Mathematical Sciences: Commutative Algebras

9402015--主要研究人员将研究仿射代数上投射模的结构和模的有效生成。他将继续寻找射影模分裂的障碍和理想的有效生成的工作，类似于拓扑学中向量丛分段的障碍理论。造成这些障碍的一些因素是Chow集团。还必须探索其他不变量。所使用的工具来自交集理论、代数K-理论和标准交换代数。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。

9402015 Murthy The principal investigator will study the structure of projective modules and efficient generation of modules over affine algebras. He will continue his work on finding obstructions to splitting of projective modules and efficient generation of ideals, analogous to obstruction theory for sections of vector bundles in topology. Some of the ingredients for these obstructions are Chow groups. Other invariants have to be explored. The tools used are from intersection theory, algebraic K-theory, and standard commutative algebra. 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.