Hilbert-Samuel function and that of the irreducible components. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:11:49Z http://mathoverflow.net/feeds/question/94302 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94302/hilbert-samuel-function-and-that-of-the-irreducible-components Hilbert-Samuel function and that of the irreducible components. Franklin 2012-04-17T16:20:48Z 2012-04-17T21:06:59Z <p>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? </p> <p>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.</p> <p>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.</p> <p><strong>Edit:</strong> 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 the maximal ideal, and $M:=R$, if I understood their definition of $Assh$ this only gives me information about those components $p$ having $dim(R/p)=dim(M)=dim(R)$.</p> http://mathoverflow.net/questions/94302/hilbert-samuel-function-and-that-of-the-irreducible-components/94317#94317 Answer by Youngsu for Hilbert-Samuel function and that of the irreducible components. Youngsu 2012-04-17T19:48:23Z 2012-04-17T19:48:23Z <p>There is an associativity formula for Hilbert-Samuel multiplicity which says, the multiplicity is the sum of the multiplicities of the irreducible components in your concrete case, i.e. $e_0(m,R/I) = \sum e_0(m,R/I_k)$. You may want to take a look at Theorem 14.7 of Matsumura, commutative ring theory for the general statement. But I don't know the answer to the relationship between Hilbert-Samuel functions.</p>