A User’s Guide to Spectral Sequences by John McCleary

By John McCleary

Spectral sequences are one of the such a lot dependent and strong equipment of computation in arithmetic. This ebook describes probably the most very important examples of spectral sequences and a few in their such a lot awesome functions. the 1st half treats the algebraic foundations for this kind of homological algebra, ranging from casual calculations. the guts of the textual content is an exposition of the classical examples from homotopy thought, with chapters at the Leray-Serre spectral series, the Eilenberg-Moore spectral series, the Adams spectral series, and, during this re-creation, the Bockstein spectral series. The final a part of the publication treats functions all through arithmetic, together with the idea of knots and hyperlinks, algebraic geometry, differential geometry and algebra. this is often a good reference for college kids and researchers in geometry, topology, and algebra.

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If A* = H* (M; Q) for M, a finitedimensional manifold, x(A*) is the classical Euler-Poincaré characteristic, x (M), of the manifold. When A* is a graded algebra, aspects of the algebra structure are reflected in the Poincaré series. For example, consider A' = Q[x] where x is of degree n. Since A* is a free commutative algebra on one generator, all of the powers of x persist in A* and generate it as a graded vector space. It is easy to see then that P(A* ,t) = 1 1 — tn =1+tn ± ± ± More generally, if B* = Q[xi, X2,...

Since x(E(**) = x(H*), it suffices to show that x(E * ) = x(E4*1 ) for all i. Recall that dimk = dimk (ker diP q) + dimk (im diPq). We compute (-1)n dirnk xe(E:'*) = —E E n=0 CBI EP'q ) [(-1)n dimk (ker dr) ± (-1)n dimk (im cri ' q )1 p-Fq=n =E[ E n=0 (-1) n dimk(ker dr) + (-1) n diMk (iM dri 's)1 r±s=n p+q=n = dinik (ker di° '° ) + E[ E (_i)dim k (kerdr) n=1 p±q=n E (_l)dirn k (iM (-1) n dirnk ( e Eix n=0 = Xt(E1) + >7 di" ) r±s=E r±s=n-1 p+q=n ( - r+s=f ) ± (_ r-ks=f 1) 6 diMk (inl di" ). 3. Exploiting further structure 17 Let f go to infinity; this incorporates the extra term into the limit and so X(E:' * ) = X(Ei+i) As an application of the example, if one wants to compute the Euler characteristic for a manifold M, the job is simplified if there is a spectral sequence converging to H* (M; k).

This provides a foundation for the generalizations that appear in later chapters. 1 Definitions and basic properties We begin by generalizing our First Definition and identifying the basic components of a spectral sequence. 1. A differential bigraded module over a ring R, is a collection of R-modules, {EP}, where p and q are integers, together with an R-linear mapping, d: E* , * E* , *, the differential, of bidegree (s,1— s) or (—s, s-1), for some integer s, and satisfying d o d = O. 1. Definitions and basic properties 29 point (p, q) in the Cartesian plane.

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