Studies on homological and intersection theory questions over local rings

点击翻译按钮获取中文摘要

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.