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Second Leray Spectral Sequence of Relative Hypercohomologyhypercohomology (Generalized Mayer-Vietoris Mayer-Vietoris Sequence)

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Second Leray Spectral Sequence of Relative Hypercohomologyhypercohomology (Generalized Mayer-Vietoris Mayer-Vietoris Sequence) 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;<p; let 5r*~* be a single sheaf 5':J FE S(X);8(X); let (Rqf*)pre(5)(Rqf* )pre(~) Harvard University Vniversity during the fall of 1969. Our second Leray 1 Harvard during be the presheaf defined by W NI'+ Hq(f-'(W),'I); Hq(f- (W),':J); and let spectral sequence of relative hypercohomology has important (Rqf*)(5r) (Rqf* )(~) be the sheaf associated to this presheaf. Then in this applications in 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 = <pX and ~*7* = ':J57 in the the-the­ also presented in the same seminar, "Zeta Matrices of an Al­ orem, we have that Hyq(X,Hyq(X,~) 7) = (Rqf*(R f*)(5). )(':J). gebraic Family," in 1969.) COROLLARY 5. If F is a continuous map from X to Y, F* 1. The second Leray spectral sequence of EF Co+(8(X)),Co+(e(X)), and VU = V = <p,X, then we have a spectral se-se­ hypercohomology modulo an open subset quence of ref. 3, chapter 111,III, section 2. Proof of the theorem. We use the theorem of the spectral THEOREM. Let X and Y be topological spaces; let f be a con­con- sequence of a composite 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 = <P, then COROLLARY If identity map and if U 4, be open subsets in X such that V'U' C U and V' C V; and letlet ~*J* EE,qE~,q = HP(X,V,Jiq(':J*))HP(X,V,yiq(j7*)) with the abutment HO(X,V,~*),Hn(X,V,J*), in a exact q q EG Co+(8(X)).Co+(&P(X)). Then there isis induced a long 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 = <p,X, then HO(VHn(V,V',l*),VI, ':J *) (inain The publication costs of this article were defrayed in part by ~agepage -----..30 HO(UHn(u n(a V,Uv,u,' n(n V/,~*)v/,5*) charge payment. This article must therefore be hereby marked "ad­ad­ vertisement" in accordance with 1818 U. S. C. §1734 solelysolely to indicate dn this fact.fact. -o Hn+l(U U V,U' U V',*) * * *. 0 0v 0 0 I 0 t HO(U', :/) tI H0(U', a7) Qc 1~~~pPi i o -+ HO(U'UV', :I) E9 0. ,HHO(U'n(U' V',:I)V',y) -+° 0 HO(V',H0(V',Y) :I) 4 1 1 JlO(U,:I)Ho(u, X) Q 1 P2 ci2 o0 ---.iH HO(UHOWUV, UV,:I) Y) -O E& H0(UflV,)HO(U n V,:I) ----+-O 0 HO(V,H°(V, :I)X) 1 o - 1I - HO(H°(U,U',.Y) U,U', :I) QO uv,U'UV'-,:I)~ ,o 1 I o0 -+-o HO(UHo(UUVU'UV',Y) E& -----.
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