Dualizing complexes and systems of parameters

Abstract

In [4] and [5] M. Hochster introduced the notion of amiability for a system of parameters in a local noetherian ring (see also the definition in 3). The existence of amiable systems of parameters is the key step for Hochster’s construction of big Cohen-Macaulay modules for rings with prime characteristic. M. Hochster showed that there are amiable systems of parameters if the local ring is a modulefinite local extension domain of an integrally closed Cohen-Macaulay local domain R,, whose fraction field is separable over the fraction field of R,, . In Theorem 2 we obtain the existence of amiable systems of parameters for local rings which admit a dualizing complex. In Proposition 5 we give examples of local rings which have no amiable system of parameters. In 1 we begin with the definition of a dualizing complex and refer to Theorem 1 from [7] which is essential in the proof of our Theorem 2. In 2 we show some properties of dualizing complexes. We obtain in Theorem 1 that the associated primes of a local ring A with dualizing complex are determined by the annihilators ai = Ann Hmi(A), where Hmi(A) denotes the i-th local cohomology module of A with support in {m}. After some further results on the annihilators a, we give in Theorem 3 an interpretation of V(a, .** a,-,) if A has a dualizing complex and dim A = n.