ar X iv : m at h / 02 09 36 5 v 1 [ m at h . A C ] 2 6 Se p 20 02 RESIDUES FOR AKIZUKI ’ S ONE - DIMENSIONAL LOCAL DOMAIN

For a one-dimensional local domain CM constructed by Akizuki, we find residue maps which give rise to a local duality. The completion of CM is described using these residue maps. Injective hulls of a given module are all isomorphic. For this reason, people often speak of the injective hull to indicate its “uniqueness”. However, isomorphisms between these injective hulls are not canonical. In fact, they are a part of the structure of the given module. For instance, a local duality for a power series ring [2, (5.9)] is interpreted as an isomorphism between two injective hulls one given by local cohomology and another by continuous homomorphisms. This isomorphism is induced by a residue map, which was not observed from the viewpoint of “uniqueness” of injective hulls. In this article, our philosophy is taken up again by a Noetherian local ring CM constructed by Akizuki [1]. Although CM behaves beyond geometric expectation, we can still define certain maps, which give rise to a local duality as an identification of local cohomology classes and continuous homomorphisms. These maps, also called residue maps, determine all endomorphisms of an injective hull of the residue field of CM . So we are able to describe the completion of CM . We recall Akizuki’s construction. Let A be a discrete valuation ring with the maximal ideal m = tA, let  be its completion, and let K (resp. K̂) be the quotient field of A (resp. Â). Assume that there is an element z = a0 + a1t n1 + a2t n2 + · · · ∈  (ai ∈ A \ m) transcendental over A with the condition nr ≥ 2nr−1 + 2 (r ≥ 1) on exponents, where n0 = 0. Let zr = ar + ar+1t nr+1−nr + · · · (r ≥ 0) and C = A[t(z0 − a0), {(zi − ai) }i=0]. CM is defined to be the localization of C at the maximal ideal M generated by t and t(z0 − a0). Akizuki [1] showed that CM is a one-dimensional Noetherian local domain, whose normalization is not a finite CM -module. The quotient field of CM is K(z), which equals (CM )t as t is a system of parameter of CM . We can use the exact sequence (1) 0 → CM localization −−−−−−−→ (CM )t → H 1 MCM (CM ) →