mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Bass conjecture says that certain algebraic ''K''-groups are supposed to be finitely generated. The conjecture was proposed by
Hyman Bass
Hyman Bass (; born October 5, 1932) finitely generated Z-algebra ''A'', the groups ''K'''''n''(''A'') are finitely generated (''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 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 ...
, the groups ''K''''n''(''A'') are finitely generated (''K''-theory of finitely generated locally free ''A''-modules).
* For any
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
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* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
Spec Spec may refer to:
*Specification (technical standard), an explicit set of requirements to be satisfied by a material, product, or service
**datasheet, or "spec sheet"
People
* Spec Harkness (1887-1952), American professional baseball pitcher
...
(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
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 ...
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
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
over
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s and the spectrum of the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
in a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
.
The (non-regular) ring ''A'' = Z
, y
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