The global dimension of the algebra of the monoid of all partial functions on an n-set as the algebra of the EI-category of epimorphisms between subsets

作者：Stein, Itamar^*

来源：Journal of Pure and Applied Algebra, 2019, 223(8): 3515-3536.

DOI：10.1016/j.jpaa.2018.11.015

We prove that the global dimension of the complex algebra of the monoid of all partial functions on an n-set is n - 1 for all n >= 1. This is also the global dimension of the complex algebra of the category of all epimorphisms between subsets of an n-set. In our proof we use standard homological methods as well as combinatorial techniques associated to the representation theory of the symmetric group. As part of the proof, we obtain a partial description of the Cartan matrix of these algebras.