Editing 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 number]]s, 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 ring|Noetherian]], [[commutative ring|commutative]], [[regular local ring]] and let ''P'' and ''Q'' be [[prime ideal]]s of ''R''. Serre defined the [[intersection multiplicity]] of ''R''/''P'' and ''R''/''Q'' by means of their [[Tor functor]]s. Below, <math>\ell_R(M)</math> denotes the [[length of a module|length]] of the module <math>M</math>, and we assume for the remainder of the article that 

:<math>
\ell _R((R/P)\otimes_R (R/Q)) < \infty.
</math>

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

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

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.)

==Dimension inequality==
{{main|Serre's inequality on height}}

: <math>\dim(R/P) + \dim(R/Q) \le \dim(R)</math>

Serre proved this for all regular local rings. 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.

==Nonnegativity==

: <math>\chi (R/P,R/Q) \ge 0</math>

This was proven by [[Ofer Gabber]] in 1995.

==Vanishing==

If

: <math>\dim (R/P) + \dim (R/Q) < \dim (R)\ </math>

then

: <math>\chi (R/P,R/Q) = 0.\ </math>

This was proven in 1985 by [[Paul C. Roberts]], and independently by [[Henri Gillet]] and [[Christophe Soulé]].

==Positivity==

If

: <math>\dim (R/P) + \dim (R/Q) = \dim (R)\ </math>

then

: <math>\chi (R/P,R/Q) > 0.\ </math>

This remains open.

==See also==
*[[Homological conjectures in commutative algebra]]

==References==
* {{Citation | last1=Serre | first1=Jean-Pierre | title=Local algebra | series=Springer Monographs in Mathematics | publisher=Springer| location=Berlin |mr=1771925  | year=2000 | pages=106–110 | doi=10.1007/978-3-662-04203-8| isbn=978-3-642-08590-1 }}
* {{Citation | last=Roberts | first=Paul | title=The vanishing of intersection multiplicities of perfect complexes | journal=Bulletin of the American Mathematical Society | publisher= Bull. Amer. Math. Soc. 13, no. 2|mr=0799793  | year=1985 | volume=13 | issue=2 | pages=127–130 | doi=10.1090/S0273-0979-1985-15394-7| doi-access=free }}
* {{Citation | last=Roberts | first=Paul | title=Recent developments on Serre's multiplicity conjectures: Gabber's proof of the nonnegativity conjecture | publisher=  L' Enseign. Math. (2) 44, no. 3-4|mr=1659224   | year=1998 | pages=305–324}}
* {{Citation | last=Berthelot | first=Pierre | title=Altérations de variétés algébriques (d'après A. J. de Jong) | publisher=  Séminaire Bourbaki, Vol. 1995/96, Astérisque No. 241 |mr=1472543   |  year=1997 | pages=273–311}}
* {{Citation | last1=Gillet | first1=H. |  last2=Soulé | first2=C. |title=Intersection theory using Adams operations.  | journal=Inventiones Mathematicae | publisher=   Invent. Math. 90, no. 2 |mr=0910201   | year=1987 | volume=90 | issue=2 | pages= 243–277 | doi=10.1007/BF01388705| bibcode=1987InMat..90..243G | s2cid=120635826 }}
* {{Citation | last1=Gabber | first1=O. | title=Non-negativity of serre's intersection multiplicities | publisher=Exposé à L’IHES | year=1995}}

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[[Category:Commutative algebra]]
[[Category:Intersection theory]]
[[Category:Conjectures]]
[[Category:Unsolved problems in mathematics]]