A note on (co)homologies of algebras from unpunctured surfaces

TL;DR

This paper extends the study of Jacobian algebras from unpunctured surfaces by computing their cyclic (co)homology and Hochschild homology, providing geometric interpretations related to surface triangulations.

Contribution

It introduces the calculation of cyclic (co)homology and Hochschild homology for these algebras, linking algebraic invariants to surface triangulation features.

Findings

Computed cyclic (co)homology of the algebras

Determined Hochschild homology dimensions

Provided geometric interpretations of the homology dimensions

Abstract

In a previous paper, the author compute the dimension of Hochschild cohomology groups of Jacobian algebras from (unpunctured) triangulated surfaces, and gave a geometric interpretation of those numbers in terms of the number of internal triangles, the number of vertices and the existence of certain kind of boundaries. The aim of this note is computing the cyclic (co)homology and the Hochschild homology of the same family of algebras and giving an interpretation of those dimensions through elements of the triangulated surface.