非诺特几何
Nonnoetherian geometry
摘要：
介绍了有限克鲁尔维数的非诺特交换代数的几何理论。特别是，我们建立了新的概念规范化和高度：描绘（一个特殊的诺特覆盖环）和几何余维。由此产生的几何是具有正维点的代数簇，因此本质上是非局部的。这些概念也产生了新的等价刻画noetherianity主要是几何。然后，我们考虑一个应用程序的最大维数的单模在每个顶点是一维的代数。证明了$A$的顶角环都是同构的当且仅当$A$是Noether模，当且仅当$A$的中心$Z$是Noether模，当且仅当$A$是一个非线性生成的$Z$ -模.进一步证明了$Z$可由一个由其环生成的交换代数来刻画.最后，我们给出一个例子，其中投射维数和几何余维数，而不是高度，是一致的。
英文摘要：
We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of $A$ are all isomorphic if and only if $A$ is noetherian, if and only if the center $Z$ of $A$ is noetherian, if and only if $A$ is a finitely generated $Z$-module. Furthermore, we show that $Z$ is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
