Applications of dualizing complexes to Buchsbaum rings

Abstract

Let (A, m, k) denote a local noetherian ring with d = dim A admitting a normalized dualizing complex D’. By well-known theorems on local duality A is Cohen-Macaulay ring if and only if the (-d)th place truncated dualizing complex r-,D’ is isomorphic to the zero complex in the derived category D(A). (See Section 1 for more precise statements.) In the following we consider a class of local rings for which the truncated dualizing complex t-,D’ is isomorphic to a complex of vector spaces over A/m = k. There have been several efforts to get results about local rings which generalize the Cohen-Macaulay property, see, for instance [7,13,14]. A connection to our consideration is given in the following sense: In 2.3 we prove one of our main, theorems: the complex t-,D’ of A is isomorphic to a complex of vector spaces over k if and only if A is Buchsbaum ring in the sense of [ 13, 141. They are those rings for which the difference L(A/d) e,(x; A) between length and multiplicity of a system of parameters x does ndt depend on x. See also 2.1 for a brief summary of essential facts about Buchsbaum rings and relation to a conjecture of D. A. Buchsbaum. In [7] M. Hochster and J. L. Roberts have studied purity of the Frobenius map and related things to show that the local cohomology modules of certain rings of such type behave nicely. Applying our criterion 3.1 to this situation, we can show in 4.1 that such rings have a truncated dualizing complex t-,D’ which is isomorphic to a complex of vector spaces over k. Following the examples given in [7] we get a number of Buchsbaum rings. For instance: Let Vc IP~ be a projective variety defined by an ideal which is generated by square-free monomials, then V is arithmetically Buchsbaum if and only if V is a Cohen-Macaulay variety. Our Theorem 2.3 is related to a result of R. Kiehl in [8]. Using the “surjective map” criterion of [ 141 he showed that the canonical module of A is Buchsbaum if t-,D’ is isomorphic to a complex of vector spaces over k. This yields in [8] that the local rings at the cusps of Hilbert modular groups are factorial Buchsbaum rings which are not Cohen-Macaulay. 61 OOOl-8708/82/040061-17$05.00/O