open subset in Y. Let us denote by Co+(8(X))Co+(Y(X)) the 2) and that the relative rel~tive hypercohomology is a system of derived category of positive cochain complexes of sheaves of abelian functors on Co+(8(X))Co+ (@(X)) (ref. 1,1.1,1, I.1, pp.pp.117-118). 117-118). We must first groups on X. For each open subset W in Y and F* E show that HO(Y,V,Hy(X,U,J(*))HO(Y,V,H~(X,U,~*)) = HO(Xf-'(V)HO(X,f- 1(V) U USA) U,~*) for Co+(8(X)), Hyq(X,V,~*) Co+(e(X)), define Hyq(X,U,5*) to be the sheaf associated to ~* 1 V,~*)). V* EF Co+(8(X)).Co+(eV(X)). This is true because W NI'+H°(f-I(W), .H°(j-l(W), the presheaf (W NI'+ Hq(f-Hq(f-l(W),f-1(W)(W),f-l(W) n u,5I*)). Then there f- 1(W) U,':J*)U, is obviously a sheaf. It remains to show that is induced a first-quadrant cohomological spectral sequence f-'(W) n V) obviously H~(X,U,.1*)Ho(X,U,Y*) is flask for an injective object .1*5* in Co+ (8(X))(&(X)) with EE,qEq = HP(Y,V,H~(X,V,~*))HP(Y,V,HI(X,U,5(*)) abutting to Ho(X,f-l(V)Hn(X,f-l(V) -Le.,i.e., a cochain complex .1*X such that 50JO -+ S.11 -+ •••.. -+- .1n+5f+ 1 UV,~*).UU,5*). -+ •••... is exact, such that .1i5i is injective, all j ~> 0 and such that Note. We call this spectral sequence "the second Leray j ker(.1o X,-+ .11) is injective (ref. 1, 1.1, Theorem 2). But spectral sequence of relative hypercohomology" because there ker(°0 51) (ref. H~(X,U,.1*) = H~(X,U,.1) in which = so that J is is another well-known Leray spectral sequence of relative hy­ Ho(X,U,*) Hy(X,U,J) .19 JiO(.1*),]fO(5*), .1 injective (ref. 1, 1.1,I. Theorem 2). Therefore it remains to show percohomology (ref. 1, 1.6;I.6; pp. 151-152), which we now call (ref. 1, 1, 2). that H~(X, is flask when .Y is an injective sheaf. If W is an the first Leray spectral sequence of relative hypercohomology. that Ho(X,U,5)U,.1) .1 injective it to show that r(Y,H~(X,U,':J)) This second Leray spectral sequence of relative hypercoho­ open subset in Y, then suffices to sh.ow that r(Y,H°(X,U,5)) -+ r(W,H~(X,U,.1))F is an Let SI.s . F mology is very general (but is skew in generality with respect (W,H°(X,U,J)) is an epimorphism. Let E 1 = r(W,H~(X,U,.1)) and let S2 F to the first Leray spectral sequence of relative hypercohomol­ HO(j-l(W),f-Ho0f-'(W),f-1(W)(W) n U,:I)UJ) F(WHO(XUJ)) and S2 E to the Leray spectral sequence hypercohomol- U. Then there exists F ogy as defined in ref. 1, 1.6, pp. 151-152). r(U,.1)F(U,5) be the zero section of U. Then there exists Sss3 E 1 1 = SI and U = S2. As special cases of the theorem we have the following co­ r(U U f-'(W),5)f- (W),.1) such that sslf-s3Jf'(W)(W) = s, and ssls31 U = S2 we s such that s 1 = rollaries. [If[If f = ididx,x, we write Hq(X,U,~*)Hq(X,U,j7*) for Hence we have s EeF r(X,.1) (X,J) such that s I (U U f-f '(W))(W)) = SsS3 because:lbecause 5 is flask. Then s EF HO(X,U,:I)H0(X,U,J) maps into SI.s . H~(X,U,~*).]HI (X,U,J ). ] COROLLARY 1. If f is an identity map, then EE,qESq = 2. The generalized Mayer-Vietoris sequence HP(X,V,Hq(X,V,~*)) with the abutment HO(X,V U V,~*). HP(X,V,Hq(X,U,5f*)) with the abutment Hn(X,V U US*). THEOREM. Let X be a topological space; letlet V,U, V',U', V, and V' COROLLARY 2. If f is an identity map and if V = exact sequence which Jiq(~*)]jq(5f*) = ker(':Jker(5fq -+ ~q+l)/Im(':Jq-l5rq+l)/Im(jq-l -+ ':J q). ). Note that Corollary 2 is the second spectral sequence of relative hyper­ , ddn-1n- 1 . pOpn HO(U,V',~*)Hn(U,U r) cohomology as defined in ref. 1,1.6, p. 141, equation 2. Thus, cohomology as defined in ref. 1, I.6, p. 141, equation Thus, •.... • • • -----..