Density functions for epsilon multiplicity and families of ideals

A density function for an algebraic invariant is a measurable function on $R$ which measures the invariant on an $R$-scale. This function carries a lot more information related to the invariant without seeking extra data. It has turned out to be a useful tool, which was introduced by the third author in Trivedi [Trans. Amer. Math. Soc. 370 (2018), no. 12, 8403–8428], to study the characteristic $p$ invariant, namely Hilbert–Kunz multiplicity of a homogeneous $m$-primary ideal. Here, we construct density functions $f_{A,\left\{I_n\right\}}$ for a Noetherian filtration ${\left\{I_n\right\}}_{n\in N}$ of homogeneous ideals and $f_{A,\left\{\stackrel{\sim }{I^n}\right\}}$ for a filtration given by the saturated powers of a homogeneous ideal $I$ in a standard graded domain $A$. As a consequence, we get a density function $f_{\epsilon \left(I\right)}$ for the epsilon multiplicity $\epsilon \left(I\right)$ of a homogeneous ideal $I$ in $A$. We further show that the function $f_{A,\left\{I_n\right\}}$ is continuous everywhere except possibly at one point, and $f_{A,\left\{\stackrel{\sim }{I^n}\right\}}$ is a continuous function everywhere and is continuously differentiable except possibly at one point. As a corollary, the epsilon density function $f_{\epsilon \left(I\right)}$ is a compactly supported continuous function on $R$ except at one point, such that ${\int }_{R_{⩾0}}{f}_{\epsilon \left(I\right)}=\epsilon \left(I\right)$. All the three functions $f_{A,\left\{I^n\right\}}$, $f_{A,\left\{\stackrel{\sim }{I^n}\right\}}$ and $f_{\epsilon \left(I\right)}$ remain invariant under passage to the integral closure of $I$. As a corollary of this theory, we observe that the ‘rescaled’ Hilbert–Samuel multiplicities of the diagonal subalgebras form a continuous family.