Normal Ideals in Regular Rings
S. Huckaba, C. Huneke
We study the Cohen-Macaulay property of the Rees algebra of a normal ideal of a regular local ring. In particular we characterize this property in the 3-dimensional case, assuming the ideal has height 2, is unmixed, and is generically a complete intersection. Our characterization yields concrete examples of normal ideals whose Rees algebras are not Cohen-Macaulay, thus settling a recent conjecture of Vasconcelos. We also provide a two-dimensional algebraic proof of a version (due to Sancho de Salas) of a vanishing theorem proved by Grauert and Riemenschnieder, plus an explicit example illustrating that the version fails in dimension 3. This example complements a family of such examples constructed geometrically by Cutkosky.