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Mod-2 (co)homology of an abelian group
- Publication Year :
- 2018
-
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 $H^*(A,\mathbb F_p)\cong \Lambda((A/p)^\vee)\otimes {\sf Sym}(({}_pA)^\vee).$ 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 $\tilde \beta:{}_2A\to A/2.$ 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 {\sf Sym}^i(({}_2A)^\vee).$
- Subjects :
- Mathematics - K-Theory and Homology
Mathematics - Algebraic Topology
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1810.12728
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s10958-021-05200-0