The asymptotic Ass

Asymptotic prime divisors and analytic spreads

“…A recent paper of Brodmann [1] shows that A(n) is constant for n large. In [5] that constant is denoted A* = A*(I).…”

“…In general it is difficult to explicitly determine A * for a given ideal I, although in [5,Corollary 22] this is done for R a 2-dimensional normal domain. This paper will discuss a concept related to A*, namely yl*.…”

“…Our first lemma is essentially a restatement of [5,Proposition 18] in a more efficient manner. Lemma 1.…”

“…As I E P c P', IT E P' and so in the form ring of /, T/IT, P'/IT is a minimal prime. According to [5,Proposition 18], in order to show that P E A* we need only show that P'/IT is a relevant prime. Being a minimal prime, P'/IT is homogeneous (thus P' is homogeneous as stated), and so if it is not relevant then clearly P' = P + Ix + I2x2 + .…”

“…By [5,Proposition 18], in the form ring T/IT there is a minimal prime, call it P'/IT, with (P'/IT) n (R/I) = P/L To prove the lemma, we must only show that in T, height P' = 1. We go to the Rees ring T + x_1Ä[x_1], and consider the prime P' + x~xR[x~x].…”