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