arXiv 1702.013452017-02-04math.AC

环上代数的张量积的局部维数论

Local dimension theory of tensor products of algebras over a ring

Samir Bouchiba

摘要：

本文的主要目的是建立一个研究任意环$R$上代数的张量积的维数论的一般框架。实际上，我们是在翻译A. Grothendieck和R.夏普和随后开发的A. Wadsworth关于域$k$上代数的张量积的Krull维数推广到任意环$R$上代数的一般情形。为此，我们引入并研究了环$R$上的可扩AF-环的概念。这个概念自然地推广了A. Wadsworth在\cite{W}中推广到任意环上的代数。证明了Wadsworth定理表示环上代数的张量积的纤维环的局部性质。此外，给定一个三元环$(R,A,B)$由两个$R$ -代数$A$和$B$，使得$A\otimes_RB\neq \{0\}$，我们引入固有的概念$(R,A,B)$的一个$B$的AF-环，它允许计算的Krull维数的所有纤维环所考虑的张量积$A\otimes_RB$。作为应用，我们给出了当$A$和$B$是$R$ -代数且$A$是零维时$A\otimes_RB$的Krull维数的计算公式，以及当$A$是整数环$\mathbb{Z}$上的一个可扩AF-环且$B$是任意环时$A\otimes_{\mathbb{Z}}B$的Krull维数的计算公式.这使我们能够回答一个问题的豪尔赫Matinez评估Krull维数的$A\otimes_{\mathbb{Z}}B$时，$A$是一个布尔环。实际上，我们证明了如果$A$和$B$是环，使得$A\otimes_{\mathbb{Z}}B$不是平凡的，并且$A$是布尔环，那么dim $(A\otimes_{\mathbb{Z}}B)=\mbox {dim}\Big (\displaystyle {\frac B{2B}}\Big )$。

英文摘要：

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 A. Grothendieck and R. Sharp and subsequently developed by A. 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 fibred AF-ring over a ring $R$. This concept extends naturally the notion of AF-ring over a field introduced by A. Wadsworth in \cite{W} to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fibre 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 $A\otimes_RB\neq \{0\}$, we introduce the inherent notion to $(R,A,B)$ of a $B$-fibred AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product $A\otimes_RB$. As an application, we provide a formula for the Krull dimension of $A\otimes_RB$ when $A$ and $B$ are $R$-algebras with $A$ is zero-dimensional as well as for the Krull dimension of $A\otimes_{\mathbb{Z}}B$ when $A$ is a fibred AF-ring over the ring of integers $\mathbb{Z}$ with nonzero characteristic and $B$ is an arbitrary ring. This enables us to answer a question of Jorge Matinez on evaluating the Krull dimension of $A\otimes_{\mathbb{Z}}B$ when $A$ is a Boolean ring. Actually, we prove that if $A$ and $B$ are rings such that $A\otimes_{\mathbb{Z}}B$ is not trivial and $A$ is a Boolean ring, then dim$(A\otimes_{\mathbb{Z}}B)=\mbox {dim}\Big (\displaystyle {\frac B{2B}}\Big )$.

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摘要：

英文摘要：

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