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How to obtain a relation between the Hilbert-Samuel function of the local ring at a point of a reduced, but not necessarily irreducible variety, and the Hilbert-Samuel functions of the corresponding local rings of its irreducible components?

More concrete. R, a regular local Noetherian ring, complete if you wish, I an ideal in R that is the intersection of some prime ideals I_k such that there are no embedded primes.

I am looking for a formula relating the HS function of R/I and those of R/I_k? The filtration is with respect to the maximal ideal of R.

Edit: Info: There is also an analogous formula for Hilbert functions (analogous to the associativity formula for multiplicity). Proposition 3.2 in Equimultiplicity and blowing up, by Herrmann, Ikeda and Orbanz. $H^{(i)}[\underline{x},a,M]=\sum_{p∈Assh(M/aM)}e(\underline{x},R/p)H^{(i)}[aR_p,M_p]$, where $M$ is finitely generated $R$-module, $a$ and ideal in $R$, and $\underline{x}$ a multiplicity system for $M/aM$. If I put $R$ as my ring , $R/I$, $a$ as my the maximal ideal$I$, , and $M:=R$, if I understood their definition of $Assh$ this only gives me information about those components $p$ having $dim(R/p)=dim(M)=dim(R)$.

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How to obtain a relation between the Hilbert-Samuel function of the local ring at a point of a reduced, but not necessarily irreducible variety, and the Hilbert-Samuel functions of the corresponding local rings of its irreducible components?

More concrete. R, a regular local Noetherian ring, complete if you wish, I an ideal in R that is the intersection of some prime ideals I_k such that there are no embedded primes.

I am looking for a formula relating the HS function of R/I and those of R/I_k? The filtration is with respect to the maximal ideal of R.

Edit: Info: There is also an analogous formula for Hilbert functions (analogous to the associativity formula for multiplicity). Proposition 3.2 in Equimultiplicity and blowing up, by Herrmann, Ikeda and Orbanz. $H^{(i)}[\underline{x},a,M]=\sum_{p∈Assh(M/aM)}e(\underline{x},R/p)H^{(i)}[aR_p,M_p]$, where $M$ is finitely generated $R$-module, $a$ and ideal in $R$, and $\underline{x}$ a multiplicity system for $M/aM$. If I put $R$ as my ring, $a$ as my ideal $I$, and $M:=R$, if I understood their definition of $Assh$ this only gives me information about those components $p$ having $dim(R/p)=dim(M)=dim(R)$.

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# Hilbert-Samuel function and that of the irreducible components.

How to obtain a relation between the Hilbert-Samuel function of the local ring at a point of a reduced, but not necessarily irreducible variety, and the Hilbert-Samuel functions of the corresponding local rings of its irreducible components?

More concrete. R, a regular local Noetherian ring, complete if you wish, I an ideal in R that is the intersection of some prime ideals I_k such that there are no embedded primes.

I am looking for a formula relating the HS function of R/I and those of R/I_k? The filtration is with respect to the maximal ideal of R.