Cohomology of polynomial functors on free groups (Q288658)

Let \(k\) be a ring, \(\mathcal C\) a small category and \(d\) a positive integer. Write \(\mathcal F(\mathcal C,k)\) for the category of functors \(F:\mathcal C\to k\)-\textbf{Mod} from \(\mathcal C\) to the category of left \(k\)-modules. The authors investigate the full subcategory \(\mathcal F_d(\mathcal C,k)\) of \(\mathcal F(\mathcal C,k)\) whose objects are the polynomial functors of degree at most \(d\). They focus on the case when \(\mathcal C\) has a zero object and finite coproducts.NEWLINENEWLINELet \(\mathbf{gr}\) denote the category of free groups of finite rank. The main result in the article under review asserts that, given two functors \(F,G\) in \(\mathcal F_d(\mathbf{gr},k)\), the inclusion functor \(\mathcal F_d(\mathbf{gr},k)\to\mathcal F(\mathbf{gr},k)\) induces an isomorphism \(\mathrm{Ext}^*_{\mathcal F_d(\mathbf{gr}},k)(F,G)\to\mathrm{Ext}^\ast_{\mathcal F(\mathbf{gr},k)}(F,G)\).NEWLINENEWLINEThe proof relies on: {\parindent=0.6cm\begin{itemize} \item[--] a vanishing property (of cohomology), and \item[--] properties of the filtration of a ring of a direct product of free groups by the powers of its augmentation ideal. NEWLINENEWLINE\end{itemize}}NEWLINENEWLINEFurthermore, the result can be extended to the category of free monoids of finite ranks instead of the category \textbf{gr}.NEWLINENEWLINEIn the last section, the authors give applications of their results to various functors. E.g., if \(\mathfrak a\) denotes the abelianisation functor \(\mathbf{gr}\to\mathbf{Ab}\), they show that its \(n\)th tensor power \(\mathfrak a^{\otimes n}\) has homological dimension \(d-n\) in \({\mathcal F_d(\mathbf{gr},\mathbb Z)}\) for \(0<n\leq d\).