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APPENDIX: Hypercohomology and Spectral Sequences 147 APPENDIX: Hypercohomology and spectral sequences 147 APPENDIX: Hypercohomology and spectral sequences 1. In this appendix, we list some formal properties of cohomology of complexes that we are using throughout these notes. However we do not pretend making a complete account on this topic. In particular, we avoid the use of the derived category, which is treated to a broad extend in the literature (see [60], [29], [7], [8], [33], [31]). 2. Through this section X is a variety over a commutative ring k. 3. We consider complexes Fe of sheaves of a-modules, where a is a sheaf of commutative rings. For example a = 7l , a = k or a = structure sheaf of X. Any map of a-modules a: Fe --+ ge between two such complexes induces a map of cohomology sheaves: where 1£i(:p) is the sheaf associated to the presheaf U ker r(U, Fi) ---t r(U, Fi+l) f---> im r(U, Fi-l ) ---t r(U, Fi) in the given topology. One says that a is a quasi-isomorphism if 1£i(a) is an isomorphism for all i. 4. We will only consider complexes Fe which are bounded below, that means Fi = 0 for i sufficiently negative. 5. Example: the analytic de Rham complex. X is a complex manifold. Then the standard map 148 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems from the constant sheaf <C to the analytic de Rham complex Ox is a quasi­ isomorphism as by the so called "Poincare lemma" 6. Example: the tech complex. Let U = {Ua; a E A}, for A c 1N, be some open covering of the variety X defined over k. To a bounded below complex Fe one associates its Cech complex ge defined as follows. gi := E9C a (U, F i- a ) a2':0 where II Here, for any Uao ... aa := Uao n ... n Uaa {! denotes the open embedding and for any sheaf F, one writes (!*F for the sheaf associated to the presheaf U f-+ f(Uao ... aa n U, F). As Fi = 0 for i < < 0, the direct sum in the definition of 9 has finitely many summands. The differential Ll of ge is defined by Ll(s) = (-1)i8s + dps for s E Ca(U,Fi- a), where 8 is the Cech differential defined by and dp is the differential of Fe. Then the natural map defined by Fi ~ II (!*Filu" = CO(U,Fi) aEA is a quasi-isomorphism. APPENDIX: Hypercohomology and spectral sequences 149 To show this one considers first a single sheaf F (see [30], III 4.2) and then one computes that, whenever one has a double complex r r r -----+ K1,i-l -----+ K1,i -----+ K 1,i+1 -----+ r r r rd vert -----+ KO,i-l -----+ KO,i -----+ KO,i+l -----+ r r r -----+ F i - 1 -----+ Fi -----+ Fi+l -----+ ... -----+ dhor such that Fi --+ Ke,i is a quasi-isomorphism for all i, then Fe --+ Ve is a quasi-isomorphism as well, where V e is the associated double complex: a a with differential (-1) i dvert + dhor . 7. If Fe and ge are two complexes of O-modules bounded below one defines the tensor product Fe @ ge by a with differential from Fa @ gi-a to F a+1 @ gi-a EB Fa @ gi+1-a. given by d(Ja @ gi-a) = dfa @ gi-a + (_l)a fa @ dgi- a. As both Fe and ge are bounded below, a takes finitely many values, and (Fe @ ge) is a complex of O-modules bounded below. 8. If in 7 we assume moreover that locally the O-modules Fa and gb are free and dFa-l as well as dgb- 1 are subbundles for all a and b, then locally one has some decomposition Fa dFa-l EB 1{a(Fe) EB F 'a gb dgb- 1 EB 1{bwe) EB g'b where d: F'a --+ dFa d: (JIb --+ d(Jb 150 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems are isomorphisms. In particular, :Fa 0 gb d(:Fa- 1 0 dg b- 1 + :Fa- 1 01ib + 1ia 0 gb-l) EB 1ia(:Fe) 01ibWe) EB (d:Fa- 1 + 1ia(:Fe)) 0 g'b EB pa 0 (dg b- 1 + 1ibwe)) EB pa 0 g'b and therefore one has the Kiinneth decomposition a 9. The map a : :Fe ~ Ie is called an injective resolution of :Fe if Ie is a complex of V-modules bounded below, a is a quasi-isomorphism, and the sheaves Ii are injective for all i, Le.: is surjective for any injective map A ~ B of sheaves of V-modules. It is an easy fact that if V is a constant commutative ring, for example V = 'll, V = k, then every complex of V-modules which is bounded below admits an injective resolution (see [33], (6.1)). 