Duality and Tilting for Commutative DG Rings
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 $A$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 pseudonoetherian). CohenMacaulay 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 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2013
 DOI:
 10.48550/arXiv.1312.6411
 arXiv:
 arXiv:1312.6411
 Bibcode:
 2013arXiv1312.6411Y
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Commutative Algebra;
 Mathematics  KTheory and Homology;
 Primary: 13D09;
 Secondary: 13D07;
 18G10;
 16E45
 EPrint:
 This version: 52 pages. Improved discussion of tilting DG modules (removal of an unnecessary finiteness condition). The author has decided not to submit this version of the paper to a peer reviewed journal