Quillen–Suslin theorem
Encyclopedia
The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem
in commutative algebra
about the relationship between free module
s and projective module
s over polynomial rings. It states that every finitely generated projective module
over a polynomial ring
is free
.
Geometrically, finitely generated projective modules correspond to vector bundle
s over affine space
, and free modules to trivial vector bundles. Affine space
is topologically contractible, so it admits no non-trivial topological vector bundles. A simple argument using the exponential exact sequence
and the d-bar Poincaré lemma shows that it also admits no non-trivial holomorphic vector bundles. Jean-Pierre Serre
, in his 1955 paper "Faisceaux algébriques cohérents", remarked that the equivalent question was not known for algebraic vector bundles: "It is not known if there exist projective A-modules of finite type which are not free." Here A is a polynomial ring
over a field, that is, A = k[x1, ..., xn].
To Serre's dismay, this problem quickly became known as Serre's conjecture. (Serre wrote, "I objected as often as I could [to the name].") The statement is not immediately obvious from the topological and holomorphic cases, because these cases only guarantee that there is a continuous or holomorphic trivialization, not an algebraic trivialization. Instead, the problem turns out to be extremely difficult. Serre made some progress towards a solution in 1957 when he proved that every finitely generated projective module over a polynomial ring over a field was stably free, meaning that after forming its direct sum with a finitely generated free module, it became free. The problem remained open until 1976, when Daniel Quillen and Andrei Suslin
independently proved that the answer was affirmative. Quillen was awarded the Fields Medal
in 1978 in part for his proof of the Serre conjecture. Leonid Vaseršteĭn
later gave a simpler and much shorter proof of the theorem which can be found in Serge Lang's Algebra.
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
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...
about the relationship between free module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
s and projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
s over polynomial rings. It states that every finitely generated projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
over a polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
is free
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
.
Geometrically, finitely generated projective modules correspond to vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s over affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
, and free modules to trivial vector bundles. Affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
is topologically contractible, so it admits no non-trivial topological vector bundles. A simple argument using the exponential exact sequence
Exponential sheaf sequence
In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry.Let M be a complex manifold, and write OM for the sheaf of holomorphic functions on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are...
and the d-bar Poincaré lemma shows that it also admits no non-trivial holomorphic vector bundles. 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:...
, in his 1955 paper "Faisceaux algébriques cohérents", remarked that the equivalent question was not known for algebraic vector bundles: "It is not known if there exist projective A-modules of finite type which are not free." Here A is a polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
over a field, that is, A = k[x1, ..., xn].
To Serre's dismay, this problem quickly became known as Serre's conjecture. (Serre wrote, "I objected as often as I could [to the name].") The statement is not immediately obvious from the topological and holomorphic cases, because these cases only guarantee that there is a continuous or holomorphic trivialization, not an algebraic trivialization. Instead, the problem turns out to be extremely difficult. Serre made some progress towards a solution in 1957 when he proved that every finitely generated projective module over a polynomial ring over a field was stably free, meaning that after forming its direct sum with a finitely generated free module, it became free. The problem remained open until 1976, when Daniel Quillen and Andrei Suslin
Andrei Suslin
Andrei Suslin is a Russian mathematician who has made major contributions to the field of algebra, especially algebraic K-theory and its connections with algebraic geometry. He is currently a Trustee Chair and Professor of mathematics at Northwestern University.He was born on December 27, 1950,...
independently proved that the answer was affirmative. Quillen was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1978 in part for his proof of the Serre conjecture. Leonid Vaseršteĭn
Leonid Vaserštein
Leonid Vaseršteĭn is a Russian-American mathematician, currently Professor of Mathematics at Penn State University. His research is focused on algebra and dynamical systems...
later gave a simpler and much shorter proof of the theorem which can be found in Serge Lang's Algebra.