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Strongly $n$-AIR-tilting modules

Weiqing Cao, Jiaqun Wei

Abstract

We introduce the notion of (strongly) $n$-AIR-tilting modules, which is a high dimension version of support $τ$-tilting modules. The relations between them and $n$-silting modules and $n$-quasi-tilting modules, as well as generalized two-term silting complexes, are investigated. Our results particularly suggest a way to negate the rank question for silting complexes.

Strongly $n$-AIR-tilting modules

Abstract

We introduce the notion of (strongly) -AIR-tilting modules, which is a high dimension version of support -tilting modules. The relations between them and -silting modules and -quasi-tilting modules, as well as generalized two-term silting complexes, are investigated. Our results particularly suggest a way to negate the rank question for silting complexes.
Paper Structure (6 sections, 17 theorems, 12 equations)

This paper contains 6 sections, 17 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of ${{\mathcal{D}}}_{\sigma_{_T}}$, ${{\mathcal{D}}}_{\sigma_{_T},k}$, ${{\mathcal{D}}}_{\sigma_{_A}}$
  4. $n$-AIR-tilting modules
  5. Generalized two-term silting complexes
  6. Examples and questions

Key Result

Lemma 3.1

Let $T\in{{\mathrm{mod}}}A$ and $\sigma_{_{T}}$ be an $(n+1)$-projective presentation of $T$. The class ${{\mathcal{D}}}_{\sigma_{_{T}}}\subseteq {{\mathrm{KerExt}}}^{i}_{A}(T,-)$ for $1\leq i \leq n$. And the class ${{\mathcal{D}}}_{\sigma_{_{T}},k}\subseteq{{\mathrm{KerExt}}}_A^{i}(T,-)$ for all $

Theorems & Definitions (23)

  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Lemma 3.4
  • Definition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Definition 4.4
  • Proposition 4.5
  • Definition 4.6
  • ...and 13 more