On Hausdorff completions of commutative rings in rigid geometry
If \(A\) is a noetherian ring and \(I\) an ideal of \(A\), the lemma of Artin-Rees ensures that the \(I\)-adic completion \(\hat{A}\) of \(A\) satisfies many nice properties. In the article under review, the authors define several classes of non-necessarily-noetherian rings that still behave well. Let us mention only some of them. In the following, \(A\) is a ring endowed with an adic topology defined by a finitely generated ideal \(I\). The ring \(A\) is said to be pseudo-adhesive if it is noetherian outside \(I\) (i.e. Spec\((A) \setminus V(I)\) is noetherian) and if finitely generated \(A\)-modules have bounded \(I\)-torsion. If \(A\) is such a ring, there are canonical isomorphisms \(M\otimes \hat{A} \simeq \hat{M}\) for finitely generated \(A\)-modules \(M\), the completion \(\hat{A}\) is flat over \(A\) and a local criterion of flatness holds in the following form. Let \(B\) be a pseudo-adhesive \(A\)-algebra with respect to \(IB\) and assume that \(1+IB\) sits inside the Jacobson radical of \(B\). Then, a finitely generated \(B\)-module \(M\) is flat over \(A\) if and only if \(M/I^kM\) is flat over \(A/I^k\) for any \(k\). Examples of pseudo-adhesive rings are rigid-noetherian rings: adic (i.e. Hausdorff complete) rings that are noetherian outside \(I\). Those rings enjoy additional nice properties: finitely generated modules over them are complete and their submodules are closed. The authors also introduce refined versions of their conditions. A ring \(A\) is said to be adhesive if it is noetherian outside \(I\) and if finitely generated \(A\)-modules have finitely generated \(I\)-torsion. Then come a universal version (polynomial rings over \(A\) are adhesive) and a topologically universal version (restricted power series rings over \(A\) are universally adhesive). In the last case, if \(A\) is also assumed to be \(I\)-torsion free, then, for any topologically finitely presented \(A\)-algebra \(B\), the structure sheaf of Spf\((B)\) is coherent. As an example, the authors prove a valuation ring \(V\) of non-zero height which is Haudorff and complete with respect to the \(a\)-adic topology defined by some \(a\in\mathfrak{m}_{V}\setminus\{0\}\) is topologically universally adhesive.
scientific article; zbMATH DE number 5969529