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Auslander correspondence for truncated connective DG-algebras

Nao Mochizuki

TL;DR

This work develops a higher-dimensional Auslander correspondence in the setting of proper connective DG-algebras by leveraging $d$-extended module categories. It defines representation-finite and Auslander $d$-truncated DG-algebras via extriangulated dimensions of their extended module categories and establishes a bijection between Morita classes of rep-finite $d$-truncated DG-algebras and Auslander $d$-truncated DG-algebras through endomorphism DG-algebras of additive generators. The construction provides a DG-analogue of the classical Auslander correspondence (the case $d=1$) and aligns with Iyama’s higher Auslander framework by translating module-category phenomena into the DG/extended-module context. The reverse direction recovers a rep-finite $d$-truncated DG-algebra from an Auslander one using the endomorphism algebra of a projective-generator, with a corroborating Morita-equivalence mechanism. Collectively, the results unify representation-finite theory, extriangulated global/dominant dimensions, and DG-endomorphism constructions, broadening the applicability of Auslander-type correspondences to truncated DG-analytic settings.

Abstract

We establish a $d$-dimensional Auslander correspondence for $d$-truncated proper connective DG-algebras via $d$-extended module categories. A $d$-truncated proper connective DG-algebra $Γ$ is called Auslander if its $d$-extended module category is a $d$-Auslander extriangulated category. Our main theorem gives a one-to-one correspondence: for any $d$-truncated proper connective DG-algebra $Λ$ whose $d$-extended module category has finitely many indecomposable objects, the DG-endomorphism algebra of an additive generator is Auslander, and conversely every Auslander $d$-truncated proper connective DG-algebra arises in this way.

Auslander correspondence for truncated connective DG-algebras

TL;DR

This work develops a higher-dimensional Auslander correspondence in the setting of proper connective DG-algebras by leveraging -extended module categories. It defines representation-finite and Auslander -truncated DG-algebras via extriangulated dimensions of their extended module categories and establishes a bijection between Morita classes of rep-finite -truncated DG-algebras and Auslander -truncated DG-algebras through endomorphism DG-algebras of additive generators. The construction provides a DG-analogue of the classical Auslander correspondence (the case ) and aligns with Iyama’s higher Auslander framework by translating module-category phenomena into the DG/extended-module context. The reverse direction recovers a rep-finite -truncated DG-algebra from an Auslander one using the endomorphism algebra of a projective-generator, with a corroborating Morita-equivalence mechanism. Collectively, the results unify representation-finite theory, extriangulated global/dominant dimensions, and DG-endomorphism constructions, broadening the applicability of Auslander-type correspondences to truncated DG-analytic settings.

Abstract

We establish a -dimensional Auslander correspondence for -truncated proper connective DG-algebras via -extended module categories. A -truncated proper connective DG-algebra is called Auslander if its -extended module category is a -Auslander extriangulated category. Our main theorem gives a one-to-one correspondence: for any -truncated proper connective DG-algebra whose -extended module category has finitely many indecomposable objects, the DG-endomorphism algebra of an additive generator is Auslander, and conversely every Auslander -truncated proper connective DG-algebra arises in this way.
Paper Structure (9 sections, 49 theorems, 57 equations, 1 figure, 1 table)

This paper contains 9 sections, 49 theorems, 57 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. d-truncated DG-algebras and d-extended module categories
  3. The notion of d-extended module categories
  4. The notion of presentations and copresentations
  5. Auslander correspondence for d-truncated DG-algebras
  6. Homological dimensions of d-truncated DG-algebras
  7. Auslander d-truncated DG-algebras
  8. The correspondence from (i) to (ii)
  9. The correspondence from (ii) to (i)

Key Result

Theorem 1

There is a bijection between the following two classes: The correspondence from (1) to (2) is given by taking the endomorphism algebra $\mathop{\mathrm{\mathrm{End}}}\nolimits{\Lambda}{M}$ of an additive generator $M$ of $\mathop{\mathrm{\mathrm{mod}}}\nolimits \Lambda$ for a finite-dimensional algebra $\Lambda$ of finite representation type.

Figures (1)

  • Figure 1: The Auslander--Reiten quiver of $\mathcal{H}^2_{\Gamma}$ where $\Gamma$ is the Auslander $2$-truncated DG-algebra associated to $\Lambda$ in the previous example. The red numbers present the vector dimension of cohomologies of degree $-1$.

Theorems & Definitions (131)

  • Theorem 1: aus1971
  • Definition 2: $=$Definition \ref{['def:extended-mod']}
  • Definition 3: $=$Definition \ref{['dfn:auslander']}
  • Definition 4: $=$Definition \ref{['dfn:morita']}
  • Theorem 5: $=$Theorem \ref{['thm:main']}
  • Corollary 6: $=$Corollaly \ref{['cor:main']}
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Lemma 1.4
  • ...and 121 more