2023/05/18 by Friedlander, Eric M.
#20C20 #20G05 #20G10 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2305.10921
We investigate infinite dimensional modules for an affine group scheme \mathbb G of finite type over a field of positive characteristic p. For any subspace X ⊂ \mathcal O(\mathbb G) of the coordinate algebra of \mathbb G, we consider the abelian subcategory Mod(\mathbb G,X) ⊂ Mod(\mathbb G) of ``X-comodules" and the left exact functor (-)X: Mod(\mathbb G) → Mod(\mathbb G,X) which is right adjoint to the inclusion functor. We employ ``ascending converging sequences" \ Xi \ of subspaces of \mathcal O(\mathbb G) to provide functorial filtrations \ MXi \ of each \mathbb G-module M. A \mathbb G-module M is injective if and only if each MXi is an injective Xi-comodule for some (or, equivalently, for all) such \ Xi \. We consider the explicit ascending converging sequence \ \mathcal O(\mathbb G)≤ d,ϕ \ of finite dimensional subcoalgebras of \mathcal O(\mathbb G) depending upon a closed embedding ϕ: \mathbb G \hookrightarrow GLN. Of particular interest to us are mock injective \mathbb G-modules, modules whose support varieties are empty. Restrictions of a \mathbb G-module to each \mathcal O(\mathbb G)≤ d,ϕ provide new invariants for \mathbb G-modules. For cofinite \mathbb G-modules M, we explore the the growth of d ↦ M_\cal O(\mathbb G)≤ d,ϕ.