Extension between functors from groups
Résumé
We compute Ext-groups between tensor powers composed by the abelianization functor. More precisely, we compute the groups $Ext^*_{\F(\gr)}(T^n \circ \abe, T^m \circ \abe)$ where $T^n$ is the n-th tensor power functor and $\abe$ is the abelianization functor from the category of free groups to abelian groups. These groups are shown to be non-zero if and only if $*=m-n \geq 0$ and $Ext^{m-n}_{\F(\gr)}(T^n \circ \abe, T^m \circ \abe)=\Z[Surj(m,n)]$ where $Surj(m,n)$ is the set of surjections from the set having $m$ elements to the one having $n$ elements. We make explicit the action of symmetric groups on these groups and the Yoneda and external products. We deduce from these computations those of other Ext-groups between functors from groups such as symmetric and exterior powers. Combining these computations with a recent result of Djament we obtain explicit computations of stable homology of automorphism groups of free groups with coefficients given by particular contravariant functors.
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