The Category of Firm Modules for Nonunital Monomial Algebras

Abstract

Let k be a field and X a set and P be a set of words over X. Consider the free nonunital k-algebra over X generated by the nonempty words over X and let R be the quotient of this algebra modulo the ideal generated by the words in P. R is called a “nonunital monomial algebra”. A right R-module M is said to be “firm” if M⊗ R R → M given by m ⊗ r↦ mr is an isomorphism. In this article we prove that if R is a nonunital monomial algebra, the category of firm modules is Grothendieck.