Studies on homological and intersection theory questions over local rings

这个建议解决了交换代数和代数几何中的几个经典和新兴问题。交换代数中的许多问题首先由Auslander、Peskine-Szpiro和Serre研究，它们将模的同调性质与其他不变量（如深度或维数）联系起来。最常被利用的性质之一允许模具有有限的投影维数。最近，人们发现，在具有足够好的奇点的环上，一个更弱的条件就足够了：在约化的有理Grothendieck群中为零。广义地说，PI将继续研究奇点的类型，从而使经典问题可以扩展。这种方法同时产生了其他几个项目。其中之一将研究一个新的，渐近版本的塞尔的交叉多重性在当地完成的交叉。另一个项目将调查当地版本的哈茨霍恩的猜想对周群的光滑射影超曲面及其后果分裂的向量丛。正在研究的问题自然在于交叉的代数几何，交换代数和代数K理论。代数几何研究多项式方程解的形状和性质。交换代数更关注一般对象，如函数环和它们上的模。这两个领域已经被研究了很长时间，并且仍在快速发展，新发现的连接到许多数学和科学领域，如物理学，统计学和编码理论。这个项目将利用代数K理论的一个相对较新的观点，它研究了环的微妙不变量，以理解几何和代数对象的良好特性。

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.