Arithmetic degree and associated graded modules

Abstract

We prove that the arithmetic degree of a graded or local ring is bounded above by the arithmetic degree of any of its associated graded rings with respect to ideals $I$ in $A$. In particular, if $Spec(A)$ is equidimensional and has an embedded component in dimension $i$, then the normal cone of $Spec(A)$ along $V(I)$ has an embedded component in dimension $i$ too.

13 pages