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

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

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.