Serre's multiplicity conjectures

In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain 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, which Serre sought to address. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra.
Let R be a Noetherian, commutative, regular local ring and let P and Q be prime ideals of R. Serre defined the intersection multiplicity of R/''P and R''/Q by means of their Tor functors. Below, denotes the length of the module, and we assume for the remainder of the article that
Serre defined the intersection multiplicity of R/''P and R''/Q by the Euler characteristic-like formula:
In order for this definition to provide a good generalization of the classical intersection multiplicity, one would want that certain classical relationships would continue to hold. Serre singled out four important properties, which became the multiplicity conjectures, and are challenging to prove 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.

This was proven by Ofer Gabber in 1995.

If
then
This was proven in 1985 by Paul C. Roberts, and independently by Henri Gillet and Christophe Soulé.

If
then
This remains open.