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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.

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
Full Text: DOI
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