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

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 S k be a strongly locally noetherian k -linear Grothendieck category. For a commutative noetherian k -algebra R , let S R denote the category of R -objects in S k obtained through a non-commutative base change by R of the abelian category S k . First, we establish Grothendieck’s Vanishing Theorem for any object M in S R . Further, if R is local and S k is Hom-finite, we prove Non-vanishing Theorem for any finitely generated flat object M in S R .