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2 results

  • 14 - Perfect and Tilting DG Modules over NC DG Rings

  • Amnon Yekutieli, Ben-Gurion University of the Negev, Israel
  • Book:
    Derived Categories
    Published online:
    15 November 2019
    Print publication:
    19 December 2019, pp 373-423

  • Summary

    We begin with a study of algebraically perfect DG modules over a NC (noncommutative) DG ring A. There is a theorem giving several conditions on a DG A-module L that are equivalent to being algebraically perfect; one of them is that L is a compact object of D(A). When A is a ring, we prove that L is algebraically perfect iff it is isomorphic, in D(A), to a bounded complex of finitely generated projective A-modules. In Section 14.2 we prove a general Derived Morita Theorem.From Section 14.3 to the end of this chapter we assume that the DG rings in question are K-flat over the base ring. Section 14.3 contains some basic constructions of derived functors between categories of DG bimodules. Next, in Section 14.4, we define tilting DG bimodules. Among other results, we prove that a DG B-A-bimodule T is tilting iff T is a compact generator on the B side and it has the NC derived Morita property on the B side. We also prove the Rickard--Keller Theorem.

    In Section 14.5 we introduce the NC derived Picard group of a ring A. The structure of this group is calculated when A is either local or commutative.

Derived Categories
  • Amnon Yekutieli
  • Published online:
    15 November 2019
    Print publication:
    19 December 2019
  • There have been remarkably few systematic expositions of the theory of derived categories since its inception in the work of Grothendieck and Verdier in the 1960s. This book is the first in-depth treatment of this important component of homological algebra. It carefully explains the foundations in detail before moving on to key applications in commutative and noncommutative algebra, many otherwise unavailable outside of research articles. These include commutative and noncommutative dualizing complexes, perfect DG modules, and tilting DG bimodules. Written with graduate students in mind, the emphasis here is on explicit constructions (with many examples and exercises) as opposed to axiomatics, with the goal of demystifying this difficult subject. Beyond serving as a thorough introduction for students, it will serve as an important reference for researchers in algebra, geometry and mathematical physics.