Koszul, Ringel and Serre duality for strict polynomial functors. (Q2845016)

This is a very informative `report on recent work of Chałupnik and Touzé', from the perspective of the author. After introducing strict polynomial functors as representations of a category of divided powers, the paper makes explicit a monoidal structure on the category of degree \(d\) strict polynomial functors over the commutative ring \(k\). By transport of structure this provides a tensor product on the category of polynomial representations of degree \(d\) of \(\text{GL}_n\), when \(n\geq d\). The author remarks that it `seems this tensor product has not been noticed before, despite the fact that polynomial representations of the general linear groups have been studied for more than a hundred years.'NEWLINENEWLINE Other subjects that are covered are Koszul duality for strict polynomial functors, Ringel duality for Schur algebras, a Serre duality functor in terms of Koszul duality. Here the key is that a left derived tensor product \(\wedge^d\otimes^{\mathbf L}\wedge^d\) of the exterior power functor \(\wedge^d\) with itself equals the symmetric power functor \(S^d\).NEWLINENEWLINE The treatment is very efficient and avoids making unnecessary hypotheses. Thus the strict polynomial functors do not always take projective modules as values. And the derived categories are sometimes unbounded derived categories.