Stable equivalence for some categories with radical square zero

Abstract

For certain abelian categories with radical square zero, containing artin rings with radical square zero as a special case, we give a way of constructing hereditary abelian categories stably equivalent to them, i.e. such that their categories modulo projectives are equivalent categories. Introduction. Let C be an additive category where idempotents split. Denote as in [2] by Mod C the category of contravariant additive functors from C to abelian groups (called C-modules), and by mod C the full subcategory of finitely presented functors. We shall further assume that each object in C is a finite direct sum of indecomiposable objects in C, and that the endomorphism ring Rc of an indecomposable object C is a division ring modulo its radical, so that a decomposition of an object in C into a finite direct sum of indecomposable objects is unique. Assume also that D = mod C is abelian and that the simple C-modules are finitely presented. All the above conditions on C and D = mod C will be assumed throughout the paper. Important cases where the above assumptions hold are D = mod A, the category of finitely generated (left) modules over a left artin ring A, and the case where D is a dualizing R-variety for a commutative artin ring R (see [4] for definition). Let D/P denote the category D modulo projectives (see [4]). We say that D = mod C and D' = mod C' are (projectively) stably equivalent if D/P and D'/P are equivalent categories. For dualizing R-varieties this coincides with our previous definition of stable equivalence [4]. In [4] we discussed Loewy length for dualizing R-varieties D = mod C, denoted by LL(D), coinciding with the ordinary Loewy length in the case of rings. We showed in [5] that if D = mod C is a dualizing R-variety with LL(D) < 2, there is a hereditary dualizing R-variety stably equivalent to D. In this paper we shall obtain this result in a completely different way. Our method here is interesting because we at the same time get a direct way of constructing a hereditary dualizing R-variety stably equivalent to a dualizing Received by the editors August 16, 1974. AMS (MOS) subject classifications (1970). Primary 16A46, 16A62; Secondary 18A25, 18E10. Copyright i 1975. American Mathematical Society