AF-Domains and Their Generalizations

Abstract

In this article, we are concerned with the study of the dimension theory of tensor products of algebras over a field k. We introduce and investigate the notion of generalized AF-domain (GAF-domain for short) and prove that any k-algebra A such that the polynomial ring in one variable A X is an AF-domain is in fact a GAF-domain, in particular any AF-domain is a GAF-domain. Moreover, we compute the Krull dimension of A ⊗ k B for any k-algebra A such that A X is an AF-domain and any k-algebra B generalizing the main theorem of Wadsworth in [16].

Krull Dimension of Tensor Products of Pullbacks

“…On the one hand, we handle the above-mentioned problem set in [6] and compute dim(A ⊗ k B) when A is a pullback arising from the above construction and B is an arbitrary k-algebra. On the other hand, we prove that the answer to the question (Q) set in [5] is affirmative for such a pullback construction A. Besides, our main result, Theorem 2.8, is, in particular, an important step towards determining a general formula for dim(A ⊗ k B) in the case where A[n] is an AF-domain for some positive integer n and B is an arbitrary k-algebra.…”

“…We begin by recalling from [3], [5], [6] and [20] the following useful results. The following easy result is probably well known.…”

“…Finally, recall that, if A is a k-algebra and n ≥ 0 is an integer, then the polynomial ring Next, we announce the main theorem of this paper. It gives an answer to the abovementioned problem set in [6] as well as to the question (Q) of [5] and represents an important step towards determining a formula for dim(A ⊗ k B) when A[n] is an AF-domain for some positive integer n, and B is an arbitrary k-algebra. 2) If M ⊆ p, then…”

“…Also, we proved that (Q) has a positive answer in the case where A is one-dimensional. Our main result in [5] tackles the case n = 1 of (Q). It computes dim(A ⊗ k B) for a k-algebra A such that A[X] is an AF-domain and for an arbitrary k-algebra B generalizing Wadsworth's main theorem [20,Theorem 3.7] and further asserts that A is a GAF-domain.…”

“…

This paper is concerned with the study of the dimension theory of tensor products of algebras over a field k. We answer an open problem set in [6] and compute dim(A ⊗ k B) when A is a k-algebra arising from a specific pullback construction involving AF-domains and B is an arbitrary k-algebra. On the other hand, we deal with the question (Q) set in [5] and show, in particular, that such a pullback A is in fact a generalized AF-domain.

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