Homological Conjectures in Commutative Algebra: A Conference in Honor of Paul C. Roberts

这次会议将与保罗·罗伯茨的六十岁生日同时举行，将讨论与他的工作相关的交换代数和代数几何领域，包括交换诺etherian环的局部同调行为，交理论和k理论。特别值得注意的是，罗伯茨率先将Baum、Fulton和MacPherson开发的局部陈类理论应用于交换代数中长期存在的开放性问题的解决。其中包括M. Auslander的零因子猜想，Peskine和Szpiro的新交点猜想，Serre关于交点多重性消失的猜想。许多其他问题，如直接和猜想，仍有待解决。我们强烈鼓励研究生和博士研究生的参与，并强调开放性问题。将要考虑的各种问题可以考虑如下。当一个人在许多未知数中有许多方程时，他可以从代数的角度来考虑解，通过形成一个称为环的抽象系统，其中方程被强制保持，或者从几何的角度来考虑，作为一个可能包含在高维空间中的集合。后一种构造推广了绘制方程的概念。当一个人有关于两个解集的信息时，他想要了解它们的交集：两组方程的公共解。特别是，它的维度，它代表了一个人在选择一个解决方案时的自由度，是非常有趣的。所考虑的许多问题可以这样考虑：给定关于两个解集的代数信息，例如，相应环的特殊性质，关于交集可以得到什么定量和定性的结论？会议的一个主要焦点将是在这个领域非常微妙和深刻的定理和猜想。

This conference, which will occur at the same time as the sixtieth birthday of Paul Roberts, will deal with areas of commutative algebra and algebraic geometry related to his work, including the local homological behavior of commutative Noetherian rings, intersection theory, and K-theory. Of particular note is the application, pioneered by Roberts, of the theory of local Chern classes developed by Baum, Fulton, and MacPherson to the solution of long standing open questions in commutative algebra. These include M. Auslander's zerodivisor conjecture, the new intersection conjecture of Peskine and Szpiro, and Serre's conjecture on the vanishing of intersection multiplicities. Many other problems, such as the direct summand conjecture, remain to be solved.The participation of graduate students and recent Ph.D.s will be strongly encouraged, and open questions will be emphasized. The kinds of questions that will be considered may be thought of as follows. When one has many equations in many unknowns, one can think of the solutions algebraically, by forming an abstract system called a ring in which the equations are forced to hold, or geometrically, as a set that is contained in a possibly high dimensional space. The latter construction generalizes the notion of graphing an equation. When one has information about two solution sets, one wants to understand their intersection: the common solutions of both sets of equations. In particular, its dimension, which represents the number of degrees of freedom one has in choosing a solution, is of great interest. Many of the questions considered can be thought of as follows: given algebraic information about two solution sets, e.g., special properties of the corresponding rings, what quantitative and qualitative conclusions can one reach about the intersection? A major focus of the conference will be on very subtle and deep theorems and conjectures in this area.