Mathematical Sciences: Homological Questions in Commutative Algebra

该奖项支持对某些不变量的继续研究，称为局部陈氏字符，为noether环上的模定义。虽然这些不变量的许多性质最近被发现，但仍有许多悬而未决的问题。研究的主线是在局部环上模的自由解析中由矩阵定义的不变量，这些不变量与交换代数中的经典结果之间的关系导致了对它们的新解释，尽管它们在许多情况下仍然是推测性的，但它们的计算方法更容易。另一项研究关注结构本身的细节，并给出了交叉多重性的替代定义，除其他外，这为Serre的积极性猜想提供了一种新的方法。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。

This award supports the continued study of certain invariants, known as local Chern characters, defined for modules over Noetherian rings. While numerous properties of these invariants have recently been discovered, there are still many open questions. The main line of study concerns invariants defined by matrices in a free resolution of a module over a local ring, where relations between these invariants and classical results in commutative algebra have led to new interpretations of them, which, although they are still conjectural in many cases, give much easier methods for their computation. Another line of research concerns the details of the construction itself, and gives alternative definitions for intersection multiplicities, leading, among other things, to a new approach to Serre's positivity conjecture. 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.