Homological properties of the module of differentials

TL;DR

This paper discusses homological properties of the module of differentials in noetherian local rings, exploring conjectures related to complete intersections and presenting known results, ideas, and conjectures in this area.

Contribution

It provides an accessible, self-contained account of key results and ideas concerning the homological conjectures of Vasconcelos, including Herzog's own conjecture on the cotangent complex.

Findings

Proves the conjecture for characteristic zero rings with rational Poincaré series

Presents strongest known results on Vasconcelos's conjectures at the time

Introduces ideas not previously published in the literature

Abstract

These notes were produced by J\"urgen 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 Poincar\'e series. Sadly J\"urgen 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.