Intersection Theorem for DG-modules

TL;DR

This paper explores the Intersection Theorem within the context of DG-modules over commutative noetherian local DG-rings, extending classical results and solving conjectures in the derived category setting.

Contribution

It establishes the DG-version of key classical theorems and conjectures, including the amplitude inequality, New Intersection Theorem, Krull's principle ideal theorem, Minamoto's conjecture, and conjectures related to Cohen-Macaulay and Gorenstein rings.

Findings

Proved the DG-setting of the amplitude inequality

Solved Minamoto's conjecture completely

Established DG-versions of the Bass and Vasconcelos conjectures

Abstract

Let A be a commutative noetherian local DG-ring with bounded cohomology. The Intersection Theorem for DG-modules is examined and some of its applications are provided. The first is to prove the DG-setting of the amplitude inequality, New Intersection Theorem and Krull's principle ideal theorem. The second is to solve completely the Minamoto's conjecture in [Israel J. Math. 242 (2021) 1-36]. The third is to show the DG-version of the Bass conjecture about Cohen-Macaulay rings and the Vasconcelos conjecture about Gorenstein rings.