Structure theorems for Gorenstein ideals of codimension four with small number of generators
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Tymoteusz Chmiel
Abstract
In this article we study minimal free resolutions of Gorenstein ideals of codimension four, using methods coming from representation theory. We introduce families of higher structure maps associated with such resolution, defined similarly to the codimension three case. As our main application, we prove that every Gorenstein ideal of codimension four minimally generated by six elements is a hyperplane section of a Gorenstein ideal of codimension three, strengthening a result by Herzog–Miller and Vasconcelos–Villarreal. We state analogous conjectural results for ideals minimally generated by seven and eight elements.
Abstract
In this article we study minimal free resolutions of Gorenstein ideals of codimension four, using methods coming from representation theory. We introduce families of higher structure maps associated with such resolution, defined similarly to the codimension three case. As our main application, we prove that every Gorenstein ideal of codimension four minimally generated by six elements is a hyperplane section of a Gorenstein ideal of codimension three, strengthening a result by Herzog–Miller and Vasconcelos–Villarreal. We state analogous conjectural results for ideals minimally generated by seven and eight elements.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- Wolmer Vasconcelos, a tribute IX
- Vasconcelos Bibliography (1964–2024) XVII
- The early years: flatness, projectivity, and R-sequences 1
- Joint work with Wolmer Vasconcelos. The early years 41
- Homological properties of the module of differentials 71
- Hilbert coefficients and Sally modules: a survey of Vasconcelos’ contributions 97
- Graph rings and ideals: Wolmer Vasconcelos’ contributions 133
- Degrees: Vasconcelos contributions 203
- The books of Wolmer V. Vasconcelos 241
- Finding nilpotents with Wolmer Vasconcelos 261
- Rees algebras over residual intersections 279
- Structure theorems for Gorenstein ideals of codimension four with small number of generators 297
- Sifted degrees of the equations of the Rees module and their connection with the Artin–Rees numbers 333
- Reembeddings of special border basis schemes 353
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- Rings with q-torsionfree canonical modules 405
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- Wolmer Vasconcelos, a tribute IX
- Vasconcelos Bibliography (1964–2024) XVII
- The early years: flatness, projectivity, and R-sequences 1
- Joint work with Wolmer Vasconcelos. The early years 41
- Homological properties of the module of differentials 71
- Hilbert coefficients and Sally modules: a survey of Vasconcelos’ contributions 97
- Graph rings and ideals: Wolmer Vasconcelos’ contributions 133
- Degrees: Vasconcelos contributions 203
- The books of Wolmer V. Vasconcelos 241
- Finding nilpotents with Wolmer Vasconcelos 261
- Rees algebras over residual intersections 279
- Structure theorems for Gorenstein ideals of codimension four with small number of generators 297
- Sifted degrees of the equations of the Rees module and their connection with the Artin–Rees numbers 333
- Reembeddings of special border basis schemes 353
-
- Rings with q-torsionfree canonical modules 405
