zbMATH — the first resource for mathematics

Taylor towers for \(\Gamma\)-modules. (English) Zbl 0997.18008
A \(\Gamma\)-module is a functor from the category of finite pointed sets to the category of \(R\)-modules for a commutative ring with unit \(R\). In this paper, an explicit construction for the Taylor tower of a \(\Gamma\)-module is given. This tower is analogous to Goodwillie’s Taylor tower for functors of topological spaces. The author shows, by generalizing a construction of Pirashvili, that the homology of \(P_nF[1]\), the \(n\)th term in the Taylor tower of a \(\Gamma\)-module \(F\) evaluated at \([1]=\{0,1\}\), is the same as \(\text{Tor}^{\Gamma}_*(t^n,F)\). Here \(t^n:\Gamma ^{\text{op}}\rightarrow R{\text{-Mod}}\) is the functor that takes a finite pointed set \(X_+\) to the dual of the free \(R\)-module generated by all non-empty subsets of \(X\) of cardinality less than or equal to \(n\). Similarly, the functor \(D_nF=\text{cone}_{*+1}(P_nF\rightarrow P_{n-1}F)\) satisfies \(H_*D_nF[1]\cong \text{Tor}^{\Gamma}_*(\theta ^n,F)\) for a functor \(\theta ^n:\Gamma ^{\text{op}}\rightarrow R{\text{-Mod}}\).
As an example, the author determines \(\text{Tor}_*^{\Gamma}(\theta ^n,F)\) over a field \(K\) of characteristic \(0\) for choices of \(F\) that enable her to recover the higher-order Hochschild homology of \(K[x]/x^2\) with coefficients in \(K\) for even orders. For an arbitrary field \(k\), she constructs a spectral sequence that converges to \(\text{Tor}_*^{\Gamma}(\theta ^n,F)\) and uses this to calculate \(\text{Tor}_*^{\Gamma}(\theta ^2, F)\) for some specific values of \(F\). She also indicates how these results can be used to obtain information about the stable homology of Eilenberg-MacLane spaces over \(\mathbb{F}_2\).

18G10 Resolutions; derived functors (category-theoretic aspects)
55P65 Homotopy functors in algebraic topology
18A25 Functor categories, comma categories
18G40 Spectral sequences, hypercohomology
Full Text: DOI Numdam EuDML
[1] Homotopy theory of \(Γ\)-spaces, spectra, and bisimplicial sets, 658, 80-150, (1978), Springer · Zbl 0405.55021
[2] Homologie nicht-additiver funktoren. anwendungen., Annales de l’Institut Fourier (Grenoble), 11, 201-312, (1961) · Zbl 0098.36005
[3] Homology theory for multiplicative systems, Transactions of the AMS, 71, 294-330, (1951) · Zbl 0043.25403
[4] On the groups \(H(π, n),\) II, Annals of Mathematics, 60, 49-139, (1954) · Zbl 0055.41704
[5] Calculus. I: The first derivative of pseudoisotopy theory · Zbl 0741.57021
[6] Calculus. II: Analytic functors · Zbl 0776.55008
[7] Calculus. III: The Taylor series of a homotopy functor · Zbl 1067.55006
[8] Simplicial homotopy theory, (1999), Birhäuser · Zbl 0949.55001
[9] Complexe cotangent et déformations II, 283, (1972), Springer · Zbl 0238.13017
[10] Taylor towers for functors of additive categories, Journal of Pure and Applied Algebra, 137, 253-284, (1999) · Zbl 0929.18007
[11] Deriving calculus with cotripels, (1999) · Zbl 1028.18004
[12] Cohomology of algebraic theories, Journal of Algebra, 137, 253-296, (1991) · Zbl 0724.18005
[13] Transformation groups and algebraic K-theory, 1408, (1989), Springer · Zbl 0679.57022
[14] Kan extension and stable homology of Eilenberg and mac Lane spaces, Topology, 35, 883-886, (1996) · Zbl 0858.55006
[15] Dold-kan type theorem for \(Γ\)-groups, Mathematische Annalen, 318, 277-298, (2000) · Zbl 0963.18006
[16] Hodge decomposition for higher order Hochschild homology, Annales Scientifiques de l’École Normale Supérieure, 33, 151-179, (2000) · Zbl 0957.18004
[17] Dissertation, 332, (2000), Universität Bonn · Zbl 0967.55015
[18] Localisation homotopique et foncteurs entre espaces vectoriels, (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.