Finite complete intersection algebras and the completeness radical (Q1372642)

Let \(A\) be a ring, and let \(B\) be a finite \(A\)-algebra. If \(B\) is of the form \(A [X_1, \dots, X_n]/(f_1, \dots, f_n)\) then we say that \(B\) is a complete intersection over \(A\). We show that such an algebra is projective as an \(A\)-module and Gorenstein as an \(A\)-algebra. Under the condition that \(A\) is noetherian we show that the finite \(A\)-algebras of the form \(A[[X_1, \dots, X_m]]/(g_1, \dots, g_m)\) are exactly those complete intersections \(B\) over \(A\) for which one has \(B=A \cdot 1+ \sqrt {\text{cr} (A) \cdot B}\). Here \(\text{cr}(A)\) denote the largest ideal of \(A\) with respect to which \(A\) is complete. This ideal, which we call the completeness radical of \(A\), satisfies the usual radical axioms.