On the injectivity of blowing-up ring morphisms
Let \(R\) be a commutative ring and let \(I=(x_1,\ldots ,x_r)\) be an ideal of \(R\) minimally generated by \(r\) elements. Let \(\alpha_I :{\mathbf S}(I) \rightarrow {\mathbf R}(I)\) be the canonical graded morphism from the symmetric algebra of \(I\) onto the Rees algebra of \(I\), and let \(\alpha_{I,n}\) be its \(n\)th graded component for \(n \geq 2\). If \(\alpha_I\) is an isomorphism, meaning \(I\) is of linear type, then \(I\) is also projectively of linear type, meaning \(\text{Proj}({\mathbf R}(I)) \rightarrow \text{Proj}({\mathbf S}(I))\) is also an isomorphism. However it is known that the converse need not hold. For a fixed \(n\geq 2\), Tchernev asked for conditions under which \(\alpha_{I,n}\) being an isomorphism implies \(\alpha_{I,p}\) is an isomorphism for \(2 \leq p \leq n\). The main goal of this paper is to provide such a set of conditions. The main theorem states that if for the ideal \(J=(x_1,\ldots ,x_{r-1})\) one has that \(\alpha_{J,p} :{\mathbf S}(J) \rightarrow {\mathbf R}(J)\) is an isomorphism for all \(2 \leq p \leq n\), then \(\alpha_{I,n}\) being an isomorphism implies that \(\alpha_{I,p}\) is an isomorphism for all \(2 \leq p\leq n\). In particular, if \(J\) is of linear type, then \(I\) is of linear type if and only if \(I\) is projectively of linear type. The authors also provide an example showing that for each fixed \(n\geq 2\), there is an ideal \(I\) for which \(\alpha_{I,p}\) is an isomorphism for all \(p\geq n\), but \(\alpha_{I,n}\) is not an isomorphism.
scientific article; zbMATH DE number 5378372
scientific article