Extensions between functors from groups

Christine Vespa

Published 2015-11-10Version 1

We compute Ext-groups between tensor powers composed by the abelianization functor. More precisely, we compute the groups $Ext^*\_{\mathcal{F}(\textbf{gr})}(T^n \circ \mathfrak{a}, T^m \circ \mathfrak{a})$ where $T^n$ is the n-th tensor power functor and $\mathfrak{a}$ 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 $Ext^{m-n}\_{\mathcal{F}(\textbf{gr})}(T^n \circ \mathfrak{a}, T^m \circ \mathfrak{a})=\mathbb{Z}[\Omega(m,n)]$ where $\Omega(m,n)$ 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 use this computation in order to obtain other computations of Ext-groups between functors from groups.