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On silting complexes associated to n-silting modules

Michal Hrbek, Jiangsheng Hu, Rongmin Zhu

Abstract

We show that any (n+1)-term silting complex whose intermediate cohomology vanishes gives rise to an n-silting module, as recently introduced by Mao. Specializing to commutative noetherian rings, we show that this assignment induces a bijection on the respective equivalence classes. Furthermore, we prove in the same setting that the n-silting modules always correspond to a tilting complex, that is, the associated t-structure is of derived type. We use this to exhibit new examples of tilting complexes in the setting of Commutative Algebra and also to show that the finite type property for n-silting modules, as formulated by Mao, can in general fail.

On silting complexes associated to n-silting modules

Abstract

We show that any (n+1)-term silting complex whose intermediate cohomology vanishes gives rise to an n-silting module, as recently introduced by Mao. Specializing to commutative noetherian rings, we show that this assignment induces a bijection on the respective equivalence classes. Furthermore, we prove in the same setting that the n-silting modules always correspond to a tilting complex, that is, the associated t-structure is of derived type. We use this to exhibit new examples of tilting complexes in the setting of Commutative Algebra and also to show that the finite type property for n-silting modules, as formulated by Mao, can in general fail.
Paper Structure (12 sections, 16 theorems, 53 equations)

This paper contains 12 sections, 16 theorems, 53 equations.

Table of Contents

  1. Preliminaries
  2. Basic notation
  3. $n$-tilting modules and $n$-cotilting modules
  4. Silting and cosilting complexes
  5. Silting complexes as tilting modules
  6. $n$-silting modules and $n$-cosilting modules
  7. $n$-silting modules and silting complexes
  8. The case of commutative noetherian rings
  9. Commutative Algebra setup
  10. Examples over local rings with vanishing intermediate cohomology
  11. More general construction
  12. Failure of finite type for $n$-silting modules

Key Result

Lemma 1.1

bhm25 A complex $X \in \mathbf{K}^{b}(\mathrm{Proj}(A))$ is silting if and only if the following two conditions hold: A complex $X \in \mathbf{K}^{b}(\mathrm{Inj}(A))$ is cosilting if and only if the following two conditions hold:

Theorems & Definitions (34)

  • Lemma 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 24 more