Canonical modules of Rees algebras

We compute the canonical class of certain Rees algebras. Our formula generalizes that of Herzog and Vasconcelos. Its proof relies on the fact that the formation of the canonical module commutes with subintersections in important cases. As an application we treat the classical determinantal ideals and the corresponding algebras of minors. A considerable part of Wolmer Vasconcelos’ work has been devoted to Rees algebras, in particular to their divisorial structure and the computation of the canonical module (see [HSV1, HSV2, HV, MV, Va]). In this paper we give a generalization of the formula of Herzog and Vasconcelos [HV] who have computed the canonical module under more special assumptions. We show that [ω R ] = [IR]+ t ∑ i=1 (1−htpi)[Pi] for ideals I in regular domains R essentially finite over a field for which the Rees algebra R = R(I) is normal Cohen–Macaulay and whose powers have a special primary decomposition. In the formula the Pi are the divisorial prime ideals containing IR, and pi = Pi ∩R. The condition on the primary decomposition can be expressed equivalently by the requirement that the restriction of the Rees valuation vPi to R coincides with the pi-adic valuation. While our hypotheses are far from the most general case, which may very well be intractable, the formula covers many interesting ideals, for example the classical determinantal ideals. As an application we can therefore compute the canonical classes of their Rees algebras, and also those of the algebras generated by minors. This paper have been inspired by the work of Bruns and Conca [BC2] where the case of the determinantal ideals of generic matrices and Hankel matrices has been treated via initial ideals. The formula above can be proved with localization arguments only if there is no containment relation between the pi. It was therefore necessary to investigate the behavior of the canonical module under subintersections, and, as we will show for normal algebras essentially of finite type over a field, its formation does indeed commute with taking a subintersection.