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Iyama-Solberg correspondence for exact dg categories

Xiaofa Chen

Abstract

We generalize the notions of $d$-cluster tilting pair and $d$-Auslander exact dg category to $d$-precluster tilting triple and $d$-minimal Auslander--Gorenstein exact dg category. We give a bijection between equivalence classes of $d$-precluster tilting triples and equivalence classes of $d$-minimal Auslander--Gorenstein exact dg categories. Our bijection generalizes Iyama--Solberg correspondence for module categories.

Iyama-Solberg correspondence for exact dg categories

Abstract

We generalize the notions of -cluster tilting pair and -Auslander exact dg category to -precluster tilting triple and -minimal Auslander--Gorenstein exact dg category. We give a bijection between equivalence classes of -precluster tilting triples and equivalence classes of -minimal Auslander--Gorenstein exact dg categories. Our bijection generalizes Iyama--Solberg correspondence for module categories.
Paper Structure (8 sections, 19 theorems, 61 equations)

This paper contains 8 sections, 19 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Iyama--Solberg correspondence
  3. Refinement to an $\infty$-equivalence
  4. Connection to exact categories
  5. Exact case
  6. Abelian case
  7. Iyama--Solberg's correspondence
  8. Connection to higher Auslander--Solberg correspondence for exact categories

Key Result

Theorem 1.1

The map $(A, M)\mapsto \Gamma=\mathrm{End}_{A}(M)$ induces a bijective correspondence between

Theorems & Definitions (47)

  • Theorem 1.1: Iyama--Solberg correspondence
  • Definition 1.2: Definition \ref{['def:d-precluster tilting triple']}
  • Theorem 1.3: Theorem \ref{['thm:Iyama--Solberg correspondence']}
  • Definition 2.1: GorskyNakaokaPalu23
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6: Chen23a
  • ...and 37 more