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Minimal ${A}_{\infty}$-algebras of endomorphisms: The case of $d\mathbb{Z}$-cluster tilting objects

Gustavo Jasso, Fernando Muro

TL;DR

The paper develops a derived, $A_ rightarrow ext{∞}$-theoretic framework to classify endomorphism algebras of $doldsymbol{Z}$-cluster tilting objects, linking them to twisted $(d+2)$-periodic algebras via minimal $A_ rightarrow ext{∞}$-structures and universal Massey products.A central tool is the enhanced $A_ rightarrow ext{∞}$-obstruction theory, which identifies obstructions to extending partial $A_ rightarrow ext{∞}$-structures with Hochschild cohomology classes (notably the Gerstenhaber square of universal Massey products) and uses an extended Bousfield–Kan spectral sequence to organize these obstructions.The work proves a derived Auslander–Iyama correspondence: under suitable hypotheses, the derived endomorphism dg algebra of a basic $doldsymbol{Z}$-cluster tilting object is determined up to quasi-isomorphism by the pair consisting of a basic Frobenius algebra $oldsymbol{oldsymbol{ extLambda}}^{0}$ and its invertible bimodule $oldsymbol{oldsymbol{ extLambda}}^{-d}$, with a converse construction via a restricted universal Massey product.

Abstract

The Derived Auslander--Iyama Corresponence, a recent result of the authors, provides a classification up to quasi-isomorphism of the derived endomorphism algebras of basic $d\mathbb{Z}$-cluster tilting objects in $\operatorname{Hom}$-finite algebraic triangulated categories in terms of a small amount of algebraic data. In this note we highlight the role of minimal $A_\infty$-algebra structures in the proof of this result, as well as the crucial role of the enhanced $A_\infty$-obstruction theory developed by the second-named author.

Minimal ${A}_{\infty}$-algebras of endomorphisms: The case of $d\mathbb{Z}$-cluster tilting objects

TL;DR

The paper develops a derived, $A_ rightarrow ext{∞}$-theoretic framework to classify endomorphism algebras of $doldsymbol{Z}$-cluster tilting objects, linking them to twisted $(d+2)$-periodic algebras via minimal $A_ rightarrow ext{∞}$-structures and universal Massey products.A central tool is the enhanced $A_ rightarrow ext{∞}$-obstruction theory, which identifies obstructions to extending partial $A_ rightarrow ext{∞}$-structures with Hochschild cohomology classes (notably the Gerstenhaber square of universal Massey products) and uses an extended Bousfield–Kan spectral sequence to organize these obstructions.The work proves a derived Auslander–Iyama correspondence: under suitable hypotheses, the derived endomorphism dg algebra of a basic $doldsymbol{Z}$-cluster tilting object is determined up to quasi-isomorphism by the pair consisting of a basic Frobenius algebra $oldsymbol{oldsymbol{ extLambda}}^{0}$ and its invertible bimodule $oldsymbol{oldsymbol{ extLambda}}^{-d}$, with a converse construction via a restricted universal Massey product.

Abstract

The Derived Auslander--Iyama Corresponence, a recent result of the authors, provides a classification up to quasi-isomorphism of the derived endomorphism algebras of basic -cluster tilting objects in -finite algebraic triangulated categories in terms of a small amount of algebraic data. In this note we highlight the role of minimal -algebra structures in the proof of this result, as well as the crucial role of the enhanced -obstruction theory developed by the second-named author.

Paper Structure

This paper contains 6 sections, 3 theorems, 69 equations, 1 figure.

Table of Contents

  1. Hochschild cohomology of graded algebras
  2. Minimal $A_\infty$-algebras and universal Massey products
  3. Minimal models of dg algebras
  4. The $d$-sparse case
  5. The Derived Auslander--Iyama Correspondence
  6. Enhanced $A_\infty$-obstruction theory

Key Result

Theorem 1

Let $d\geq1$ and assume that the ground field $\mathbf{k}$ is perfect.Or, more generally, that the quotient of the finite-dimensional algebra $\mathcal{T}(G,G)$ by its Jacobson radical is separable. Let $\mathcal{T}$ be a $\operatorname{Hom}$-finite idempotent-complete algebraic triangulated categor in particular, $\mathcal{T}$ admits a unique dg enhancement in the sense of BK90. Moreover, again u

Figures (1)

  • Figure 1: Range of definition of the extended Bousfield--Kan spectral sequence with $r=5$. The region hashed NE-SW consists of vector spaces, the region hashed SE-NW consists of abelian groups, with the exception of the large dots that are plain pointed sets. We depict a differential that 'jumps' from the range of definition of the classical Bousfield--Kan spectral sequence (hashed SE-NW) to its extended part.

Theorems & Definitions (14)

  • Theorem 1: Mur22 for $d=1$ and JKM22 for $d\geq 1$
  • remark 1.1
  • remark 1.2
  • remark 2.1
  • remark 2.2
  • example 2.3
  • remark 2.4
  • remark 2.5
  • theorem 3.1: JKM22
  • remark 3.2
  • ...and 4 more