Grothendieck’s Vanishing and Non-vanishing Theorems in an Abstract Module Category

Abstract

In this article, we prove Grothendieck’s Vanishing and Non-vanishing Theorems of local cohomology objects in the non-commutative algebraic geometry framework of Artin and Zhang. Let k be a field of characteristic zero and ${\mathscr {S}}_{k}$ be a strongly locally noetherian k-linear Grothendieck category. For a commutative noetherian k-algebra R, let ${\mathscr {S}}_R$ denote the category of R-objects in ${\mathscr {S}}_k$ obtained through a non-commutative base change by R of the abelian category ${\mathscr {S}}_{k}$. First, we establish Grothendieck’s Vanishing Theorem for any object ${\mathscr {M}}$ in ${\mathscr {S}}_{R}$. Further, if R is local and ${\mathscr {S}}_{k}$ is Hom-finite, we prove Non-vanishing Theorem for any finitely generated flat object ${\mathscr {M}}$ in ${\mathscr {S}}_R$.