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Analytic spread of a pregraduation. (English) Zbl 1058.13001

Fontana, Marco (ed.) et al., Commutative ring theory and applications. Proceedings of the fourth international conference, Fez, Morocco, June 7–12, 2001. New York, NY: Marcel Dekker (ISBN 0-8247-0855-5/pbk). Lect. Notes Pure Appl. Math. 231, 107-116 (2003).
Let \((A,\mathfrak m)\) be a Noetherian local ring with \(k:=A/\mathfrak m\) infinite, \(F=(I_n)_{n\geq 0}\) a filtration on \(A\), i.e., a descending chain of ideals of \(A\) with \(I_0=A\) and \(I_iI_j\subseteq I_{i+j}\), \(\forall i,j\), and let \(R_+(F):=\bigoplus_{n\geq 0}I_nt^n\subseteq A[t]\) be the Rees algebra of \(A\) with respect to \(F\). The elements \(a_1,\dots ,a_r\in I_1\) are “analytically independent” with respect to \(F\) if whenever \(f\in A[X_1,\dots ,X_r]\) is a homogeneous polynomial of (arbitrary) degree \(s\) such that \(f(a_1,\dots ,a_r)\in{\mathfrak m}I_s\) then all the coefficients of \(f\) are in \(\mathfrak m\). Let \(\ell (F)\) denote the largest \(r\) for which such a sequence of elements exists. D. G. Northcott and D. Rees [Proc. Camb. Philos. Soc. 50, 145–158 (1954; Zbl 0057.02601)] showed that if \(I\) is a proper ideal of \(A\) and \(F\) is the \(I\)-adic filtration then \(\ell (F)\) equals the analytic spread of \(I\), i.e., the Krull dimension of \(R_+(F)/{\mathfrak m}R_+(F)\).
The author remarks that if \(a_1,\dots a_r\in I_1\) are analytically independent with respect to a filtration \(F\), then, \(\forall p\geq 1\), \(a_1^p,\dots ,a_r^p\in I_p\) are analytically independent with respect to the filtration \(F^{(p)}:=(I_{pn})_{n\geq 0}\). He defines the “regular analytic spread” \({\ell}^a(F)\) of \(F\) as \(\text{sup}\{ {\ell}(F^{(p)})\mid p\geq 1\} \) and shows that if \(F\) is a Noetherian filtration, i.e., if \(R_+(F)\) is a Noetherian ring, then \({\ell}^a(F)\) equals the Krull dimension of \(R_+(F)/{\mathfrak m}R_+(F)\).
Furthermore, the author investigates the properties of the regular analytic spread in a more general context, for an arbitrary commutative ring \(A\), replacing the filtration \(F\) by what he calls a “pregraduation”, i.e., by a family of ideals \(G=(G_n)_{n\in{\mathbb Z}}\) such that \(G_0=A\) and \(G_iG_j\subseteq G_{i+j}\), \(\forall i,j\), and \(\mathfrak m\) by an arbitrary ideal \(J\) of \(A\).
For the entire collection see [Zbl 1027.00012].

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13A02 Graded rings
13E05 Commutative Noetherian rings and modules
13H99 Local rings and semilocal rings

Citations:

Zbl 0057.02601
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