On the Serre functor in the category of strict polynomial functors (Q6655953)

This paper studies the Serre functor in the derived category of the category \(\mathcal{P}_{d}\) of strict polynomial functors of degree \(d\) over a field of positive characteristic. The existence of Serre functor in this context follows from general theory, and it was really constructed in [\textit{H. Krause}, Compos. Math. 149, No. 6, 996--1018 (2013; Zbl 1293.20046)]. This paper investigates its interplay with various structures existing on \(\mathcal{P}_{d}\) (Frobenius twist, affine subcategories, blocks).\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] shows that the Poincaré type formulas for Ext\ groups in \(\mathcal{P}_{d}\) obtained in [\textit{M. Chałupnik}, Proc. Am. Math. Soc. 144, No. 3, 963--970 (2016; Zbl 1375.18012); \textit{R. Ksouri}, Appl. Categ. Struct. 24, No. 3, 283--314 (2016; Zbl 1344.18013)] are a consequence of the interplay between \(\boldsymbol{S}\) and the Frobenius twist (Corollary 2.4). In general, a Serre functor produces Poincaré duality in Ext groups when it acts on some object as the shift functor, which exactly happens for some Frobenius twisted strict polynomial functor (Proposition 2.3).\N\N\item[\S 3] studies a Serre functor in the derived category \(\mathcal{DP}_{d}^{afi}\) of the category \(\mathcal{P}_{d}^{afi}\) of \(i\)-affine strict polynomial functors of degree \(d\).\N\N\item[\S 4] introduces the semiblock decomposition of \(\mathcal{DP}_{d}^{afi}\), which is a collection of reflective subcategories of \(\mathcal{DP}_{d}^{afi}\) indexed by the set of blocks in \(\mathcal{P}_{d}\).\N\N\item[\S 5] focuses on the basic semiblocks, i.e., the subcategories of \(\mathcal{DP}_{d}^{afi}\) which correspond to the blocks containing a single simple object. It is established (Theorem 5.1) that\N\NTheorem. For any basic Young diagram \(\lambda\), the category \(\mathcal{DP}_{d}^{afi,b}\) is Calabi-Yau of dimension \(2d(p^{i}-1)\), the category \(\mathcal{DP}_{d}^{afi}\) is weak Calabi-Yau of dimension \(2d(p^{i}-1)\).\N\end{itemize}