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Silting theory and derived base change

Riku Fushimi

Abstract

For finite-dimensional algebras over a field, Koenig and Yang established a bijection between silting complexes and simple-minded collections in the bounded derived category, with further contributions by many authors in various settings. In this paper, we work over a commutative complete local noetherian ring $(R,\m,k)$ rather than over a field and establish a bijection in this more general setting. As an application of this generalization, we construct a bijection between silting complexes over a noetherian $R$-algebra $Λ$ and silting complexes over $Λ\ten^\LL_RS$ for any morphism of commutative complete local noetherian rings $(R,\m,k)\to(S,\n,k)$. This result generalizes some known results on silting complexes over noetherian algebras.

Silting theory and derived base change

Abstract

For finite-dimensional algebras over a field, Koenig and Yang established a bijection between silting complexes and simple-minded collections in the bounded derived category, with further contributions by many authors in various settings. In this paper, we work over a commutative complete local noetherian ring rather than over a field and establish a bijection in this more general setting. As an application of this generalization, we construct a bijection between silting complexes over a noetherian -algebra and silting complexes over for any morphism of commutative complete local noetherian rings . This result generalizes some known results on silting complexes over noetherian algebras.
Paper Structure (10 sections, 20 theorems, 19 equations)

This paper contains 10 sections, 20 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Triangulated categories
  4. Dg algebras
  5. Silting-Simple-minded collection correspondence
  6. Main results
  7. Silting objects
  8. Simple-minded collections
  9. Bijections
  10. Relation to Previous Results

Key Result

Theorem A

(=thm:ST) Let $A$ be a non-positive locally finitely generated dg $R$-algebra. Then there is a bijection between isomorphism classes of basic silting complexes over $A$ and isomorphism classes of simple-minded collections in $\mathop{\mathrm{\mathcal{D}^b_{fl}}}\nolimits(A):=\{X\in\mathcal{D}(A)\mid

Theorems & Definitions (43)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.9
  • ...and 33 more