2013/12/22 by Amnon Yekutieli, Yekutieli, Amnon · 2 citations
Mathematics · #16E45 #18G10 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #K-Theory and Homology (math.KT) #Primary: 13D09 #Rings, Modules, and Algebras #Secondary: 13D07 #math.AC #math.AG #math.KT #msc:13D07 #msc:13D09 #msc:16E45 #msc:18G10
paper · pdf · doi:10.48550/arxiv.1312.6411
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
openalex publication_date 2013/12/22 · arxiv created 2016/03/22 · arxiv updated 2016/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 := Spec H0(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:... [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.