Derived categories of complete intersections and Hochschild cohomology

本建议涉及交换代数中衍生范畴和相关三角化范畴的结构，部分原因是有限维代数的表示理论和有限群的上同调的发展。研究的重点是派生类别的定性方面（例如，支持品种）和定量方面（在Rouquier的意义上的维度）。Hochschild上同调代数在这些研究中扮演了关键角色，因为它作用于派生范畴，并捕获了关于它的重要同调信息，特别是在完全交环上。在研究拓扑空间（或其他几何对象）时，考虑如何用更简单的对象（如圆盘或球体）构建拓扑空间通常是有用的。然后，可以将数值不变量附加到构建过程中（例如，需要多少步骤？），以获得对象复杂性的度量。令人惊讶的是，类似的考虑在更离散的代数结构世界中也被证明是非常有效的。提案中提出的问题试图从这个角度研究称为交换环的代数小工具，它自然地出现在数学和物理的各个分支中。

This proposal concerns the structure of the derived categories and associated triangulated categories arising in commutative algebra, and is partly motivated by developments in the representation theory of finite dimensional algebras, and in the cohomology of finite groups. The research focuses both on qualitative aspects of the derived category (for example, support varieties) and on quantitative aspects (for dimension, in the sense of Rouquier). The Hochschild cohomology algebra emerges as a critical player in these investigations, for it acts on the derived category and to capture significant homological information about it, especially over complete intersection rings.In studying a topological space (or other geometric objects) it is often useful to consider how it can be built out of simpler objects, like discs or spheres. One can then attach numerical invariants to the building process (for example, how many steps are required?) to obtain a measure of the complexity of the object. Surprisingly, similar considerations turn out to be extremely fruitful also in the more discrete world of algebraic structures. The questions addressed in the proposal seek to investigate algebraic gadgets called commutative rings, which arise naturally in various branches of mathematics and physics, from this perspective.