On the structure of double complexes

By J. Stelzig. Journal of the London Mathematical Society, 2018

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We study consequences and applications of the folklore statement that every double complex over a field decomposes into so‐called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of ‘universal’ quasi‐isomorphism and the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi‐isomorphism (and some variants).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d399ce8f914a271b2ec54c7dfec5897ec607a1dc