Variation of Hilbert coefficients

Abstract

For a Noetherian local ring ( R , m ) (\mathbf {R}, \mathfrak {m}) , the first two Hilbert coefficients, e 0 e_0 and e 1 e_1 , of the I I -adic filtration of an m \mathfrak {m} -primary ideal I I are known to code for properties of R \mathbf {R} , of the blowup of Spec ⁡ ( R ) \operatorname {Spec}(\mathbf {R}) along V ( I ) V(I) , and even of their normalizations. We give estimations for these coefficients when I I is enlarged (in the case of e 1 e_1 in the same integral closure class) for general Noetherian local rings.