New progress on Grothendieck duality, explained to those familiar with category theory and with algebraic geometry

Abstract

Much has been written about Grothendieck duality. This survey will make the point that most of this literature is now obsolete: there is a brilliant 1968 article by Verdier with the right idea on how to approach the subject. Verdier's article was largely forgotten for two decades until Lipman resurrected it, reworked it and developed the ideas to obtain the right statements for what had before been a complicated theory.

For the reader's benefit, Sections 1 through 5, which present the (large) portion of the theory that can nowadays be obtained from formal nonsense about rigidly compactly generated tensor triangulated categories, are all post‐Verdier. The major landmarks in developing this approach were a 1996 article by me which was later generalized and improved on by Balmer, Dell'Ambrogio and Sanders, and a much more recent article of mine about improvements to the Verdier base‐change theorem and the functor 𝑓! . Section 6 is where Verdier's 1968 ideas still play a pivotal role, but in the cleaned‐up version due to Lipman and with new, short and direct proofs.