Waldhausen’s regular coherent groups: torsionfree non-examples and behaviour under taking products?

Waldhausen defined a group $G$ to be regular coherent, if for all regular noetherian rings $R$ the group algebra $RG$ is regular coherent. (see Waldhausen – Algebraic $K$-Theory of generalized free products III, IV, Section 19 or below for more details).

Unsurprisingly this class of groups is mentioned regularly when the vanishing of his Nil-terms is discussed. But – apart from Waldhausen’s original list of groups that are regular coherent and the fact that all regular coherent groups are torsion-free – I could not find more information about which groups are and which are not regular coherent. In particular I am interested in the following two questions:

  1. Is there an example of a torsion-free group, that is not regular coherent?
  2. If $G_1,G_2$ are regular coherent, is $G_1\times G_2$ regular coherent?

More details on the definition:

A ring $R$ is called regular coherent if any finitely presented (right) $R$-module has a finite resolution by finitely generated projective $R$-modules.

$R$ is called regular noetherian if it is regular coherent and any finitely generated $R$-module is already finitely presented.

Answer

Attribution
Source : Link , Question Author : Svenja Knopf , Answer Author : Community

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