Schemes over symmetric monoidal categories and torsion theories

Abstract

Let (C,⊗,1) be an abelian symmetric monoidal category satisfying certain conditions and let X be a scheme over (C,⊗,1) in the sense of Toën and Vaquié. In this paper, we construct torsion theories on the categories OX-Mod and QCoh(X) respectively of OX-modules and quasi-coherent sheaves on X, when X is Noetherian and integral over (C,⊗,1). Thereafter, we study these torsion theories with respect to the quasi-coherator QX:OX-Mod⟶QCoh(X) that is right adjoint to the inclusion iX:QCoh(X)⟶OX-Mod. Finally, we obtain an alternative description of the quasi-coherator QX(F) as a subsheaf of F, when F∈OX-Mod satisfies certain conditions. Along the way, we present further results on the notions of “Noetherian” and “integral” for schemes over (C,⊗,1) that we believe to be of independent interest.