Homology and Cohomology over Commutative Rings

阿夫拉莫夫正在调查的问题所产生的边界代数，代数几何和代数拓扑。 其基本研究对象是反映代数簇几何性质的交换Noether环的同调不变量。 特别注意将集中在几何意义的消失Hochschild上同调，它有限代作为一个代数，并在消失的安德烈-奎伦同源。 用几何方法研究完全交上的经典同调和上同调的渐近消失，这是数学领域的一个项目，称为交换代数和代数几何。 交换代数和代数几何可以被认为是研究许多未知数中许多方程的解，通常情况下，解不是唯一的。 解的集合可以被几何地观察，但它通常最好被编码到定义在这个集合上的函数族中。 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.