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.
Serre singled out four important properties, which became the multiplicity conjectures, and are challenging to prove in the general case.
(The statements of these conjectures can be generalized so that R/P and R/Q are replaced by arbitrary finitely generated modules: see Serre's Local Algebra for more details.)