2 1.THE LERAY-SERRE SPECTRAL SEQUENCE In fact, one has not to go to infinity in this situation. Since E ^ = 0 for i 0 or j 0, the arrows dr starting or ending at (z, j) are trivial maps for r large enough (e.g. r max{i, j}). Thus, the modules E^J become stationary after some time. However, there is one big problem: in general, no information is given about the arrows c?2, fife,...! Thus, it seems to be impossible to determine E^'-7, E4"7,.... Here, the multiplicative structure of the cohomology becomes important. The arrows dr act as derivations, and this makes it (often) feasible to determine all terms in the spectral sequence. The material of this chapter can be found in McCleary [69], Borel [9, 12], Spanier [90], Whitehead [113], and Fomenko-Fuchs-Gutenmacher [35]. Throughout this chapter, R is a principal ideal domain. l.A. Additive structure 1.1. GRADED AND BIGRADED MODULES A graded R-module is a direct sum zGZ of i?-modules M% indexed by the integers. Similarly, a bigraded R-module is a direct sum M = M" = 0 M^J tj'GZ of i?-modules M 2 J , indexed by pairs of integers. The elements of M2 or M2'-7 are called homogeneous of degree i or bidegree (i, j), respectively. A graded or bigraded module is of finite type if the Ml or M M are finitely generated. A submodule N C M # is graded if JV = 0 - G Z N\ where Nl = N n M \ 1.2. TOTAL GRADINGS AND TENSOR PRODUCTS Associated to a bigraded module M** is the graded module M* = Tot(M) which is graded by the total degree, Tot(M)*= 0 M3'k. j-\-k=i A typical example for a bigraded module is a tensor product of graded modules. Put (MS)NfJ =Ml®Nj. Then the corresponding graded module is (M®Ny= 0 (Mj®Nk). j-\-k=i Another example is obtained from nitrations. 1.3. FlLTRATIONS AND ASSOCIATED GRADINGS A filtration of a module M is a collection of submodules C F 2 M C F ' _ 1 M C ¥Z~2M C F 2 _ 3 M C The filtration is bounded if F°M = M, and convergent if p | ^ o F ' ^ = °- I f M = M* is graded, then one requires that the submodules in the filtration are
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