Hochschild and ordinary cohomology rings of small categories |
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Authors: | Fei Xu |
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Institution: | UMR 6629 CNRS/UN, Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 Rue de la Houssinière, 44322 Nantes, France |
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Abstract: | Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C|=BC. We study the ring homomorphism HH∗(kC)→H∗(|C|,k) and prove it is split surjective, using the factorization category of Quillen D. Quillen, Higher algebraic K-theory I, in: Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin, 1973, pp. 85-147] and certain techniques from functor cohomology theory. This generalizes the well-known theorems for groups and posets. Based on this result, we construct a seven-dimensional category algebra whose Hochschild cohomology ring modulo nilpotents is not finitely generated, disproving a conjecture of Snashall and Solberg N. Snashall, Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. 88 (3) (2004) 705-732]. |
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Keywords: | Hochschild cohomology ring Ordinary cohomology ring Category algebra Category of factorizations Left Kan extension Finite EI-categories Finite generation Nilpotent element |
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