Diagrammatic Hochschild cohomology via cohomology of categories, and incidence algebras

2025/12/17 by Caputi, Luigi, Vaccarino, Francesco
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2512.15835

Abstract

In this paper, we study the Hochschild cohomology of diagrams of algebras introduced by Gerstenhaber and Schack and provide computations for filtrations of incidence algebras. Our aims are threefold: firstly, we revisit and explore the connection between the Gerstenhaber-Schack complexes and the Baues-Wirsching cohomology of categories. Secondly, we analyse the behaviour of (diagrammatic) Hochschild cohomology in the context of homological epimorphisms of algebras. Thirdly, we study diagrams of incidence algebras. In particular, as a main application, we compute the diagrammatic Hochschild cohomology of diagrams of incidence algebras associated to finite filtrations of simplicial complexes.

Citations

• Lax functoriality of Hochschild cochain complex
• Categorical non-properness in wrapped Floer theory
• The integral Hodge conjecture for two-dimensional Calabi-Yau categories
• Cyclic Gerstenhaber-Schack cohomology
• The Gerstenhaber-Schack complex for prestacks
• Self-dual and quasi self-dual algebras
• The cohomology of lambda-rings and Psi-rings
• A Hochschild Cohomology Comparison Theorem for prestacks
• Cohomology of diagrams of algebras
• Applications of the change-of-rings spectral sequence to the computation of Hochschild cohomology
• (Co)Homology theories for oriented algebras

Related

• Hochschild cohomology of some finite category algebras as simplicial cohomology
• Hochschild cohomology of abelian categories and ringed spaces
• Hochschild cohomology with support
• Posets, their Incidence Algebras and Relative Operads, and the Cohomology Comparison Theorem
• Cohomology of diagram algebras