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

Yadira Valdivieso-Díaz

doi:10.1142/s0219498818500883

Abstract

In a previous paper, the author computed 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 to compute the cyclic (co)homology and the Hochschild homology of the same family of algebras and to give an interpretation of those dimensions through elements of the triangulated surface.

Highlights

A surface with marked points, or a surface, is a pair (S, M ), where S is a compact connected Riemann surface with boundary, and M is a non-empty finite subset of S containing at least one point from each connected component of the boundary of S
We compute three differenthomologies of those gentle algebras coming from unpunctured surfaces, namely: Hochschild homology and cyclic homology and cohomology, and we show that there is a combinatorial interpretation of thosehomologies through the elements of the surface and the triangulation
In the first part of this section, we recall some definitions and notations of path algebras and surfaces with marked points and we include the computation of the Hochschild homology given by Skoldberg in [10] for completeness

Summary

Introduction

A surface with marked points, or a surface, is a pair (S, M ), where S is a compact connected Riemann surface with (possibly empty) boundary, and M is a non-empty finite subset of S containing at least one point from each connected component of the boundary of S. Let AT be the algebra associated to the triangulated surface (S, M, T) and denote by int(T) the set of internal triangles of T.

Results

Conclusion