Given a 0-dimensional scheme X in a projective space P n K over a field K, we characterize the Cayley-Bacharach property of X in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley-Bacharach schemes by Dedekind's formula for the conductor and the complementary module, we study schemes with minimal Dedekind different using the trace of the complementary module, and we prove various results about almost Gorenstein and nearly Gorenstein schemes.

Given a 0-dimensional scheme in a projective n-space P n over a field K, we study the Kähler differential algebra Ω R X /K of its homogeneous coordinate ring R X . Using explicit presentations of the modules Ω m R X /K of Kähler differential m-forms, we determine many values of their Hilbert functions explicitly and bound their Hilbert polynomials and regularity indices. Detailed results are obtained for subschemes of P 1 , fat point schemes, and subschemes of P 2 supported on a conic.

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Given a 0-dimensional scheme X in a n-dimensional projective space P n K over an arbitrary field K, we use Liaison theory to characterize the Cayley-Bacharach property of X. Our result extends the result for sets of Krational points given in [7]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of X when X has the Cayley-Bacharach property.Theorem 1.1. Let W be a set of points in P n K which is a complete intersection, let X ⊆ W, let Y = W \ X, and let I W , I X and I Y denote the homogeneous vanishing ideals of W, X and Y in P = K[X 0 , ..., X n ], respectively. Set α Y/W = min{i ∈ N | (I Y /I W ) i = 0 }. Then the following conditions are equivalent.(a) X is a Cayley-Bachrach scheme. (b) A generic element of (I Y ) α Y/W does not vanish at any point of X. (c) We have I W : (I Y ) α Y/W = I X .

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