Serre's multiplicity conjectures
In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection numbers, around 1949, there had been a question of how to provide a more flexible and computable theory.
Let R be a regular local ring and P and Q be prime ideals of R. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra. Serre defined the intersection multiplicity of R/P and R/Q by means of the Tor functors of homological algebra, as
This requires the concept of the length of a module, denoted here by, and the assumption that
If this idea were to work, however, certain classical relationships would presumably have to continue to hold. Serre singled out four important properties. These then became conjectures, challenging in the general case.Dimension inequality
Serre proved this for all regular local rings. He established the following three properties when R is either of equal characteristic or of mixed characteristic and unramified, and conjectured that they hold in general.Nonnegativity
This was proven by Ofer Gabber in 1995.Vanishing
If
then
This was proven in 1985 by Paul C. Roberts, and independently by Henri Gillet and Christophe Soulé.Positivity
If
then
This remains open.
