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The Cayley-Bacharach property, which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine algebras over arbitrary base fields. We present characterizations and explicit algorithms for checking the Cayley-Bacharach property directly, via the canonical module, and in combination with the property of being a locally Gorenstein ring. Moreover, we characterize strict Gorenstein rings by the Cayley-Bacharach property and the symmetry of their affine Hilbert function, as well as by the strict Cayley-Bacharach property and the last difference of their affine Hilbert function.
Given a 0-dimensional scheme $mathbb{X}$ in a projective space $mathbb{P}^n_K$ over a field $K$, we characterize the Cayley-Bacharach property of $mathbb{X}$ in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate
In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley-Bacharach condition. In particular, by bounding the number of points satisfying the Cayley-Bacharach condition, we force them to lie on uni
In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~ref{111015a} where a short li
We introduce 2-partitionable clutters as the simplest case of the class of $k$-partitionable clutters and study some of their combinatorial properties. In particular, we study properties of the rank of the incidence matrix of these clutters and prope
We answer a question of Celikbas, Dao, and Takahashi by establishing the following characterization of Gorenstein rings: a commutative noetherian local ring $(R,mathfrak m)$ is Gorenstein if and only if it admits an integrally closed $mathfrak m$-pri