Castelnuovo–Mumford regularity and Ratliff–Rush closure.

We establish strong relationships between the Castelnuovo–Mumford regularity and the Ratliff–Rush closure of an ideal. Our results have several interesting consequences on the computation of the Ratliff–Rush closure, the stability of the Ratliff–Rush filtration, the invariance of the reduction number, and the computation of the Castelnuovo–Mumford regularity of the Rees algebra and the fiber ring. In particular, we prove that the Castelnuovo–Mumford regularity of the Rees algebra and of the fiber ring are equal for large classes of monomial ideals in two variables, thereby verifying a conjecture of Eisenbud and Ulrich for these cases. [ABSTRACT FROM AUTHOR]