Geodesic
Mod-$2$ (co)homology of an abelian group
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 35, Tome 484 (2019), pp. 72-85 Cet article a éte moissonné depuis la source Math-Net.Ru

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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).$ 
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     author = {S. O. Ivanov and A. A. Zaikovskii},
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S. O. Ivanov; A. A. Zaikovskii. Mod-$2$ (co)homology of an abelian group. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 35, Tome 484 (2019), pp. 72-85. http://geodesic.mathdoc.fr/item/ZNSL_2019_484_a5/

[1] H.-J. Baues, “Quadratic functors and metastable homotopy”, J. Pure Appl. Algebra, 91:1–3 (1994), 49–107 | DOI | MR | Zbl

[2] A. K. Bousfield, “On the $p$-adic completions of nonnilpotent spaces”, Trans. Amer. Math. Soc., 331:1 (1992), 335–359 | MR | Zbl

[3] K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, 87, 1994 | MR

[4] H. Cartan, “Algèbre d'Eilenberg MacLane et homotopie”, Matematika, 3:5 (1959), 3–50

[5] H. Cartan, “Sur les groupes d'Eilenberg-MacLane $H(\pi,n)$”, Proc. Nat. Acad. Sci. USA, 40 (1954), 467–471 ; 704–707 | DOI | MR | Zbl | Zbl

[6] L. Evens, “The cohomology ring of a finite group”, Trans. Amer. Math. Soc., 101 (1961), 224–239 | DOI | MR | Zbl

[7] M. Hartl, C. Vespa, “Quadratic functors on pointed categories”, Adv. Math., 226 (2011), 3927–4010 | DOI | MR | Zbl

[8] A. Hatcher, Algebraic topology, Cambridge University Press, 2002 | MR | Zbl

[9] S. O. Ivanov, “On Bousfield's problem for solvable groups of finite Prüfer rank”, J. Algebra, 501 (2018), 473–502 | DOI | MR | Zbl

[10] R. Mikhailov, Polynomial functors and homotopy theory, Progress Math., 311, 2016 ; arXiv: 1202.0586 | MR

[11] J. Milnor, J. C. Moore, “On the structure of Hopf algebras”, Ann. Math., 81:2 (1965), 211–264 | DOI | MR | Zbl

[12] K. Newman, “A correspondence between bi-ideals and sub-Hopf algebras in cocommutative Hopf algebras”, J. Algebra, 36:1 (1975), 1–15 | DOI | MR | Zbl

[13] N. E. Steenrod, D. B. A. Epstein, Cohomology operations, Ann. Math. Studies, 50, Princeton University Press, 1962 | MR | Zbl

[14] M. Takeuchi, “A correspondence between Hopf ideals and sub-Hopf algebras”, Manuscripta Math., 7 (1972), 251–270 | DOI | MR | Zbl