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Triangular Matrix Categories I: Dualizing Varieties and Generalized One-point Extensions

机译：Triangular Matrix Categories I: Dualizing Varieties and Generalized One-point Extensions

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摘要

Following Mitchell's philosophy, in this paper we define the analogous of the triangular matrix algebra to the context of rings with several objects. Given two preadditive categories U and T and M is an element of Mod(U circle times T-op) we construct the triangular matrix category. Lambda:= T 0 M U. First, we prove that Mod(Lambda) is equivalent to a comma category (Mod(T), GMod(U)) which is induced by a functor G : Mod(U) -> Mod(T). One of our main results is that if U and T are dualizing K-varieties and M is an element of Mod(U circle times T-op) satisfies certain conditions then Lambda : = T 0 M U is a dualizing variety (see Theorem 6.10). In particular, mod(Lambda) has Auslander-Reiten sequences. Finally, we apply the theory developed in this paper to quivers and give a generalization of the so called one-point extension algebra.

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