Some theorems on generation of ideals in affine algebras (Q802607)

Let A be an affine ring over a field k. Consider the following statements: (i) Every maximal ideal of A is a complete intersection. (ii) A is reduced and if \(I\subset A\) is any ideal which is a local complete intersection of height \(=\dim A\), then I is a complete intersection. (iii) If P is a rank n projective module over A, \(n=\dim A,\) then \(P\cong Q\oplus A,\) where Q is a rank (n-1)-projective module. (iv) Let M be any finitely generated module over A. Let \(\mu_ p(M)\) denote the minimal number of generators of \(M_ p\) as an \(A_ p\)-module where p is any prime ideal of A. Define \(e(M)=\max \{\mu_ p(M)+\dim A/p;\quad \dim A/p<\dim A\}.\) Then e(M)\(\geq the\) minimal number of generators of M. The author proves, among other things, that (i)\(\Leftrightarrow (ii)\Leftrightarrow (iii)\Leftrightarrow (iv)\) if k is algebraically closed.