A negative answer to a question of Bass

In this companion paper to arXiv:0802.1928 we provide an example of an isolated surface singularity $R$ over a number field such that $K_0(R) = K_0(R[t])$ but $K_0(R) \neq K_0(R[t_1,t_2])$. This answers, negatively, a question of Bass.Comment: The paper was previously part of arXiv:0802.192

Variation of Hilbert Coefficients

For a Noetherian local ring (\RR, \m), the first two Hilbert coefficients, $e_0$ and $e_1$, of the $I$-adic filtration of an \m-primary ideal $I$ are known to code for properties of \RR, of the blowup of \spec(\RR) along $V(I)$, and even of their normalizations. We give estimations for these coefficients when $I$ is enlarged (in the case of $e_1$ in the same integral closure class) for general Noetherian local rings