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Approximability and Rouquier dimension for noncommutative algebras over schemes

Timothy De Deyn, Pat Lank, Kabeer Manali Rahul

Abstract

This work is concerned with approximability (à la Neeman) and Rouquier dimension for triangulated categories associated to noncommutative algebras over schemes. Amongst other things, we establish that the category of perfect complexes of a Noetherian quasi-coherent algebra over a separated Noetherian scheme is strongly generated if, and only if, there exists an affine open cover where the algebra has finite global dimension. As a consequence, we solve an open problem posed by Neeman. Further, as a first application, we study the existence of generators for Azumaya algebras.

Approximability and Rouquier dimension for noncommutative algebras over schemes

Abstract

This work is concerned with approximability (à la Neeman) and Rouquier dimension for triangulated categories associated to noncommutative algebras over schemes. Amongst other things, we establish that the category of perfect complexes of a Noetherian quasi-coherent algebra over a separated Noetherian scheme is strongly generated if, and only if, there exists an affine open cover where the algebra has finite global dimension. As a consequence, we solve an open problem posed by Neeman. Further, as a first application, we study the existence of generators for Azumaya algebras.
Paper Structure (12 sections, 20 theorems, 24 equations)

This paper contains 12 sections, 20 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Regularity
  3. Azumaya algebras
  4. Notation
  5. Generation & approximation
  6. Generation
  7. Approximation
  8. Algebras over schemes
  9. Key ingredients
  10. Applications
  11. Regularity
  12. Azumaya algebras

Key Result

Theorem A

(see Theorem thm:coherent_alg_perf_strong_gen) Let $X$ be a quasi-compact separated scheme and let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_X$-algebra. The following are equivalent: Any of these equivalent conditions imply that $\operatorname{perf}(\mathcal{A})$ has finite Rouquier dimension.

Theorems & Definitions (67)

  • Theorem A
  • Corollary B
  • Corollary C
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 57 more