Modules Over Commutative Rings and Cohomology Operations

在最近的工作中，Avramov成功地利用乘法结构研究了交换局部环上的自由分解问题。借助于分次李代数，他证明了许多模具有最大可能渐近规模的极小自由和内射分解。他利用同伦理论证明了Quillen关于余切复形刚性的一个长期存在的猜想，并将其应用于局部完全交同态问题。他产生了描述Gulliksen和Eisenbud定义的运算的上同调作用的代数簇，并与Buchweitz一起用它们找到了Ext和Tor的意外增长和消失性质。Avramov建议开发用于环论和模论的系统乘积和其他高阶(Co)同调结构，并将其应用于通常只涉及同调不变量的特定问题。交换代数和代数几何可以被认为是研究许多未知的许多方程的解，当解通常不是唯一的时。然后可以从几何角度查看这组解，但可以使用编码在称为交换环的抽象代数对象中编码有关方程的所有相关信息。这两种观点以美丽而富有成效的方式相互作用。

ABSTRACT, AVRAMOVIn 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.