Modules Over Commutative Rings and Cohomology Operations

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

ABSTRACT, AVRAMOV In recent work Avramov has successfully used multiplicative structures to study problems on free resolutions over commutative local rings. With the help of graded Lie algebras he exhibited many modules with minimal free and injective resolutions of maximal possible asymptotic size. Using homotopy theory he proved a long standing conjecture of Quillen on the rigidity of the cotangent complex and applied it to problems on locally complete intersection homomorphisms. He produced algebraic varieties that describe the action on cohomology of operations defined by Gulliksen and Eisenbud, and with Buchweitz used them to find unexpected growth and vanishing properties of Ext and Tor. Avramov proposes to develop systematic products and other higher order (co)homological structures for use in ring theory and module theory, and to apply it to specific problems, that usually involve only homological invariants. 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 one can use instead encode all the pertinent information about the equations in abstract algebraic objects called commutative rings. The two points of view interact in beautiful and productive ways.