10. From now on we assume that V is a constant commutative ring. Let :Fe be a complex of V-modules, bounded below. One defines the hypercohomology group ]Ha(x, Fe), to be the V-module ]Ha(X :Fe) .= ker r(X,Ia) -+ r(x, Ia+1) , . im r(X,Ia-l) -+ r(X,Ia) . One verifies that this definition does not depend on the injective resolution choosen (see [30] III, 1.0.8). In particular, if a : :Fe ~ ge is a quasi-isomorphism, then a induces an isomorphism of the hypercohomology groups: (by taking an injective resolution Ie of ge which is also an injective resolution of :Fe). 11. By definition, Ha(X,I) = 0 for a > 0 if I is an injective sheaf. We will verify in (A.28) that if a : Fe ~ ge is a quasi-isomorphism and if Ha(x, gi) = 0 APPENDIX: Hypercohomology and spectral sequences 151 for all a > 0 and all i, then We call we, (J) an acyclic resolution of :P in this case. 12. IH transforms short exact sequences of complexes of a-modules which are bounded below into long exact sequences of a-modules. 13. We assume now that the complex Fe of a-modules, bounded below, has sub­ complexes ... C Filti-l C Filti C ... C Fe (or ... c Fz'lti C F'lti-lz c ... c Fe) such that and such that Fitti = 0 for i < < 0 (or Fitti = 0 for i» 0). One says that Fe is filtered by the sub complexes Fitti (or Filti ). Via (A.12), this filtration defines a filtration on the hypercohomology groups: This just means that the group IHa(X, Fe) has subgroups ... C Filti_lIHa(X,:p) C FiltiIHa(X,Fe) C ... C IHa(X,Fe) (or ... C FiltiJHa(X:p) C FiW-1IHa(X,Fe) C ... C IHa(X,Fe)). 152 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems We pass from increasing to decreasing filtrations by setting 14. We define Gri := Filti/Filti- l . One has a diagram of exact sequences 0 0 1 1 0 ------. Filti- l Filti ------. Gr i ------. 0 1 1 identity F· F· 1 1 o ------. Gr i ------. F· / Filti-l ------. F· /Filti ------. 0 1 1 0 0 which gives via (A.12): ker(Ha (Gri)-->Ha+I (Filti_I)) ker(Ha(Gri)-->Ha('p / Filti_I))' We define Obviously E~-i,i is a subquotient of E;-i,i. 15. The formations of spectral sequences has the aim to compute Eg;;i,i only in terms of E~,t, by filtering the terms JH a+1(Filti_l ) and JHa(F· / Filti-d ap­ pearing in the description (A.14) by the induced filtrations: Filtl JHa+1(Filti_d := im(JHa+1(Filtl) ----+ JHa+1(Filti_d) APPENDIX: Hypercohomology and spectral sequences 153 for 1:S i-I and for I ~ i-I, and by computing the corresponding graded quotients. 16. If we assume that for some a, the filtration FiltiJHa(:P) is exhausting, that is: then is the corresponding graded group. In particular, assume that JHa (.7='.) and E~-i,i are free O-modules (where 0 is 7l or k), and that JHa(.p) is of finite rank. Then If E~-i,i is also free, then rankoJHa(.P) :S I:i rankoE~-i,i and one has equality if and only if Er:;;i,i = E~-i,i for all z. 17. Example: One step filtration: 0= Filts- 1 c Filts = :P. Then JHa(.p) = JHa(Gr s) = E~-s,s = E~-s,s. 18. Example: Two steps filtration: 0= Filts- 2 C Filts- 1 c Fitts =:r Then one has the exact sequence o ----t G r s -1 ----t F· ----t Gr s ----t 0 and with 154 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems 19. Define the differential d2 as the connecting morphism of the vertical exact sequence on the right hand side (or of the above horizontal exact sequence) of the diagram 0 0 1 1 0--+ GTi-l --+ Filtd Filti- 2 --+ GTi --+ 0 II 1 1 0 --+ GTi-l --+ Filti+1/ Filti- 2 --+ Filti+1/ Filti- 1 --+ 0 1 1 identity GTi+l GTi+l 1 1 0 0 and the V-module E~-i,i by k Ea-i,i ~ E a-i+2,i-l Ea-i,i _ er 2 2 3 - . 2 d2 ' , . E a-.-,.'+1 E a -.,. 1m 2 ---+ 2 For c = 1,2 and the c-steps filtration one has Ea-i,i _ Ea-i,i 00 - e+l' as we saw in (A.17) and (A.18), and the filtration on Ha(.p) is exhausting. 20. The right vertical exact sequence of the diagram (A.19) gives a surjection and the middle horizontal sequence induces a morphism Replacing i by i-I, one obtains maps d' Ha+1(G ) a( .
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