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Zapiski Nauchnykh Seminarov POMI, 2019, Volume 484, Pages 72–85
(Mi znsl6859)
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Mod-$2$ (co)homology of an abelian group
S. O. Ivanova, A. A. Zaikovskiiab a Laboratory of Modern Algebra and Applications, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178 Russia
b St. Petersburg Department of Steklov Mathematical Institute
Abstract:
It is known that for a prime $p\ne 2$ there is the following natural description of the homology algebra of an abelian group $H_*(A,\mathbb F_p)\cong \Lambda(A/p)\otimes \Gamma({}_pA)$ and for finitely generated abelian groups there is the following description of the cohomology algebra of $H^*(A,\mathbb F_p)\cong \Lambda((A/p)^\vee)\otimes \mathsf{Sym}(({}_pA)^\vee).$ We prove that there are no such descriptions for $p=2$ that “depend” only on $A/2$ and ${}_2A$ but we provide natural descriptions of $H_*(A,\mathbb F_2)$ and $H^*(A,\mathbb F_2)$ that “depend” on $A/2,$ ${}_2A$ and a linear map $\widetilde \beta\colon {}_2A\to A/2.$ Moreover, we prove that there is a filtration by subfunctors on $H_n(A,\mathbb F_2)$ whose quotients are $\Lambda^{n-2i}(A/2)\otimes \Gamma^i({}_2A)$ and that for finitely generated abelian groups there is a natural filtration on $H^n(A,\mathbb F_2)$ whose quotients are $ \Lambda^{n-2i}((A/2)^\vee)\otimes \mathsf{Sym}^i(({}_2A)^\vee).$
Key words and phrases:
homological algebra, algebraic topology, abelian group homology, Eilenberg–MacLane space, Hopf algebra, divided power algebra.
Received: 11.09.2019
Citation:
S. O. Ivanov, A. A. Zaikovskii, “Mod-$2$ (co)homology of an abelian group”, Problems in the theory of representations of algebras and groups. Part 35, Zap. Nauchn. Sem. POMI, 484, POMI, St. Petersburg, 2019, 72–85
Linking options:
https://www.mathnet.ru/eng/znsl6859 https://www.mathnet.ru/eng/znsl/v484/p72
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Abstract page: | 72 | Full-text PDF : | 48 | References: | 9 |
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