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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 \(\mathbb{F}_2\).

MSC:
18G10 Resolutions; derived functors (category-theoretic aspects)
55P65 Homotopy functors in algebraic topology
18A25 Functor categories, comma categories
18G40 Spectral sequences, hypercohomology
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