Local duality for representations of finite group schemes

Abstract

A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander-Reiten triangles for the p-local and p-torsion subcategories of the stable category, for each homogeneous prime ideal p in the cohomology ring of the group scheme.

Local Duality for the Singularity Category of a Finite Dimensional Gorenstein Algebra

“…The first one is by adjunction as I(r) = Hom k (R, k); see [10,Lemma A.2]. This is the stated Serre duality on γ r (D), for the Krull dimension of R/r is 0 and R r = R.…”

“…compatible with the localisation functor; see [10,Remark 7.1]. The result below can be interpreted as the statement that the category γ p (Gproj A), and hence also γ p (Ginj A), has a Serre functor.…”

“…When A satisfies the (Fg) condition, it follows from Corollary 5.4 that the triangulated category γ p (Gproj A), and hence also anything equivalent to it, has ARtriangles. This is explained in [10,Section 7], for which we refer the reader also for other consequences of Serre duality.…”

“…The duality statements above are modeled on, and extensions of, analogous results for representation of modules over finite group schemes established in [11]. In that context, the stable category of (finite dimensional) Gorenstein projectives is the stable category of (finite dimensional) representations.…”

Stratification for module categories of finite group schemes

“…The methods developed in this work have led to other new results concerning the structure of the stable module category of a finite group scheme, including a classification of its Hom closed colocalising subcategories [10], and to a type of local Serre duality theorem for StMod G; see [12].…”