Homological properties of the module of differentials
Jürgen Herzog, Benjamin Briggs, Srikanth B. Iyengar
TL;DR
This work surveys the homological properties of the conormal module $\,I/I^2\,$, the module of differentials $\,\Omega_{S/k}\,$, and the cotangent complex for noetherian local rings, focusing on Vasconcelos' conjectures that finite homological dimension forces a complete intersection. It develops multiple approaches, including linkage, duality via canonical modules, and the cotangent functors $T_i$, to detect when an ideal $I$ yields a locally complete intersection in $R$, and to relate these properties to the structure of $S=R/I$ (e.g., quasi-Gorenstein, Gorenstein). A central theme is Platte's theorem, which connects finite $S$-pd of $I/I^2$ to quasi-Gorensteinness, and the subsequent implications for complete intersections under equidimensional and depth conditions; the cotangent framework further links obstructions to deformations with the LCI property. The notes also discuss the role of Poincaré series and deviations in rationality questions, and summarize recent developments showing progress toward the conjectures, including criteria that force regular sequences under various hypotheses.
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 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 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.