Hochschild and ordinary cohomology rings of small categories

• Autores: Fei Xu
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 219, Nº 6, 2008, págs. 1872-1893
• Idioma: inglés
• DOI: 10.1016/j.aim.2008.07.014
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• Resumen

  • Let be a small category and k a field. There are two interesting mathematical subjects: the category algebra and the classifying space . We study the ring homomorphism 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].