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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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