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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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