Nilpotent elements in Grothendieck rings (Q580436)

Let \(M_ 1,...,M_ n\) be isomorphism classes of finitely presented modules over a commutative ring R. One forms the ring \({\mathbb{Z}}[M_ 1,...,M_ n]\) with \(\oplus\) and \(\otimes\) as addition and multiplication, and with the obvious relations. It is shown that if M and N are locally isomorphic, then there is an integer n, depending on M, N and R, such that \((M-N)^ n=0\) in \({\mathbb{Z}}[M,N]\). In some situations there is such an n depending only on R: If R has d-dimensional Noetherian maximal ideal space, or if R is the ring of continuous functions on a compact Hausdorff space of covering dimension d, one can take \(n=d+1.\) In rings of low dimension there are cancellation theorems, giving applications of the following sort: Let R be a one-dimensional reduced Noetherian ring with finite normalization, and let M and N be finitely generated R-modules. If \(\oplus ^ nM\cong \oplus ^ nN\), then \(\otimes ^ nM\cong \otimes ^ nN\).