On the injectivity of blowing-up ring morphisms
type: Item
instance of: scholarly article
title: On the injectivity of blowing-up ring morphisms
zbMATH Open document ID: 1153.13002
author: Ferran Muiños
author: Francesc Planas-Vilanova
published in: Journal of Algebra
publication date: 2008-12-05T00:00:00Z
review text: 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.
reviewed by: Susan Morey
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13D02
zbMATH DE Number: 5378372
zbMATH Keywords: symmetric algebra
zbMATH Keywords: Rees algebra
zbMATH Keywords: linear type
describes a project that uses: SINGULAR
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.07.020
OpenAlex ID: W2037628503
cites work: Hilbert function of powers of ideals of low codimension
cites work: On the torsion-freeness of the symmetric powers of an ideal
cites work: Sequences of linear type
cites work: Q3203119
cites work: Q3964688
cites work: On the symmetric and Rees algebra of an ideal generated by a d-sequence
cites work: Determinantal ideals of linear type
cites work: On equations defining Veronese rings
cites work: Intersection Theory of Linear Embeddings
cites work: On the symmetric algebra of an ideal
cites work: Q3272531
cites work: On the unboundedness of generators of prime ideals in powerseries rings of three variables
cites work: On the module of effective relations of a standard algebra
cites work: Torsion freeness of symmetric powers of ideals
cites work: On the symmetric and Rees algebras of an ideal
cites work: Q4293985
cites work: Q4680541
cites work: Rees algebras of edge ideals
Wikidata QID: Q122222110
DOI: 10.1016/J.JALGEBRA.2008.07.020
description: scientific article; zbMATH DE number 5378372