Novikov, all utilized on a daily basis by most workers in stable homotopy theory, are three. We extend this construction to ext-groups and construct a similar spectral sequence for source regular extensions (with right module coefficients). The spectral sequences of Atiyah-Hirzebruch, Adams, and Adams. Algebra 212 (2008), 2555-2569, Xu constructs a LHS-spectral sequence for target regular extensions of small categories. Weibel, An introduction to homological algebra, Cambridge Univ. Xu, On the cohomology rings of small categories, J. We provide the necessary background to understand the degeneration of theHodge spectral sequence for a smooth and proper schemeXover a eldk. A spectral sequence is a tool of homological algebra that has many applications in algebra, algebraic geometry, and algebraic topology. Nelson, Parameterized Topological Data Analysis, Ph.D. This is an expository write up and modernization of a classical result inHodge theory, developed over the duration of the 2020 REU program at the Universityof Chicago. In a sense there is really only one spectral sequence, just as there is only one concept of a long exact sequence (although each object may originate in a variety of settings), but there are many dierent named uses. Morozov, Parallel computation of persistent homology using the blowup complex Annual ACM Symposium on Parallelism in Algorithms and Architectures, 323-331, (2015).ī. examples of the kinds of things spectral sequences are good for. Suppose that T is a cochain complex which admits an decreasing ltration T F 0T F 1T F pT F p+1T, where each F pT is a sub-cochain complex of T. Lewis, Spectral Sequences and Applied Topology, Ph.D. The spectral sequence of a filtered complex At this point it seems appropriate to introduce spectral sequences in considerable generality. Hungerford, Algebra, Graduate Texts in Math. Hartshorne, Algebraic Geometry, Graduate Texts in Math. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Math. Harer, Computational Topology: An Introduction, American Mathematical Society, Providence (2010).ĭ.
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