Serre's multiplicity conjectures, named after Jean-Pierre Serre, are problems in commutative algebra motivated by the needs of algebraic geometry. In 1958, Serre realized that the classical algebraic-geometric ideas of multiplicity could be generalized using homological algebra, defining intersection multiplicity through Tor functors and singling out four important properties that became the multiplicity conjectures. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor a Noetherian commutative regular local ring R with prime ideals P and Q, and intersection multiplicity chi(R/P,R/Q) defined through Tor functors, Serre singled out four properties that became the multiplicity conjectures: the dimension inequality dim(R/P) + dim(R/Q) <= dim(R); non-negativity, chi(R/P,R/Q) >= 0; vanishing, chi(R/P,R/Q) = 0 when dim(R/P) + dim(R/Q) < dim(R); and positivity, chi(R/P,R/Q) > 0 when dim(R/P) + dim(R/Q) = dim(R). 1 Proposed Year Progress Toward ResolutionDimension inequality proved by Serre for all regular local rings; non-negativity proved by Gabber in 1995; vanishing proved by Roberts in 1985 and independently by Gillet and Soule in 1987; positivity remains open in general, with special cases proved by Dutta (2008) and Skalit (2019). 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Serre's Multiplicity Conjectures (Wikipedia)
Sources
1. Serre's Multiplicity Conjectures (Wikipedia)
Wikimedia FoundationLead section
In 1958, Serre realized the classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra.
Positivity section
The proof for this property in the general case remains open.
Lead section, four properties sentence
Serre singled out four important properties, which became the multiplicity conjectures, and are challenging to prove in the general case.
In Branch: Commutative Algebra, Lead sentence
after Jean-Pierre Serre, are certain problems in commutative algebra, motivated by the needs of algebraic geometry.
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