A MULTIPLICITY BOUND FOR GRADED RINGS AND A CRITERION FOR THE COHEN-MACAULAY PROPERTY.

Let R be a polynomial ring over a field. We prove an upper bound for the multiplicity of R/I when I is a homogeneous ideal of the form I = J+(F), where J is a Cohen-Macaulay ideal and F ∉ J. The bound is given in terms of two invariants of R/J and the degree of F. We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given. [ABSTRACT FROM AUTHOR]