Duality and Tilting for Commutative DG Rings

TL;DR

This paper develops a comprehensive framework for understanding various classes of DG modules over commutative DG rings, introducing new methods like Cech resolutions and exploring properties relevant to duality and rigidity in derived algebraic geometry.

Contribution

It defines and studies perfect, tilting, dualizing, Cohen-Macaulay, and rigid DG modules, introduces Cech resolutions for DG modules, and establishes functorial properties crucial for duality theories.

Findings

Introduced Cech resolutions for DG modules based on open covers.

Established functorial properties of Cohen-Macaulay DG modules.

Proposed conjectures on existence and uniqueness of rigid DG modules.

Abstract

We consider commutative DG rings (better known as nonpositive strongly commutative associative unital DG algebras). For such a DG ring $A$ we define the notions of perfect, tilting, dualizing, Cohen-Macaulay and rigid DG $A$-modules. Geometrically perfect DG modules are defined by a local condition on $\operatorname{Spec} \bar{A}$, where $\bar{A} := \operatorname{Spec} \, \operatorname{H}^0(A)$. Algebraically perfect DG modules are those that can be obtained from $A$ by finitely many shifts, direct summands and cones. Tilting DG modules are those that have inverses w.r.t. the derived tensor product; their isomorphism classes form the derived Picard group $\operatorname{DPic}(A)$. Dualizing DG modules are a generalization of Grothendieck's original definition (and here $A$ has to be cohomologically pseudo-noetherian). Cohen-Macaulay DG modules are the duals (w.r.t. a given dualizing DG module) of finite $\bar{A}$-modules. Rigid DG $A$-modules, relative to a commutative base ring $K$, are defined using the squaring operation, and this is a generalization of Van den Bergh's original definition. The techniques we use are the standard ones of derived categories, with a few improvements. We introduce a new method for studying DG $A$-modules: Cech resolutions of DG $A$-modules corresponding to open coverings of $\operatorname{Spec} \bar{A}$. Here are some of the new results obtained in this paper:... [truncated] The functorial properties of Cohen-Macaulay DG modules that we establish here are needed for our work on rigid dualizing complexes over commutative rings, schemes and Deligne-Mumford stacks. We pose several conjectures regarding existence and uniqueness of rigid DG modules over commutative DG rings.