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, ${\displaystyle \ell _{R}(M)}$ denotes the length of the module ${\displaystyle M}$, and we assume for the remainder of the article that

${\displaystyle \ell _{R}((R/P)\otimes _{R}(R/Q))<\infty .}$

Serre defined the intersection multiplicity of R/P and R/Q by the Euler characteristic-like formula:

${\displaystyle \chi (R/P,R/Q):=\sum _{i=0}^{\infty }(-1)^{i}\ell _{R}(\operatorname {Tor} _{i}^{R}(R/P,R/Q)).}$

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. (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.)