Homological properties of the module of differentials
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Jürgen Herzog
Abstract
These notes were produced by Jürgen Herzog to accompany his lectures in Recife, Brazil, in 1980, on the homological algebra of Noetherian local rings. They are concerned with two conjectures made by Wolmer Vasconcelos: if the conormal module of a local ring has finite projective dimension, or if the module of differentials, taken over an appropriate field, has finite projective dimension, then the ring must be a complete intersection. The notes present an accessible and self-contained account of the strongest results known at the time in connection with these problems; this includes a number of ideas that have not appeared elsewhere. In the last section, Herzog turns his attention to the cotangent complex, and conjectures himself that if the cotangent complex of a local ring has bounded homology groups, then the ring must be complete intersection. Among other results, he proves that the conjecture holds for local rings of characteristic zero over which all modules have rational Poincaré series.
Sadly, Jürgen Herzog passed away in April of 2024. The notes in this form have been prepared in his memory, newly typeset and lightly edited. A short appendix has been added to survey some of the results of the intervening decades.
Abstract
These notes were produced by Jürgen Herzog to accompany his lectures in Recife, Brazil, in 1980, on the homological algebra of Noetherian local rings. They are concerned with two conjectures made by Wolmer Vasconcelos: if the conormal module of a local ring has finite projective dimension, or if the module of differentials, taken over an appropriate field, has finite projective dimension, then the ring must be a complete intersection. The notes present an accessible and self-contained account of the strongest results known at the time in connection with these problems; this includes a number of ideas that have not appeared elsewhere. In the last section, Herzog turns his attention to the cotangent complex, and conjectures himself that if the cotangent complex of a local ring has bounded homology groups, then the ring must be complete intersection. Among other results, he proves that the conjecture holds for local rings of characteristic zero over which all modules have rational Poincaré series.
Sadly, Jürgen Herzog passed away in April of 2024. The notes in this form have been prepared in his memory, newly typeset and lightly edited. A short appendix has been added to survey some of the results of the intervening decades.
Chapters in this book
- 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
Chapters in this book
- 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
