Mathematical Sciences: Commutative Algebras

9402015 Murthy首席研究员将研究仿射代数上射影模的结构和模的有效生成。他将继续他的工作，寻找投影模块分裂的障碍和理想的有效生成，类似于拓扑中矢量束截面的障碍理论。这些障碍的一些成分是周氏菌群。我们还需要探索其他不变量。所使用的工具来自于交点理论、代数k理论和标准交换代数。本研究涉及交换代数和代数几何中的若干问题。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解决问题，这些函数形成了所谓的交换环。这种对偶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.