Affine strict polynomial functors and formality

On the Serre Functor in the Category of Strict Polynomial Functors

“…The aim of this section is to introduce a suitable version of Serre functor on the category of affine strict polynomial functors. The notion of affine strict polynomial functor was introduced in [7]. In the present paper we generalize it in two directions: firstly we consider the case of multiple twists (i.e., our category P af i d for i = 1 corresponds to the original category P af d from [7]), and secondly we allow unbounded graded objects.…”

“…We provide a categorical interpretation of this phenomenon by investigating a Serre functor on the derived category of the category DP af i d of i-affine strict polynomial functors of degree d (c.f. [7]), which is a full triangulated subcategory of DP dp i generated the i-times Frobenius twisted functors (c.f. [7]).…”

“…[7]), which is a full triangulated subcategory of DP dp i generated the i-times Frobenius twisted functors (c.f. [7]). Namely, we find certain subcategories, called "basic affine semiblocks" of DP dp i , which correspond to "basic blocks" appearing in Cor.…”

“…We start with reviewing basic properties of the categories P af i d and DP af i d . This is mainly a recollection of some facts from [7] where these concepts were introduced and adapting them to a slightly more general setting of "multiple twists" in which we work in the present article. A new ingredient is the "affine Kuhn duality" which is useful in all kinds of duality issues.…”

“…A new ingredient is the "affine Kuhn duality" which is useful in all kinds of duality issues. In order to have this piece of structure we were forced (in contrast to [7]) to allow unbounded complexes in our derived categories. In the next subsection we use the affine Kuhn duality to obtain the "affine recollement diagram" which is not used in the rest of the paper but may be of independent interest.…”