Studies on homological and intersection theory questions over local rings

局部环同调与交集理论问题的研究

• 批准号: 0834050
• 负责人: Hailong Dao
• 金额: $ 14.52万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-15 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0834050&HistoricalAwards=false
• 关键词: Studies; homological; intersection; theory; questions

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.

这一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，最早由Auslander，Peskine-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。最常被利用的性质之一是允许模具有有限的射影维度。最近，有研究表明，在具有足够好奇异性的环上，一个弱得多的条件足够：在有限生成模的约化有理Grothendieck群中为零。广义地说，PI将执行一项计划，以调查古典问题可以推广到的奇点类型。这种方法可以同时生成其他几个项目。其中一人将研究局部完全交集上Serre交多重性的一种新的渐近形式。另一个项目将研究哈特肖恩关于光滑射影超曲面的Chow群的猜想的一个局部版本及其对向量丛分裂的影响。所研究的问题自然是代数几何、交换代数和代数K-理论的交集。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，如函数环和函数环上的模。这两个领域都已经研究了很长时间，而且仍在快速发展，新发现的联系与许多数学和科学领域，如物理、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来研究环的微妙不变量，以了解几何和代数对象的良好性质。