Finitely presented modules over Leavitt algebras.
In this paper the author studies the structure of the Abelian category \(\text{fp}(L)\) of finitely presented left modules over a Leavitt algebra \(L\) of type \((1,n)\) over a field \(k\) by obtaining it as the quotient category of the category of finitely presented modules over the free algebra \(k\langle Y\rangle:=k\langle y_0,y_1,\dots,y_n\rangle\). A special subcategory \(\text{fp}(L)_{fl}\) of \(\text{fp}(L)\) of all finitely presented left \(L\)-modules of finite length turns out to be the category of torsion modules with respect to the localization \(L\to Q\). The author also obtains a category equivalence between \(\text{fp}(L)_{fl}\) and the quotient of an Abelian category of finitely presented left \(k\langle Y\rangle\)-modules of negative Euler characteristic. Finally, as an application, the author computes \(K_1(Q)\) by getting a formula which is analogous to the one for \(K_1\) of the universal field of fractions of \(k\langle X\rangle\).
categories of finitely presented modules
scientific article; zbMATH DE number 2093436
scientific article