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Homological Epimorphisms and Hochschild-Mitchell Cohomology

V. Santiago-Vargas, E. O. Velasco-Páez

TL;DR

The paper develops a framework to relate Hochschild-Mitchell cohomology across quotient constructions in K-categories via homological epimorphisms arising from strongly idempotent ideals. It proves that under suitable conditions (I idempotent and I(C,-) projective), the projection pi: C -> C/I induces a long exact sequence connecting H^i(C) and H^i(C/I). This theory is then applied to triangular matrix categories, yielding a long exact sequence between H^i(Lambda) and H^i(U) that unifies and extends classical results of Cibils and Michelena-Platzeck and provides Happel-type sequences for one-point extensions. The work further connects these cohomological relations to recollements and torsion theories, showing how certain torsion-free and TTF structures induce homological epimorphisms and derived category decompositions with wide-ranging algebraic consequences.

Abstract

In this work, we study the Hochschild-Mitchell Cohomology of triangular matrix categories. Given a triangular matrix category $Λ=\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$, we investigate the relationship of the Hochschild-Mitchell cohomologies $H^{i}(Λ)$ and $H^{i}(\mathcal{U})$ of $Λ$ and $\mathcal{U}$ respectively; and we show that they can be connected by a long exact sequence. This result extend the well-known result of Michelana-Platzeck given in [S. Michelena, M. I. Platzeck. {\it{Hochschild cohomology of triangular matrix algebras}}. J. Algebra 233, (2000) 502-525].

Homological Epimorphisms and Hochschild-Mitchell Cohomology

TL;DR

The paper develops a framework to relate Hochschild-Mitchell cohomology across quotient constructions in K-categories via homological epimorphisms arising from strongly idempotent ideals. It proves that under suitable conditions (I idempotent and I(C,-) projective), the projection pi: C -> C/I induces a long exact sequence connecting H^i(C) and H^i(C/I). This theory is then applied to triangular matrix categories, yielding a long exact sequence between H^i(Lambda) and H^i(U) that unifies and extends classical results of Cibils and Michelena-Platzeck and provides Happel-type sequences for one-point extensions. The work further connects these cohomological relations to recollements and torsion theories, showing how certain torsion-free and TTF structures induce homological epimorphisms and derived category decompositions with wide-ranging algebraic consequences.

Abstract

In this work, we study the Hochschild-Mitchell Cohomology of triangular matrix categories. Given a triangular matrix category , we investigate the relationship of the Hochschild-Mitchell cohomologies and of and respectively; and we show that they can be connected by a long exact sequence. This result extend the well-known result of Michelana-Platzeck given in [S. Michelena, M. I. Platzeck. {\it{Hochschild cohomology of triangular matrix algebras}}. J. Algebra 233, (2000) 502-525].
Paper Structure (8 sections, 25 theorems, 50 equations, 2 figures)

This paper contains 8 sections, 25 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hochschild cohomology
  4. Homological epimorphisms in functor categories
  5. Homological epimorphisms and Hochschild Mitchell-Cohomology
  6. Applications
  7. Happel's exact sequence
  8. Recollements and torsion pairs

Key Result

Proposition 2.1

Mitchelring Let $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ be three $K$-categories. Let $F\in \mathrm{Fun}_{K}(\mathcal{C},\mathcal{A})$ and $G\in \mathrm{Mod}(C^{op}\otimes_{K}\mathcal{D})$ where $\mathcal{A}$ is an abelian category with coproducts. Consider the following conditions: If one of the above conditions hold, then $F\boxtimes_{\mathcal{C}} G$ is a projective object in $\mathrm{Fun

Figures (2)

  • Figure 1: The Auslander-Reiten quiver of $\mathrmmod(A)$.
  • Figure 2: The Auslander-Reiten quiver of $\mathrmmod(A)/\mathcalI_\mathcalF$

Theorems & Definitions (56)

  • Proposition 2.1
  • Definition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Definition 3.1
  • Remark 3.2
  • ...and 46 more