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首页> 外文期刊>Journal of algebra and its applications >Local dimension theory of tensor products of algebras over a ring
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Local dimension theory of tensor products of algebras over a ring

机译：圆环上代数张量产品的局部维度理论

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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) 391-401.1 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(circle times R) B not equal {0}, we introduce the inherent notion to (R, A, B) of a B-fibered AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product. A(circle times R) B. As an application, we provide a formula for the Krull dimension of A(circle times R )B when either R or A is zero-dimensional as well as for the Krull dimension of A(circle times z )B when A is a fibered AF-ring over the ring of integers Z with nonzero characteristic and B is an arbitrary ring. This enables us to answer a question of Jorge Martinez on evaluating the Krull dimension of A(circle times z) B when A is a Boolean ring. Actually, we prove that if A and B are rings such that A(circle times z) B is not trivial and A is a Boolean ring, then dim(A(circle times z) B) = dim(B/2B).
机译：本文的主要目标是设定普通环R研究代数的张量产品尺寸理论的一般框架。实际上，我们通过Wadsworth在张量产品的Krull尺寸上翻译了Grothendieck和夏普和随后开发的理论在一个领域K上的代数在任意环R上的代数的一般设置。为此，我们介绍并研究了环R环R的纤维AF-环的概念。这一概念自然延伸到AF-Ring的概念Wadsworth介绍了[克鲁尔尺寸的张力产品的换代代数在田地，J.伦敦数学。 SOC。 19（1979）391-401.1在任意环上的代数。我们证明了Wadsworth定理表达了与环上的代数张量产品的纤维环相关的本地物业。此外，给定由两个R-Algebras A和B组成的环（R，A，B），使得（圆时r）b不等于{0}，我们介绍了（r，a，b的固有概念）B纤维的AF形圈，其允许计算所考虑的张量产品的所有光纤环的Krull尺寸。 a（圆时r）b.作为应用程序，当R或A为零的零维附上的（圈时Z）提供（圆时r）b的krull尺寸的公式（圆时z ）B当A在整数Z环上是具有非零特性的纤维的AF-环，B是任意环。这使我们能够回答Jorge Martinez的问题，当A是布尔环时，评估（圆时Z）B的Krull尺寸。实际上，我们证明，如果A和B是环，使得（圆时Z）B不是微不足道并且A是布尔环，则暗淡（A（圆次Z）B）= DIM（B / 2B）。

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