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,
CohenMacaulay and rigid DG Amodules. 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 pseudonoetherian). CohenMacaulay DG modules are the
duals (w.r.t. a given dualizing DG module) of finite Amodules. Rigid
DG Amodules, 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 Amodules: Cech resolutions of DG Amodules corresponding to open coverings of Spec A.
Here
are some of the new results obtained in this paper:
 A DG
Amodule 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 Amodules.
 CohenMacaulay DG modules
are insensitive to cohomologically surjective DG ring
homomorphisms.
 Rigid dualizing DG Amodules are unique up to
unique rigid isomorphisms.
The functorial properties of CohenMacaulay DG modules that we establish here are needed for our work on rigid dualizing complexes over commutative rings, schemes and DeligneMumford stacks.
We pose several conjectures regarding existence and uniqueness of rigid DG modules over commutative DG rings.
updated 24 March 2016