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 (Noetherian, commutative) 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

\chi (R/P,R/Q):=\sum _{i=0}^{\infty}(-1)^i\ell_R (\mathrm{Tor} ^R_i(R/P,R/Q)).

This requires the concept of the length of a module, denoted here by lR, and the assumption that

\ell _R((R/P)\otimes(R/Q)) < \infty.

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. (There are more general statements of these conjectures where R/P and R/Q are replaced by finitely generated modules: see Serre's Local Algebra for more details.)

Read more about Serre's Multiplicity Conjectures:  Dimension Inequality, Nonnegativity, Vanishing, Positivity

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