On the generators of \(\mathrm{Nil} \ K\)-groups (Q6614572)

For a commutative ring \(R\) with unity, denote the category of finitely generated projective \(R\)-modules by \(P(R)\). Let \(v\) be a nilpotent endomorphism of \(P\). The category of all pairs \((P,v)\) is denoted by \(\mathrm{Nil}(R)\). Let \(\mathrm{Nil}_0(R)\) be the kernel of the forgetful map \(K_0(\mathrm{Nil}R)) \longrightarrow K_0(P(R))=K_0(R)\) [\textit{D. R. Grayson}, J. Am. Math. Soc. 25, No. 4, 1149--1167 (2012; Zbl 1276.19003)].\N\NThe authors of the article under review show the vanishing of zeroth Nil K-groups. Precisely, they show in Theorem 3.3 that, if every finitely generated torsion free \(R\)-module is projective then \(\mathrm{Nil}_0(R)=0\). Furthermore, they investigate higher Nil K-groups and give explicit forms of generators of higher Nil K-groups \(\mathrm{Nil}_n(R)\) in term of binary complexes (Theorem 5.8).