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Sally modules of \(\mathfrak m\)-primary ideals in local rings. (English) Zbl 1188.13002

Let \((R,\mathfrak{m})\) be a noetherian ring of dimension \(d > 0\), and let \(I\) be an \(\mathfrak{m}\)-primary ideal. In this paper, a minimal reduction of \(I\) is a \(d\)-generated subideal \(J\) of \(I\) such that \(I^{r+1} = JI^r\) for all \(r \gg 0\). The Sally module of \(I\) with respect to \(J\) was defined by W. V. Vasconcelos in [Hilbert functions, analytic spread and Koszul homology. Commutative Algebra: Syzygies, Multiplicities and Birational Algebra. Contemp. Math. 159, 401–422 (1994; Zbl 0803.13012)] to be \[ S_J(I) = \bigoplus_{n=2}^\infty I^n/JI^{n-1}. \]
The author investigates the dimension and the multiplicity of Sally modules. Main results include:
Theorem 2.1(a). The dimension of \(S_J(I)\) is exactly \(d\) if and only if its multiplicity is \(e_1(\mathfrak{m}) - e_0(\mathfrak{m}) - e_1(J) + 1\). Here, \(e_i(-)\) indicates the \(i\)-th Hilbert coefficient.
Theorem 2.8. The multiplicity of a \(d\)-dimensional Sally module \(S_J(I)\) is at most \(e_1(I) - e_0(I) - e_1(J) + \lambda(R/I)\), where \(\lambda(R/I)\) is the length of \(R/I\). The equality occurs if and only if \(I\) contains \((x_1, \dots, x_{d-1}): \mathfrak{m}^\infty\), where \(x_1, \dots, x_{d-1}\) are general elements in \(J\).
As a by-product of the study, the author also obtains a sharp estimate for the multiplicity \(f_0(I)\) of the special fiber ring \(\mathfrak{F}(I)\) of \(I\), namely, \[ f_0(I) \leq e_1(I) - e_0(I) + \lambda(R/I) + \mu(I) - d + 1, \] where \(\mu(I)\) denotes the number of minimal generators of \(I\). Here, the special fiber \(\mathfrak{F}(I)\) is defined to be \(\mathfrak{R}(I) \otimes_R R/\mathfrak{m}\), where \(\mathfrak{R}(I)\) is the Rees algebra of \(I\).

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13B21 Integral dependence in commutative rings; going up, going down
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series

Citations:

Zbl 0803.13012
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References:

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