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The cardinality of a minimal basis of an ideal I is denoted ν ( I ) \nu (I) . Let A be a polynomial ring in n > 0 n > 0 variables with coefficients in a noetherian (commutative with 1 ≠ 0 1 \ne 0 ) ring R, and let M be a maximal ideal of A. In general ν ( M A M ) + 1 ⩾ ν ( M ) ⩾ ν ( M A M ) \nu (M{A_M}) + 1 \geqslant \nu (M) \geqslant \nu (M{A_M}) . This paper is concerned with the attaining of equality with the lower bound. It is shown that equality is attained in each of the following cases: (1) A M {A_M} is not regular (valid even if A is not a polynomial ring), (2) M ∩ R M \cap R is maximal in R and (3) n > 1 n > 1 . Equality may fail for n = 1 n = 1 , even for R of dimension 1 (but not regular), and it is an open question whether equality holds for R regular of dimension > 1 > 1 . In case n = 1 n = 1 and dim ⁡ ( R ) = 2 \dim (R) = 2 the attaining of equality is related to questions in the K-theory of projective modules. Corollary to (1) and (2) is the confirmation, for the case of maximal ideals, of one of the Eisenbud-Evans conjectures; namely, ν ( M ) ⩽ max { ν ( M A M ) , dim ⁡ ( A ) } \nu (M) \leqslant \max \{ \nu (M{A_M}),\dim (A)\} . Corollary to (3) is that for R regular and n > 1 n > 1 , every maximal ideal of A is generated by a regular sequence—a result well known (for all n ⩾ 1 n \geqslant 1 ) if R is a field (and somewhat less well known for R a Dedekind domain).

This paper is an examination of the connection between the classical Cayley-Bacharach theorem for complete intersections in p2 and properties of graded Gorenstein algebras. Introduction. It is known, if not well known, that the Cayley-Bacharach theorem for complete intersections in p2 is valid for 0-dimensional arithmetically Gorenstein subschemes of P'. More generally, we show that the result is valid for 0-dimensional subschemes of P' having minimal Cohen-Macaulay type compatible with their Hilbert functions. The Cayley-Bacharach theorem in the Gorenstein case is a special instance of a theorem, interesting and technically useful in its own right, relating the Hilbert functions of linked subschemes of P'. Lastly we show that the 0-dimensional, arithmetically Gorenstein, reduced subschemes of Pn are characterized by the validity of the Cayley-Bacharach theorem and the symmetry of the Hilbert function. Fixed notation. A denotes a standard N-graded k-algebra, k a field: AO= k; A = k[AI]; X (A1) < ox. We use X to denote k-linear dimension, reserving "dim" for dimension of rings or schemes, and 8 to denote multiplicity for such algebras (or degree of the corresponding projective scheme). Note that 8(A) = X(A) if dim A = 0. We use I to denote a nonzero, nonunit, homogeneous ideal of A, and J = ann I ("ann" = annihilator). We assume always that A and A/I are CM (Cohen-Macaulay) and that Ass(A/I) c Ass(A). Hence Ass(A/J) c Ass(A). (Indeed, since the 0-ideal of A is unmixed of height 0, so is the annihilator of any nonzero ideal of A.) Therefore A/J is CM if dim A < 1. In any case, A/J is CM if A is Gorenstein [PS, Proposition 1.3]. In our applications A/J will be CM for one of these two reasons. Recall that, by definition, a ring R is Gorenstein provided that RI is a Gorenstein local ring for every prime idealA, of R, and A is Gorenstein