In mathematics

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...

, especially 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...

, the Bass conjecture says that certain algebraic K-groups

Algebraic K-theory

In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....

 are supposed to be finitely generated. The conjecture was proposed by Hyman Bass

Hyman Bass

Hyman Bass is an American mathematician, known for work in algebra and in mathematics education. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus...

.

Statement of the conjecture

Any of the following equivalent statements is referred to as the Bass conjecture.

• For any finitely generated
  Finitely generated algebra
  In mathematics, a finitely generated algebra is an associative algebra A over a field K where there exists a finite set of elements a1,…,an of A such that every element of A can be expressed as a polynomial in a1,…,an, with coefficients in K...
  
   Z-algebra A, the groups K'_n(A) are finitely generated
  Finitely generated abelian group
  In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the formwith integers n1,...,ns...
  
   (K-theory of finitely generated A-modules, also known as G-theory of A) for all n ≥ 0.
• For any finitely generated Z-algebra A, that is a regular ring
  Regular ring
  In commutative algebra, a regular ring is a commutative noetherian ring, such that the localization at every prime ideal is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.Jean-Pierre...
  
  , the groups K_n(A) are finitely generated (K-theory of finitely generated locally free A-modules).
• For any scheme
  Scheme (mathematics)
  In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
  
   X of finite type over Spec
  Spectrum of a ring
  In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
  
  (Z), K'_n(X) is finitely generated.
• For any regular scheme X of finite type over Z, K_n(X) is finitely generated.

The equivalence of these statements follows from the agreement of K- and K'-theory for regular rings and the localization sequence for K'-theory.

Known cases

Daniel Quillen showed that the Bass conjecture holds for all (regular, depending on the version of the conjecture) rings or schemes of dimension ≤ 1, i.e., algebraic curves over finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

s and the spectrum of the ring of integers

Ring of integers

In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

 in a number field.

The (non-regular) ring A = Z[x, y]/x² has an infinitely generated K₁(A).

Implications

The Bass conjecture is known to imply the Beilinson–Soulé vanishing conjecture.