Homology and Cohomology over Commutative Rings

Avramov正在研究代数、代数几何和代数拓扑边界上出现的问题。基本的研究对象是反映代数簇几何性质的交换Notherian环的同调不变量。我们将特别关注Hochschild上同调消失的几何意义，以及作为代数的有限生成，以及Andre-Quillen同调的消失。我们将用几何的方法来研究完全交上的经典同调和上同调的渐近零化。这是一个被称为交换代数和代数几何的数学领域的项目。交换代数和代数几何可以被认为是研究在许多未知数中的许多方程的解，当解通常不是唯一的时。然后可以从几何角度查看解的集合，但通常最好将其编码为在该集合上定义的函数族。这种函数族的抽象形式称为交换环。

Avramov is investigating problems arising at the boundary of algebra, algebraic geometry, and algebraic topology. The basic objects of study are homological invariants of commutative noetherian rings that reflect geometric propertied of algebraic varieties. Special attention will be focused on the geometric meaning of the vanishing of Hochschild cohomology, on it finite generation as an algebra, and on the vanishing of Andre-Quillen homology. Asymptotic vanishing of classical homology and cohomology over complete intersections will be studied by geometric methods.This is a project in the areas of mathematics known as Commutative Algebra and Algebraic Geometry. Commutative algebra and algebraic geometry may be thought of as studying solutions of many equatons in many unknowns when, typically, the solution is not unique. The set of solutions can then be viewed geometrically, but it is often best encoded into a family of functions defined on this set. The abstract version of such families of functions are called commutative rings.