Local cohomology modules of a smooth \(\mathbb{Z}\)-algebra have finitely many associated primes
Let \(R\) be a Noetherian ring and \(\mathfrak a\) an ideal. The local cohomology module \(H_{\mathfrak a}^k(R)\) has infinitely many associated primes in general. However, it is known that \[ \# \text{Ass}_R H_{\mathfrak a}^k(R) < \infty \tag{1} \] if \(R\) is a regular local ring of equicharacteristic of if \(R\) is an unramified regular local ring of mixed characteristic. In the present paper, authors want to generalize this result. Assume \(R\) is a smooth algebra over a ring \(A\). If \(A\) is a field, then \(R\) is regular and hence (1) is fulfilled. The main theorem of this paper says that (1) is fulfilled if \(A = \mathbb{Z}\) or if \(A\) is a Dedekind domain such that \(A_{\mathfrak p}\) has mixed characteristic for any maximal ideal \(\mathfrak p\) of \(A\).
scientific article; zbMATH DE number 6358025
scientific article