Let R be a (Noetherian, commutative) regular local ring and P and Q be prime ideals of R. In 1961, Jean-Pierre 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 lR, 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.
Serre verified this for all regular local rings. He established the following three properties when R is unramified, and conjectured that they hold in general.
Ofer Gabber verified this, quite recently.
This was proven around 1986 by Paul C. Roberts, and independently by Gillet and Soulé.
This remains open.