Serre's multiplicity conjectures
Encyclopedia
In mathematics
, Serre's multiplicity conjectures, named after Jean-Pierre Serre
, are certain purely algebraic 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.
Let R be a (Noetherian, commutative) regular local ring
and P and Q be prime ideal
s of R. In 1958, 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
This requires the concept of the length of a module
, denoted here by lR, and the assumption that
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. (There are more general statements of these conjectures where R/P and R/Q are replaced by finitely generated modules: see Serre's Local Algebra for more details.)
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.
Ofer Gabber verified this, quite recently (see the articles by Berthelot and Roberts in the references).
then
This was proven in 1985 by Paul C. Roberts, and independently by Henri Gillet and Christophe Soulé
.
then
This remains open.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Serre's multiplicity conjectures, named after Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
, are certain purely algebraic problems, in commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, motivated by the needs of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. Since André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
's initial definition of intersection number
Intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple curves, and accounting properly for tangency...
s, 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
Regular local ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...
and P and Q be prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s of R. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
. Serre defined the intersection multiplicity of R/P and R/Q by means of the Tor functors of homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
, as
This requires the concept of the length of a module
Length of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
, denoted here by lR, and the assumption that
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. (There are more general statements of these conjectures where R/P and R/Q are replaced by finitely generated modules: see Serre's Local Algebra for more details.)
Dimension inequality
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
Residue field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field...
is not an element of the square of the maximal ideal of the local ring), and conjectured that they hold in general.
Nonnegativity
Ofer Gabber verified this, quite recently (see the articles by Berthelot and Roberts in the references).
Vanishing
If
then
This was proven in 1985 by Paul C. Roberts, and independently by Henri Gillet and Christophe Soulé
Christophe Soulé
Christophe Soulé is a French mathematician working in algebraic geometry and number theory.In 1979, he was awarded a CNRS Bronze Medal. He received the Prix J. Ponti in 1985 and the Prix Ampère in 1993. Since 2001, he is member of the French Academy of Sciences...
.
Positivity
If
then
This remains open.