# Elementary proof that $\operatorname{Ass}(I^n)$ stabilizes for a monomial ideal $I$

For my bachelor thesis I'd like to have a short "elementary" proof that $\operatorname{Ass}(I^n)$ stabilizes for large $n$ if $I$ is a monomial ideal in a polynomial ring $K[x_1, \dots, x_r]$ over some field $. The papers of Brodmann (Asymptotic stability of$\operatorname{Ass}(M / I^nM)$) and Ratliff (On prime divisors of$I^n$,$n$large) are frankly beyond me. (I have only very basic commutative algebra knowledge. I know basic properties of primary decomposition and localization, that's kind of it.) I actually found something that looks promising: McAdam, Eakin, The Asymptotic Ass However I'd like to avoid the language of graded rings altogether if that's somehow possible. Looking over the papers of Brodmann and Ratliff I could follow that$I^{n+1}:I=I^n$for large$n$implies asymptotic monotony of$\operatorname{Ass}(I^n/I^{n+1})$, which is just what I'm looking for. The question then becomes, is there an elementary proof that$I^{n+1}:I=I^n$for large$n$and$I$a monomial ideal? (I feel there should be given how easy calculating with monomial ideals is.) I would also be thankful for a conceptual explanation of the proof of this fact in general Noetherian rings as given by Ratliff or Brodmann. Most standard facts about monomial ideals in this context (formulas for colons, intersections, associated primes of$I$are monomial primes) are available. Gröbner basis methods are not available. Thanks in advance for any help you can offer! • Perhaps the bachelor thesis would even benefit from including "the language of graded rings". The result is Corollary 5 in the McAdam paper, and the proof seems accessible. – Dietrich Burde Jul 25 '13 at 20:49 • Hm. Maybe you could help me understand the proof of Remark 1(b) in McAdam's paper. It says there's a$t \geq 1$such that$(I^{n+1} : I) \cap I^t = I^n$for$n \geq t$. The proof uses the form ring of I. I'd like to "translate" that proof so there's no mention of "homogeneous Noetherian ring" or any of that wizardry. Basically I got as far as defining$B_k$as the set of monomials in$(I^{k+2} : I) \cap I^k - I^{k+1}$and define$B$as the monomial ideal generated by the union of the$B_k$. I'm pretty sure B corresponds to McAdam's$(0:R_1)\$ somehow, but I'm not sure how to finish from there. – Arno Nym Jul 27 '13 at 2:05
• Actually, nevermind. Looking again at the paper by Ratliff I found that I was basically looking for so called "superficial elements", and the book "Integral Closure of Ideals, Rings and Modules" (Swanson, Huneke) has a proof for existence of superficial elements which only depends on primary decomposition and graded prime avoidance. So I'll go with that one. – Arno Nym Jul 29 '13 at 10:21