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.
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).