Bass conjecture

In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated. The conjecture was proposed by Hyman Bass.

## Contents

* 1 Statement of the conjecture
* 2 Known cases
* 3 Implications
* 4 References

## Statement of the conjecture

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

* For any 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, the groups Kn(A) are finitely generated (K-theory of finitely generated locally free A-modules).
* For any scheme X of finite type over Spec(Z), K'n(X) is finitely generated.
* For any regular scheme X of finite type over Z, Kn(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 fields and the spectrum of the ring of integers 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.[1]

## References

1. ↑ Kahn, Bruno (2005), "Algebraic K-theory, algebraic cycles and arithmetic geometry", in Friedlander, Eric; Grayson, Daniel, Handbook of Algebraic K-theory, Berlin, New York: Springer-Verlag, pp. 351–428, doi:10.1007/3-540-27855-9_9, ISBN 978-3-540-23019-9 , Theorem 39

* Friedlander, Eric M.; Weibel, Charles W. (1999), An overview of algebraic K-theory, World Sci. Publ., River Edge, NJ, pp. 1–119 , p. 53

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