Bass conjecture
# 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.

## 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 $K_n(A)$ 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, $K_n(X)$ 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 fields and the spectrum of the ring of integers in a number field.

The (non-regular) ring A = Z[x, $y]/x^2$ has an infinitely generated $K_1(A).$

## Implications

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

