![]() Given a bigraded exact couple of modules over some ring, we determine the meaning of the -terms of its associated spectral sequence: Let and denote the limit and colimit abutting objects of the exact couple, filtered by the kernel and image objects to the associated cone and cocone diagrams. We extend this construction to ext-groups and construct a similar spectral sequence for source regular extensions (with right module coefficients). ![]() ![]() Algebra 212 (2008), 2555-2569, Xu constructs a LHS-spectral sequence for target regular extensions of small categories. In 2 spectral sequence arguments forthe right hand half plane bicomplexes were used to show that the totalisation with the product gives a Milnor additive. The examples I have in mind come from topology. Exact couples and their spectral sequences. Xu, On the cohomology rings of small categories, J. Is there an example of a useful filtration where one really computes something nontrivial also in the higher sheets? Soon it was taken up by other authors who realized its potential. I'm looking for basic examples that show the usefulness of spectral sequences even in the simplest case of spectral sequence of a filtered complex.Īll I know are certain "extreme cases", where the spectral sequences collapses very early to yield the acyclicity of the given complex or some quasi-isomorphism to another easier complex (balancing tor, for example). Spectral sequences were invented by Leray in the context of study of cohomology of sheaves.
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