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Maximal Cohen-Macaulay DG-complexes

Zachary Nason

Abstract

Let $R$ be a commutative noetherian local differential graded (DG) ring. In this paper we propose a definition of a maximal Cohen-Macaulay DG-complex over $R$ that naturally generalizes a maximal Cohen-Macaulay complex over a noetherian local ring, as studied by Iyengar, Ma, Schwede, and Walker. Our proposed definition extends the work of Shaul on Cohen-Macaulay DG-rings and DG-modules, as any maximal Cohen-Macaulay DG-module is a maximal Cohen-Macaulay DG-complex. After proving necessary lemmas in derived commutative algebra, we establish the existence of a maximal Cohen-Macaulay DG-complex for every DG-ring with constant amplitude that admits a dualizing DG-module. We then use the existence of these DG-complexes to establish a derived Improved New Intersection Theorem for all DG-rings with constant amplitude.

Maximal Cohen-Macaulay DG-complexes

Abstract

Let be a commutative noetherian local differential graded (DG) ring. In this paper we propose a definition of a maximal Cohen-Macaulay DG-complex over that naturally generalizes a maximal Cohen-Macaulay complex over a noetherian local ring, as studied by Iyengar, Ma, Schwede, and Walker. Our proposed definition extends the work of Shaul on Cohen-Macaulay DG-rings and DG-modules, as any maximal Cohen-Macaulay DG-module is a maximal Cohen-Macaulay DG-complex. After proving necessary lemmas in derived commutative algebra, we establish the existence of a maximal Cohen-Macaulay DG-complex for every DG-ring with constant amplitude that admits a dualizing DG-module. We then use the existence of these DG-complexes to establish a derived Improved New Intersection Theorem for all DG-rings with constant amplitude.

Paper Structure

This paper contains 14 sections, 17 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Background
  3. DG-Rings
  4. Derived Hom and Derived Tensor
  5. Derived Torsion and Derived Completion
  6. Dualizing DG-Modules
  7. The Improved New Intersection Theorem for DG-Rings
  8. Depth and Projective Dimension
  9. Derived Completion
  10. DG-Modules of Maximal Depth
  11. Maximal Cohen-Macaulay DG-Complexes
  12. Definition
  13. Maximal Cohen-Macaulay DG-Modules and DG-Complexes
  14. The Canonical Element Theorem for DG-Rings

Key Result

Lemma 3.1

Let $R$ be a DG-ring with an ideal $I \subset \textup{H}^0(R)$, and let $M \in \mathsf{D}^+(R)$. Then $\operatorname{depth}(I, M) \geq \inf(M)$, and equality holds if $\Gamma_I(\textup{H}^{\inf(M)}(M)) \neq 0$.

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1: Depth and Sequential Depth
  • Lemma 3.1
  • proof
  • Theorem 3.2: The Derived Auslander-Buchsbaum Formula
  • proof
  • Lemma 3.3: Nakayama's Lemma for Derived Complete DG-Modules
  • proof
  • ...and 32 more