Characterizations of zero-dimensional complete intersections

Abstract

Given a 0-dimensional subscheme \({\mathbb X}\) of a projective space \({\mathbb P}^n_K\) over a field K, we characterize in different ways whether \({\mathbb X}\) is the complete intersection of n hypersurfaces. Besides a generalization of the notion of a Cayley–Bacharach scheme, these characterizations involve the Kahler and the Dedekind different of the homogeneous coordinate ring of \({\mathbb X}\) or its Artinian reduction. We also characterize arithmetically Gorenstein schemes in novel ways and bring in further tools such as the module of regular differential forms, the fundamental class, and the Jacobian module of \({\mathbb X}\). Throughout we strive to work over an arbitrary base field K and keep the scheme \({\mathbb X}\) as general as possible, thereby improving several known characterizations.