-> HO(UHHn(U U V,UVU' ' U V/,~*)V',"7*) -----.. ~ that spectral sequence isis a special case of the theorem. H>(U COROLLARY 3. If ff isis an identity map and ifif V = ,, 5*J* = 5J a sin­ sin­ V',F*) and the negative of the restriction Hn(v,V',~*)Hn(V,Vf,"7*) -+ gle'gle injective sheaf. But then the restriction map HO(V,J) HO(U,5) Hn(u n V,U'vxr n v,,5r*). V',~*). -+;;.,HO(V.HO(U n V,J)V,5) is an epimorphism because J:J is 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 abelian category, 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"relative' hy­ whenever 1I isinjective,is injective, and if H is left exact and G is half exact, percohomology are established for punctual cohomology and re~ults then there is induced a long lon~ exact sequence 0 -+ F -+ G -+ H punctual sheaves. Of course those results [and also the first and 1 ~cond ~pectral -+RIFRIF -+D-*RIG RIG -+RIHdz RIHd -+ •••... cfn-a -1* -+RnFRnF -+RnGRnG -+ second spectral sequences ofof' relative hypercohomology (ref. 1, 1.6,I.6, 140-141) in RnHdnRnHcJn -+-Rn+ Rn+ IFF -+.-.*. • • (ref. 4,4,III, III, Exercise 5, p. 52). There­ 1, pp. 140-141) in that paper] work equally well for ordi­ (ref. p. 52). sheaf fore it suffices to show that, for J~ * E Co +(4(X)),(8(X)), the se­se- nary sheaf cohomology by the methods analogous to this quence note. . Conversely, the second Leray spectral sequence of relative - HOHO(V V' ~*) pO (U,Uf,5*)" hypercohomology and the generalized Mayer-Mayer-VietorisVietoris se­ 0-+0 HO(VHO(U U V,V'V,U.' U Vf'J*)V',~*) ~ _ EDdo quence established in this paper hold equally well for punctual HHO(V,V',~*)V , 5* cohomology and punctual sheaves by the analogue of the °(V, ~echnique a0 technique in ref. 1, Chapter 1. . HO(u n vup n v ) [ii Remark 2. The first and secondsecond' Leray spectral sequepcesequence of relative hypercohomology,.hypercohomology; the generalized Mayer-Mayer-Vietoris Vietoris is exact, in which p0pO and aOa0 are defined as in the statement of sequence, and the two spectral sequences of relative hyperco­hyperco- the theorem, and alsoalso to show that for an injective object J* in homology go through equally well to combinatorial punctual Co+(8(X))Co+ (S(X)) the sequence hypercohomology of positive cochain complexes of etale sheaves on a proscheme, as defined in ref. 5, Chapter 1. -0 HO(V,V',J*) - . p HO(U,Uf',*) .00O -+ HO(VHOWU U V,V'VU'1 U V',J*)VI',Y*) ~30 ED01 Note Added in Proof. In the theorem (resp.: Corollary 1; in Corollary in H'(X,U,~*) HO(V,V',J*)HO(V,V",a*) 3) in Section 1, the sheaf HI(X,U,9*) is concentrated on the closed subsetDsubset D = f(X - U)(resp.:U) (resp.: D'=XD= X- - U;DU; D = X­X- U)U)ofY(resp.:X;X).of Y (resp.: X; X). aa0 O Therefore, Eg'qE~,q = HP(D, D n V, H'(X,U,~*)).Hj(X,U,57*)). ~ HO(VHO(U n V,V'v,u' n V',J*)v',5*) -+- 0o [2] is exact. Equation 1 is plainly true from the definitions of p0 pO and 1. 1. Lubkin, S. (1968) Ann. Math. 87,87,105-255. 105-255. aO.a0. To prove Eq. 2, we can replace J*5* by J5 = JfO(J*),110(5*), which 2. 2. Harvey, R. (1969) Am. ].J. Math. 91,91,853-873. 853-873. 3. 3. Grothendieck, A. & Dieudonne,j. (1961) Elements de ce01'J1£trie is injective (ref. 1, 1.1,I. 1, Theorem 2). Consider the commutative Grothendieck, Dieudonne, J. (1961) Elements Geometrie the commutative Algebrique (Publ. Math. I.H.E.S. 11). diagram with exact columns shown in Fig. 1. We must show 4. 4. Carlan,Cartan, H. & Eilenberg, S. (1956) Homological Algebra (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. 5. Lubkin, S. (1967) Am. ].J. Math. 89,443-548.