Cohomology and Structure of Commutative Algebras

交换代数和代数几何可以被认为是研究许多未知数中的许多方程的解，通常，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。 同调代数带来了研究环的方法代数拓扑，发展研究几何结构。 将复杂结构分解为基本构建块的方法是作为代数的另一个分支的一部分发展起来的，称为表示论。Avramov将研究交换代数，同调代数和表示论的十字路口出现的问题。不同观点的结合使得技术和直觉在不同领域之间的转移富有成效。交换Noether环的性质将通过同调构造产生的数值、代数和几何不变量来研究。 这些包括自由决议，Hochschild上同调，稳定上同调，上同调支持品种，导出类别的模块和微分模块。 将特别注意发展有限识别标准的几何重要性质的家庭交换环。 通过同调代数与有限维代数的表示理论的联系将被研究。

Commutative algebra and algebraic geometry may be thought of as studying solutions of many equations 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. Homological algebra brings to the study of rings methods of algebraic topology, developed to study geometric structures. Methods for decomposing complicated structures into primitive building blocks are developed as part of a different branch of algebra, known as representation theory.Avramov will investigate problems arising at a crossroads of commutative algebra, homological algebra, and representation theory. The combination of different points of view allows for a fruitful transfer of techniques and intuition between fields.Properties of commutative noetherian rings will be studied through numerical, algebraic, and geometric invariants generated by homological constructions. These include free resolutions, Hochschild cohomology, stable cohomology, cohomological support varieties, derived categories of modules and of differential modules. Special attention will be paid to developing finitistic recognition criteria for geometrically important properties of families of commutative rings. Connections through homological algebra with representation theory of finite dimensional algebra will be studied.