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

IF 0.6 4区 数学 Q3 MATHEMATICS
Divya Ahuja, Surjeet Kour
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引用次数: 0

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 \({\mathscr {S}}_R\).

抽象模类中的格罗登第克消失和非消失定理
在本文中,我们在阿尔廷和张的非交换代数几何框架中证明了格罗thendieck 的局部同调对象的消失和非消失定理。设 k 是特征为零的域,且 \({\mathscr {S}}_{k}\) 是强局部诺特 k 线性格罗thendieck 范畴。对于交换的无醚 k 代数 R,让 \({\mathscr {S}}_R\) 表示通过 R 对无性范畴 \({\mathscr {S}}_{k}\) 进行非交换基变化得到的 \({\mathscr {S}}_{k}\) 中的 R 对象范畴。首先,我们为 \({\mathscr {S}}_{R}\) 中的任何对象 \({\mathscr {M}}\) 建立格罗登第克消失定理(Grothendieck's Vanishing Theorem)。此外,如果 R 是局部的,并且 \({\mathscr {S}}_{k}\) 是同无限的,我们会证明 \({\mathscr {S}}_R}\) 中任何有限生成的平面对象 \({\mathscr {M}}\) 的非消失定理。
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来源期刊
CiteScore
1.30
自引率
16.70%
发文量
29
审稿时长
>12 weeks
期刊介绍: Applied Categorical Structures focuses on applications of results, techniques and ideas from category theory to mathematics, physics and computer science. These include the study of topological and algebraic categories, representation theory, algebraic geometry, homological and homotopical algebra, derived and triangulated categories, categorification of (geometric) invariants, categorical investigations in mathematical physics, higher category theory and applications, categorical investigations in functional analysis, in continuous order theory and in theoretical computer science. In addition, the journal also follows the development of emerging fields in which the application of categorical methods proves to be relevant. Applied Categorical Structures publishes both carefully refereed research papers and survey papers. It promotes communication and increases the dissemination of new results and ideas among mathematicians and computer scientists who use categorical methods in their research.
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