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Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring R. Actually, we translate the theory initiated by Grothendieck and Sharp and subsequently developed by Wadsworth on Krull dimension of tensor products of algebras over a field k into the general setting of algebras over an arbitrary ring R. For this sake, we introduce and study the notion of a fibered AF-ring over a ring R. This concept extends naturally the notion of AF-ring over a field introduced by Wadsworth in [The Krull dimension of tensor products of commutative algebras over a field, J. London Math. Soc. 19 (1979) 391–401.] to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fiber rings of tensor products of algebras over a ring. Also, given a triplet of rings (R,A,B) consisting of two R-algebras A and B such that A⊗RB≠{0}, we introduce the inherent notion to (R,A,B) of a B-fibered AF-ring which allows...

Local dimension theory of tensor products of algebras over a ring