Bass conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A finitely generated $\mathbb Z$-algebra is a commutative ring $A=\mathbb Z[a_1,\ldots,a_r]$ with identity, for some finite $r$. Let $K'_n(A)$ denote Quillen's $n$th algebraic $K$-group of the exact category of finitely generated $A$-modules; this is also called $G_n(A)$. A group is finitely generated if a finite set generates it as an abelian group. For a regular ring $A$ (all local rings at prime ideals are regular local rings), $K'_n(A)$ agrees with $K_n(A)$, the $K$-theory of finitely generated projective modules. Here $n$ is an integer with $n\geq0$.

Statement recorded in the supplied source.

Is $K'_n(A)$ finitely generated for every finitely generated $\mathbb Z$-algebra $A$ and every $n\geq0$? Equivalently, does finite generation hold for $K_n(A)$ for every regular finitely generated $\mathbb Z$-algebra $A$ and every $n\geq0$? [S2]

Short English statement

Are all higher coherent-module K-theory groups of rings of finite type over the integers finitely generated? The equivalent regular-ring version uses projective-module K-theory.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200002 (AMR-071-0002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Thomas Geisser, Finite generation conjectures for cohomology over finite fields (2011). Introduction, p. 1. https://arxiv.org/pdf/1103.5544v1.