Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the category of cofinite complexes and modules

Reza Sazeedeh

TL;DR

This work extends Hartshorne's characterization of $\mathfrak a$-cofinite complexes to broader classes of Noetherian rings by developing a Koszul-cohomology criterion and a derived-category framework with subcategories $D(A,\mathfrak a)_{\rm cof}$, $D_{\rm cof}(A,\mathfrak a)$, and $D^0_{\rm cof}(A,\mathfrak a)$. It shows when these subcategories are thick and how abelianity of module cofiniteness categories governs this thickness, connecting cofiniteness to Ext-finiteness and Koszul cohomology, and establishing Hartshorne-type equivalences in a more general setting (including dualizing complexes and $\mathfrak a$-adic completeness). The paper also provides multiple affirmative results to Hartshorne's fourth question under various hypotheses, and proves low-dimensional cofiniteness phenomena for rings with $\dim A\le 3$ or $\dim A/({\bf x})\le 3$, generalizing Takahashi–Wakasugi and related work. Overall, it offers a unified, categorical approach to cofiniteness via Koszul methods and derived categories, with broad applicability to rings of different dimensions and structural assumptions.

Abstract

Let $A$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $A$. In this paper, we extend Hartshorne's characterization of cofinite complexes to more general classes of rings. We also determine conditions under which Hartshorne's fourth question [H1] admits an affirmative answer. Finally, we investigate the cofiniteness of complexes of $\frak a$-cofinite modules for rings of lower dimensions.

On the category of cofinite complexes and modules

TL;DR

This work extends Hartshorne's characterization of -cofinite complexes to broader classes of Noetherian rings by developing a Koszul-cohomology criterion and a derived-category framework with subcategories , , and . It shows when these subcategories are thick and how abelianity of module cofiniteness categories governs this thickness, connecting cofiniteness to Ext-finiteness and Koszul cohomology, and establishing Hartshorne-type equivalences in a more general setting (including dualizing complexes and -adic completeness). The paper also provides multiple affirmative results to Hartshorne's fourth question under various hypotheses, and proves low-dimensional cofiniteness phenomena for rings with or , generalizing Takahashi–Wakasugi and related work. Overall, it offers a unified, categorical approach to cofiniteness via Koszul methods and derived categories, with broad applicability to rings of different dimensions and structural assumptions.

Abstract

Let be a commutative noetherian ring, let be an ideal of . In this paper, we extend Hartshorne's characterization of cofinite complexes to more general classes of rings. We also determine conditions under which Hartshorne's fourth question [H1] admits an affirmative answer. Finally, we investigate the cofiniteness of complexes of -cofinite modules for rings of lower dimensions.
Paper Structure (5 sections, 35 theorems, 51 equations)

This paper contains 5 sections, 35 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. A criterion for cofiniteness of modules
  3. Characterizations of Cofinite complexes
  4. On a question of Hartshorne
  5. $\frak a$-Cofinite complexes on rings with lower dimensions

Key Result

Theorem 1

Let $A$ be a ring with a dualizing complex $\mathcal{D}$ and let $X^{\bullet}\in D^{+}(A)$. Consider the following conditions: ${\rm (1)}$ The complex $X^{\bullet}$ is $\frak a$-cofinite. ${\rm (2)}$ The complex $X^{\bullet}$ satisfies the following conditions: ${\rm (i)}$$\mathop{\mathrm{Supp}}\nol

Theorems & Definitions (72)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 62 more