Cohomology and Structure of Commutative Algebras

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候。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。同调代数为环的研究带来了代数拓扑学的方法，发展起来是为了研究几何结构。将复杂结构分解为原始构件的方法是作为代数的另一个分支--表示理论的一部分开发的。Avramov将研究在交换代数、同调代数和表示理论的十字路口出现的问题。不同观点的结合允许技术和直觉在领域之间的卓有成效的转移。交换Noether环的性质将通过由同调结构产生的数值、代数和几何不变量来研究。其中包括自由分解、Hochschild上同调、稳定上同调、上同调支撑簇、导模范畴和微分模范畴。将特别注意为交换环族的几何重要性质制定有限识别标准。本课程将研究同调代数与有限维代数表示理论之间的联系。

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.