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Analytic spread and non-vanishing of asymptotic depth. (English) Zbl 1388.13013
If $$S$$ is a noetherian ring and $$I$$ is an ideal of $$S$$ with integral closure $$\overline{I}$$, it is known that the sets of associated prime ideals $$\text{Ass}(S/\overline{I^n})$$ ($$n \geq 1$$) form an increasing sequence that eventually stabilizes at $$\overline{A}^*(I)=\bigcup_{n \geq 1} \text{Ass}(S/\overline{I^n})$$. Moreover, if $$S$$ is locally formally equidimensional and $$P \in V(I)$$, a result of S. McAdam [Proc. Am. Math. Soc. 80, 555–559 (1980; Zbl 0445.13002)] shows that $$P \in \overline{A}^*(I)$$ if and only if $$I_P$$ has maximal analytic spread, i.e., $$\ell(I_P)=\text{ht} P$$.
In this paper, the author considers the sets $$\overline{A}^*(I)$$ when $$S$$ is a polynomial ring $$k[x_1,\ldots, x_n]$$ over a field $$k$$ of characteristic zero and $$I$$ is a monomial ideal satisfying the condition
$$(*)$$ $$I$$ is a finite intersection of powers of monomial prime ideals and $$I$$ has no embedded primary components.
(Note that every square-free monomial ideal satisfies the above condition.) If we denote $$I_{[j]}=I:(I:x_j)$$ for $$j=1,\ldots, n$$, the main result of the paper shows the following: Let $$1\leq i_1 < \ldots < i_k\leq n$$ and let $$I\subseteq (x_{i_1}, \ldots, x_{i_k})$$ be a monomial ideal satisfying $$(*)$$ such that $$\sum_{j=i_1}^{i_k} [\sqrt I]_{[j]} \neq (x_{i_1},\ldots, x_{i_k})$$. Then $$(x_{i_1},\ldots, x_{i_k}) \notin \overline{A}^*(I)$$. Moreover, if $$R=S_{(x_{i_1},\ldots, x_{i_k})}$$ and the associated graded ring $$G_{IR}(R)$$ is Cohen-Macaulay, then $$\text{depth}(R/I^tR) \geq 1$$ and $$(x_{i_1},\ldots, x_{i_k}) \notin \text{Ass}(I^t/I^{t+1})$$ for $$t \gg 0$$. The author also provides concrete examples where the result can be applied.

MSC:
 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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