Title: Duality and Tilting for Commutative DG Rings
Author: Amnon Yekutieli
Publication status: arXiv:1312.6411

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 Spec A, where A is the commutative ring 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 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 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 Spec A.

Here are some of the new results obtained in this paper:
- A DG A-module is geometrically perfect iff it is algebraically perfect.
- The canonical group homomorphism DPic(A) → DPic(A) is bijective.
- The group DPic(A) acts simply transitively on the set of isomorphism classes of dualizing DG A-modules.
- Cohen-Macaulay DG modules are insensitive to cohomologically surjective DG ring homomorphisms.
- Rigid dualizing DG A-modules are unique up to unique rigid isomorphisms.

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.

updated 24 March 2016