Simple-minded systems in stable module categories. (Q2907040)

\(R\) denotes a commutative Artin ring. An \(R\)-algebra \(A\) is an Artin algebra if \(A\) is finitely generated as an \(R\)-module. \(\mathrm{mod\,}A\) denotes the category of all finitely generated left \(A\)-modules. The stable category \(\underline{\text{mod}}\,A\) of \(A\) has the same objects as \(\mathrm{mod\,}A\), and morphisms between two objects \(X,Y\) are the following quotients of \(R\)-modules: \(\underline{\Hom}_A(X,Y)=\Hom_A(X,Y)/\mathcal P(X,Y)\), where \(\mathcal P(X,Y)\) is the \(R\)-submodule of \(\Hom_A(X,Y)\) that consists of all the homomorphisms from \(X\) to \(Y\) which factor through a projective \(A\)-module.NEWLINENEWLINE The aim of this paper is to suggest and to explore a new definition of generating sets of stable categories. \textit{Z. Pogorzały} [CMS Conf. Proc. 14, 393-406 (1993; Zbl 0804.16007); Commun. Algebra 22, No. 4, 1127-1160 (1994; Zbl 0805.16008)] has already treated this problem by introduction of what he called maximal systems of orthogonal stable bricks, the main features of his systems being mutual orthogonality and maximality.NEWLINENEWLINE The present authors introduce simple minded systems (sms) that satisfy orthogonality and a generating condition that replaces maximality. Given an Artin algebra \(A\), a class of objects \(\mathcal S\) in \(\mathrm{mod}_{\mathcal P}A\) is called an sms, if the following two conditions are satisfied: (1) (the orthogonality condition) For \(S,T\in\mathcal S\), \(\underline{\Hom}_A(S,T)=0\), if \(S\) and \(T\) are different and is a division ring, if \(S=T\); (2) (generating condition) For every indecomposable, non-projective \(A\)-module \(X\), there exists a natural number \(n\) (dependent on \(X\)) such that \(X\in\langle S\rangle_n\); here \(\langle S\rangle=\langle S\rangle_1\) denotes the full subcategory of \(\mathrm{mod\,}A\) which consists of modules that are direct summands of finite direct sums of objects in \(\mathcal S\) and \(\langle\mathcal S\rangle_n=\langle\langle\mathcal S\rangle_{n-1}*\langle\mathcal S\rangle\rangle\) are obtained by appropriate iterations through indecomposable \(A\)-modules. The authors prove that the sms are always finite as well as that invariance under stable equivalence still holds (but is not straightforward). The generating assumptions that the authors use also provide a direct relation to the stable Grothendieck group. sms are always invariant under stable equivalences, thus the class of all sms is an invariant of a stable module category. The authors also describe sms of several classes of algebras along with connections to the Auslander-Reiten conjecture.