Homology and \(K\)-theory of commutative algebras: Characterization of complete intersections (Q1891763)

Let \(A = k[X_1, \ldots, X_n]/I\), where \(k\) is a field of characteristic zero and the \(X_i\) are indeterminates. Let \(HH_n (A) = \bigoplus_{0 \leq p \leq n} HH^p_n(A)\) and \(HC_n (A) = \bigoplus_{0 \leq p \leq n} HC^p_n (A)\) be respectively the Hodge decompositions of Hochschild homology and cyclic homology. The ring \(A\) is called a complete intersection if \(I\) is generated by a regular sequence of \(R\), and is called a local complete intersection if \(I_{\mathfrak p}\) is generated by a regular sequence \(R_{\mathfrak p}\) for every prime ideal \({\mathfrak p}\) of \(R\). Then it is proved (theorem 1) that \(A\) is a local complete intersection if there exists an integer \(N\) such that either \(HH^p_n (A) = 0\) for \(n > N\) and \(0 < p < n/2\) or \(HC^p_n (A) = 0\) for \(n > N\) and \(0 < p < n/2\). The converse (with \(N = 0)\) of this result is well known and has been obtained by a number of people [including the present author: \(K\)-Theory 4, No. 5, 399-410 (1991; Zbl 0731.19004)]. If in addition the \(X_i\) are of positive degree and \(I\) is homogeneous then \(I\) is a complete intersection (theorem \(1'\)). These results are applied to \(K\)-theory, one variant (corollary 3) of the result being that if \(A\) is an artinian \(k\)-algebra such that for all maximal ideals \(m\) of \(k\) the residue field of \(A\) at \(m\) is isomorphic to \(k\) \((k\) a number field) then \(A\) is a local complete intersection if and only if \(K^i_n (A) = 0\) for \(n \geq 1\) and \(i < (n + 1)/2\) (where for Quillen \(K\)-theory \(K_*(A)\) and the rational numbers \(\mathbb{Q}\), \(K_n (A) \bigotimes \mathbb{Q} = \bigoplus_{i \geq 0} K^i_n (A)\) is the decomposition induced by the Adams operations). This proves a conjecture of Beilinson and Soulé in the case of artinian algebras over a number field.