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Sally modules of m-primary ideals in local rings

Alberto Corso

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Abstract

Abstract: Given a local Noetherian ring (R, m) of dimension d> 0 and infinite residue field, we study the invariants (dimension and multiplicity) of the Sally module SJ(I) of any m-primary ideal I with respect to a minimal reduction J. As a by-product we obtain an estimate for the Hilbert coefficients of m that generalizes a bound established by J. Elias and G. Valla in a local Cohen-Macaulay setting. We also find sharp estimates for the multiplicity of the special fiber ring F(I), which recover previous bounds established by C. Polini, W.V. Vasconcelos and the author in the local Cohen-Macaulay case. Great attention is also paid to Sally modules in local Buchsbaum rings. 1.

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Abstract: Given a local Noetherian ring (R, m) of dimension d> 0 and infinite residue field, we study the invariants (dimension and multiplicity) of the Sally module SJ(I) of any m-primary ideal I with respect to a minimal reduction J. As a by-product we obtain an estimate for the Hilbert coefficients of m that generalizes a bound established by J. Elias and G. Valla in a local Cohen-Macaulay setting. We also find sharp estimates for the multiplicity of the special fiber ring F(I), which recover previous bounds established by C. Polini, W.V. Vasconcelos and the author in the local Cohen-Macaulay case. Great attention is also paid to Sally modules in local Buchsbaum rings. 1.

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Available abstract

Abstract: Given a local Noetherian ring (R, m) of dimension d> 0 and infinite residue field, we study the invariants (dimension and multiplicity) of the Sally module SJ(I) of any m-primary ideal I with respect to a minimal reduction J. As a by-product we obtain an estimate for the Hilbert coefficients of m that generalizes a bound established by J. Elias and G. Valla in a local Cohen-Macaulay setting. We also find sharp estimates for the multiplicity of the special fiber ring F(I), which recover previous bounds established by C. Polini, W.V. Vasconcelos and the author in the local Cohen-Macaulay case. Great attention is also paid to Sally modules in local Buchsbaum rings. 1.

Key concepts: Local ring, Residue field, Mathematics, Noetherian, Regular local ring, Multiplicity (mathematics), Pure mathematics, Noetherian ring

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