Modules over Local Rings

该奖项支持交换代数的研究。主要研究者将研究Notherian局部环上有限生成模的刚性问题。他将把这项工作应用于确定张量积何时为无扭转的问题。他还将致力于有限Cohen-Macaulay型局部环的分类，并试图确定这些环上不可分解的极大Cohen-Macaulay模的秩数。最后，他将研究代数簇的Picard群的算术性质以及与域的乘法结构有关的几个相关问题。这项研究涉及到交换代数和代数几何中的一些问题。代数几何研究多项式方程族的解。人们既可以研究解集的几何，也可以通过研究解集上的某些函数来代数地处理问题，这些函数构成了所谓的交换环。这种双重视角在交换代数和代数几何之间建立了紧密的联系。

This award supports research in commutative algebra. The principal investigator will study the rigidity question for finitely generated modules over a noetherian local ring. He will apply this work to the problem of determining when the tensor product is torsion-free. He will also work on the classification of local rings of finite Cohen-Macaulay type and will try to determine the ranks of the indecomposable maximal Cohen-Macaulay modules over these rings. Finally, he will study the arithmetical properties of Picard groups of algebraic varieties and several related questions concerning the multiplicative structure of fields. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.