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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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