Approximating modules by modules of finite projective dimension
Let \(\Lambda\) be a finite dimensional split left serial algebra over a field \(K\) and let \(P^\infty\) be the full subcategory of \(\Lambda\)-mod consisting of all modules of finite projective dimension. The authors establish the contravariant finiteness of \(P^\infty\) in \(\Lambda\)-mod in the sense of Auslander et al. Each finitely generated \(\Lambda\)-module \(M\) has a right \(P^\infty\)-approximation, i.e. there exists a homomorphism \(f: A\to M\) with \(A \in P^\infty\) such that each homomorphism \(B \to M\) with \(B \in P^\infty\) factors through \(f\). Moreover the authors give an explicit description of the structure of the minimal \(P^\infty\)-approximations \(A_1, \dots, A_n\) of the simple left \(\Lambda\)-modules. By a result of \textit{M. Auslander} and \textit{I. Reiten} [Adv. Math. 86, No. 1, 111-152 (1991; Zbl 0774.16006)] these modules \(A_1, \dots, A_n\) are the building blocks of the modules of finite projective dimension.
finite dimensional split left serial algebras
scientific article; zbMATH DE number 826741