Hilbert-Kunz multiplicities of powers of ideals and normalized Hilbert coefficients
We give negative answers to two questions of Smirnov on Hilbert coefficients of Frobenius powers and Hilbert-Kunz multiplicities of ordinary powers. We construct complete normal Cohen-Macaulay hypersurfaces whose maximal ideals have normalized Hilbert coefficients of every index at least three tending to minus infinity. We also construct isolated normal F-pure non-Cohen-Macaulay singularities whose maximal ideals have convergent normalized coefficients but do not satisfy the proposed eventual polynomial formula. For every Noetherian local ring of positive characteristic and dimension $d\geq2$, we prove convergence of the second normalized Hilbert coefficient and a three-term asymptotic expansion with remainder $o(n^{d-2})$. Thus coefficient convergence always holds in dimension two, and index three is the first possible failure in higher dimensions. Both questions have positive answers for every ideal primary to the maximal ideal in a local ring of finite F-representation type.