Studies on homological and intersection theory questions over local rings

本文讨论了交换代数和代数几何中的几个经典问题和新问题。交换代数中的许多问题，首先由Auslander， peskin - szpiro和Serre研究，将模的同调性质与其他不变量（如深度或维度）联系起来。最常被利用的属性之一是允许模块具有有限的射影维度。最近，在具有足够好的奇点的环上，出现了一个弱得多的条件：在有限生成模的约简有理Grothendieck群中为零。广义地说，PI将进行一个项目来研究奇点的类型，这些奇点可以扩展到经典问题。这种方法同时产生了其他几个项目。其中一个将研究局部完全相交上Serre相交多重性的一个新的渐近版本。另一个项目将研究Hartshorne关于光滑射影超曲面的Chow群猜想的局部版本及其对向量束分裂的影响。所研究的问题自然是代数几何、交换代数和代数k理论的交叉点。代数几何研究多项式方程解的形状和性质。交换代数更多地关注一般对象，例如函数环和它们上面的模块。这两个领域已经被研究了很长时间，并且仍在迅速发展，与许多数学和科学领域（如物理学、统计学和编码理论）有新的联系。该项目将利用代数k理论的一个相对较新的观点，它检查环的微妙不变量，以理解几何和代数对象的良好性质。

This proposal addresses several classical and emerging problems in commutative algebra and algebraic geometry. Many questions in commutative algebra, first studied by Auslander, Peskine-Szpiro, and Serre, relate homological properties of modules to other invariants such as depth or dimension. One of the most often exploited properties allows the modules to have finite projective dimensions. Recently, it has emerged that over rings with nice enough singularities, a much weaker condition suffices: being zero in the reduced, rational Grothendieck group of finitely generated modules. The PI will, broadly speaking, pursue a program to investigate the type of singularities for which the classical questions can be extended. This approach simultaneously generates several other projects. One of them will study a new, asymptotic version of Serre's intersection multiplicity over local complete intersections. Another project will investigate a local version of Hartshorne's conjecture on Chow groups of smooth projective hypersurfaces and its consequences on splitting of vector bundles. The questions being studied lie naturally at the intersection of algebraic geometry, commutative algebra and algebraic K-theory. Algebraic geometry studies shapes and properties of solutions of polynomial equations. Commutative algebra focuses more on general objects, such as rings of functions and modules over them. Both of these fields have been studied for a very long time and are still developing rapidly, with newly discovered connections to many areas of mathematics and sciences, such as physics, statistics, and coding theory. This project will utilize a relatively new viewpoin from algebraic K-theory, which examines subtle invariants of rings, to understand nice properties of geometric and algebraic objects.