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

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

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.

Contents

1 Dimension inequality

1 Nonnegativity
2 Vanishing
3 Positivity

Dimension inequality

$dim(R/P) + dim(R/Q) \le dim(R).$

Serre verified this for all regular local ring. 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

dim(R / P) + dim(R / Q) < dim(R),

then

χ(R / P,R / Q) = 0.

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

Positivity

dim(R / P) + dim(R / Q) = dim(R),

then

χ(R / P,R / Q) > 0.

This remains open.

Categories: Commutative algebra | Module theory | Algebraic geometry | Conjectures

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia article. Browse Wikipedia for more information.