In
mathematics, Serre's multiplicity conjectures, named after
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
, are certain purely algebraic problems, in
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, motivated by the needs of
algebraic geometry. Since
André Weil's initial definition of
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 (more than 2) curves, and accounting properly for ta ...
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 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 ide ...
and ''P'' and ''Q'' be
prime ideals 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
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 (more than 2) curves, and accounting properly for ta ...
of ''R''/''P'' and ''R''/''Q'' by means of the
Tor functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to con ...
s of
homological algebra, as
:
This requires the concept of the
length of a module In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It ...
, denoted here by
, 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 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. Frequently, ''R'' is a ...
is not an element of the square of the maximal ideal of the local ring), and conjectured that they hold in general.
Nonnegativity
:
This was proven by
Ofer Gabber
Ofer Gabber (עופר גאבר; born May 16, 1958) is a mathematician working in algebraic geometry.
Life
In 1978 Gabber received a Ph.D. from Harvard University for the thesis ''Some theorems on Azumaya algebras,'' written under the supervis ...
in 1995.
Vanishing
If
:
then
:
This was proven in 1985 by
Paul C. Roberts, and independently by
Henri Gillet and
Christophe Soulé
Christophe Soulé (born 1951) is a French mathematician working in arithmetic geometry.
Education
Soulé started his studies in 1970 at École Normale Supérieure in Paris.
He completed his Ph.D. at the University of Paris in 1979 under the sup ...
.
Positivity
If
:
then
:
This remains open.
See also
*
Homological conjectures in commutative algebra In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a ...
References
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{{DEFAULTSORT:Serre's Multiplicity Conjectures
Commutative algebra
Intersection theory
Conjectures
Unsolved problems in mathematics