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.
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 "K"1("A").
Implications.
The Bass conjecture is known to imply the Beilinson–Soulé vanishing conjecture.