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 the 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 the prime ideals of R. Serre defined the intersection multiplicity of R/P and R/Q by means of their Tor functors. Below, ℓ R ( M ) {\displaystyle \ell _{R}(M)} denotes the length of the module M {\displaystyle M} , and for the remainder of the article, we assume
ℓ R ( ( R / P ) ⊗ R ( R / Q ) ) < ∞ . {\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:
χ ( R / P , R / Q ) := ∑ i = 0 ∞ ( − 1 ) i ℓ R ( Tor i R ( R / P , R / Q ) ) . {\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 certain classical relationships to 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 2000 for more details.
Dimension inequality
dim ( R / P ) + dim ( R / Q ) ≤ dim ( R ) {\displaystyle \dim(R/P)+\dim(R/Q)\leq \dim(R)}
Serre proved this for all regular local rings using the Cohen structure theorem. He established the following three properties when R is either of equal characteristic or of mixed characteristic and unramified (which in this case means that characteristic of the residue field is not an element of the square of the maximal ideal of the local ring), and conjectured that they hold in general.
Non-negativity
χ ( R / P , R / Q ) ≥ 0 {\displaystyle \chi (R/P,R/Q)\geq 0}
This property was proven by Ofer Gabber in 1995 using A. J. de Jong's methods. Dutta demonstrated this property becomes complex when studied in the case of blow ups, especially for mixed characteristics.
Vanishing If
dim ( R / P ) + dim ( R / Q ) < dim ( R ) {\displaystyle \dim(R/P)+\dim(R/Q)<\dim(R)\ }
then
χ ( R / P , R / Q ) = 0. {\displaystyle \chi (R/P,R/Q)=0.\ }
This property was proven in 1985 by Paul C. Roberts using local Chern characters. It was independently proven by Henri Gillet and Christophe Soulé in 1987, by using K-theory. A new proof for this property using regular schemes was published by SP Dutta in 2008.
Positivity If
dim ( R / P ) + dim ( R / Q ) = dim ( R ) {\displaystyle \dim(R/P)+\dim(R/Q)=\dim(R)\ }
then
χ ( R / P , R / Q ) > 0. {\displaystyle \chi (R/P,R/Q)>0.\ }
The proof for this property in the general case remains open. The property was proved for a special case in 2008 by SP Dutta; the case being when the projective scheme of a graded ring finitely generated over its 0th component is artinian local, when one of F and G has a finite resolution by direct sum of copies of O(t) for various t. Another special case was proven by C. Skalit in 2019, for regular local rings that are essentially smooth over a two-dimensional, regular base.
See also Homological conjectures in commutative algebra
References Berthelot, Pierre (1997). Altérations de variétés algébriques (d'après A. J. de Jong) [Alterations of algebraic varieties (according to A. J. de Jong)] (in French). Séminaire Bourbaki, Vol. 1995/96, Astérisque No. 241. pp. 273–311. MR 1472543. Gabber, Ofer (1995). Non-negativity of serre's intersection multiplicities. Exposé à L’IHES. Dutta, Sankar P. (April 2008). "Intersection multiplicity of Serre on regular schemes". Journal of Algebra. 319 (4): 1530–1534. doi:10.1016/j.jalgebra.2007.10.016. Gillet, Henri; Soulé, Christophe (1987). "Intersection theory using Adams operations". Inventiones Mathematicae. 90 (2). Invent. Math. 90, no. 2: 243–277. Bibcode:1987InMat..90..243G. doi:10.1007/BF01388705. MR 0910201. S2CID 120635826. Roberts, Paul (1985). "The vanishing of intersection multiplicities of perfect complexes". Bulletin of the American Mathematical Society. 13 (2): 127–130. doi:10.1090/S0273-0979-1985-15394-7. MR 0799793. Roberts, Paul (1998). "Recent developments on Serre's multiplicity conjectures: Gabber's proof of the nonnegativity conjecture". L' Enseignement Mathématique. 44 (3–4): 305–324. MR 1659224. Serre, Jean-Pierre (2000). Local algebra. Springer Monographs in Mathematics. Berlin: Springer. pp. 106–110. doi:10.1007/978-3-662-04203-8. ISBN 978-3-642-08590-1. MR 1771925. Skalit, C. (April 2019). "Positivity of intersection multiplicity over a two-dimensional base". Journal of Pure and Applied Algebra. 223 (4): 1801–1816. arXiv:1510.05146. doi:10.1016/j.jpaa.2018.07.008.
References