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Castelnuovo-Mumford regularity and Ratliff-Rush closure. (English) Zbl 1412.13015
Let \(A\) be a standard graded algebra over a field \(k\) with the standard graded maximal ideal \(m\). Let \(I\) be an ideal of \(A\). We write \(R(I)=\bigoplus_{n\geq0}I^{n}\), the Rees algebra of \(I\) and \(F(I)=\bigoplus_{n\geq n}I^{n}/mI^{n}\), the fiber ring of \(I\). We denote \(\text{reg}(*)\), the Castelnuovo-Mumford regularity of \(*\). D. Eisenbud and B. Ulrich have the following conjecture in [Proc. Am. Math. Soc. 140, No. 4, 1221–1232 (2012; Zbl 1246.13016)]: \[ \text{reg}(R(I))=\text{reg}(F(I)) \] when \(I\) is an \(m\)-primary graded ideal generated by forms of the same degree.
The authors try to verify the conjecture with some extra assumptions. The first main result is when \(\text{depth}G(I)\geq\text{dim}A-1\) then \[ \text{reg}(R(I))=\text{reg}(F(I))=r_{J}(I) \] for every minimal prime of \(J\) of \(I\), where \(G(I)=\bigoplus_{n\geq0}I^{n}/I^{n+1}\) is the associated graded ring of \(I\) (Theorem 3.5). When \(I\subseteq A=k[x,y]\) is a monomial ideal generated by degree \(d\) such that it contains \(x^{d},x^{d-1},y^{d}\), then the conjecture is true (Theorem 4.4).
The main results of this paper is that they find \(\text{reg}(R(I))\) and \(\text{reg}(F(I))\) control the behavior of the Ratliff-Rush closure and the Ratliff-Rush filtration of \(I\). More specifically, if \(\text{dim}A=2\), they express \(\text{reg}(R(I))\) and \(\text{reg}(F(I))\) in terms of the stability of the Ratliff-Rush filtration (Theorem 2.4, 2.9 and 3.8).

MSC:
13C05 Structure, classification theorems for modules and ideals in commutative rings
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14B05 Singularities in algebraic geometry
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