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Pré-Publication, Document De Travail Année : 2015

EXTENSIONS BETWEEN FUNCTORS FROM GROUPS

Résumé

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.
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Dates et versions

hal-01224436 , version 1 (05-11-2015)
hal-01224436 , version 2 (11-04-2016)
hal-01224436 , version 3 (20-09-2016)

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Citer

Christine Vespa. EXTENSIONS BETWEEN FUNCTORS FROM GROUPS. 2015. ⟨hal-01224436v1⟩
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