The relation between Grothendieck duality and Hochschild homology

Grothendieck duality goes back to 1958, to the talk at the ICM in Edin- burgh [13] announcing the result. Hochschild homology is even older, its roots can be traced back to the 1945 article [16]. The fact that the two might be related is relatively recent. The first hint of a relationship came in 1987 in Lipman [20], and another was found in 1997 in Van den Bergh [28]. Each of these discoveries was interesting and had an impact, Lipman’s mostly by giving another approach to the computations and Van den Bergh’s especially on the development of non-commutative versions of the subject. However in this survey we will almost entirely focus on a third, much more recent connection, discovered in 2008 by Avramov and Iyengar [2] and later developed and extended in several papers, see for example [3, 19]. There are two classical paths to the foundations of Grothendieck duality, one fol- lowing Grothendieck and Hartshorne [15] and (much later) Conrad [10], and the other following Deligne [11], Verdier [29] and (much later) Lipman [21]. The accepted view is that each of these has its drawbacks: the first approach (of Grothendieck, Hartshorne and Conrad) is complicated and messy to set up, while the second (of Deligne, Verdier and Lipman) might be cleaner to present but leads to a theory where it’s not obvious how to compute anything. The point of this article is that the recently-discovered connection with Hochschild homology and cohomology (the one due to Avramov and Iyengar) changes this. It renders clearly superior the highbrow appoach to the subject, the one due to Deligne, Verdier and Lipman. Not only is it (relatively) easy to set up the machinery, the computations also become transparent. And in the process we learn that Grothendieck duality is not really about residues of meromorphic differential forms, it is about the local cohomology of the Hochschild homology. By a fortuitous accident, if f : X −→ Y is a smooth map then the top Hochschild homology happens to be isomorphic to the relative canonical bundle, and its top local cohomology is represented by meromorphic differential forms. This is the reason that, as long as we stick to smooth maps, what comes up is residues of meromorphic forms. For non-smooth, flat maps it’s Hochschild homology and maps from it that we need to study.