Cohomology and Structure of Commutative Algebras

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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.