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 are finitely generated for all n ≥ 0.
For any finitely generated Z-algebra A, that is a regular ring, the groups K_n are finitely generated.
For any scheme X of finite type over Spec, K'_n is finitely generated.
For any regular scheme X of finite type over Z, K_n 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.

Daniel Quillen showed that the Bass conjecture holds for all 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 ring A = Z/x² has an infinitely generated K₁.