Cohomology over Commutative Rings: Structure and Applications

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。通过同调构造生成的数值、代数和几何不变量来研究交换Notherian环的性质。将开发新的方法来计算这种不变量，并在抵制更传统方法的问题上进行测试。同调代数将被用来在交换环和有限维代数的理论之间建立进一步的联系，目的是传递观点并使在一个领域中发展的技术用于另一个领域。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几个方程的解，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。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。将复杂对象分解为基本构件的方法是作为代数的另一个分支，即表示理论的一部分而发展起来的。

Avramov will investigate problems arising at the crossroads of commutative algebra, homological algebra, and representation theory. Properties of commutative noetherian rings will be studied through numerical, algebraic, and geometric invariants generated by homological constructions. New methods will be developed for computing such invariants, to be tested on problems that have resisted more conventional approaches. Homological algebra will be utilized to establish further connections between the theories of commutative rings and of finite-dimensional algebras, with the goal of transferring viewpoints and adapting techiques developed in one of the fields for use in the other. Commutative algebra and algebraic geometry may be thought of as studying solutions of several 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 objects into primitive building blocks are developed as part of a different branch of algebra, known as representation theory.