Serre's multiplicity conjectures

In mathematics, Serre's multiplicity conjectures are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial rigorous 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 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

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

This requires the concept of the length of a module, denoted here by "l_R", 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.

Dimension inequality

: $$dim(R/P) + dim(R/Q) le dim(R)

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.

Nonnegativity

: $$chi (R/P,R/Q) ge 0

Ofer Gabber verified this, quite recently.

Vanishing

If

: $$dim (R/P) + dim (R/Q) < dim (R)

then

: $$chi (R/P,R/Q) = 0.

This was proven around 1986 by Paul C. Roberts, and independently by Gillet and Soulé.

Positivity

If

: $$dim (R/P) + dim (R/Q) = dim (R)

then

: $$chi (R/P,R/Q) > 0.

This remains open.

References