An Improvement on the Base-Change Theorem and the Functor $$f^!$$

Abstract

AbstractIn 2009, Lipman published his polished book (Lipman in Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Mathematics, vol. 1960, Springer, Berlin, pp 1–259, 2009), the product of a decade’s work, giving the definitive, state-of-the-art treatment of Grothendieck duality. In this article, we achieve a sharp improvement: we begin by giving a new proof of the base-change theorem, which can handle unbounded complexes and work in the generality of algebraic stacks (subject to mild technical restrictions). This means that our base-change theorem must be subtle and delicate, the unbounded version is right at the boundary of known counterexamples—counterexamples (in the world of schemes) that had led the experts to believe that major parts of the theory could only be developed in the bounded-below derived category. Having proved our new base-change theorem, we then use it to define the functor $$f^!$$ f ! on the unbounded derived category and establish its functoriality properties. In Sect. 1, we will use this to clarify the relation among all the various constructions of Grothendieck duality. One illustration of the power of the new methods is that we can improve Lipman (Lecture Notes in Mathematics, vol. 1960, Springer, Berlin, pp 1–259, 2009, Theorem 4.9.4) to handle complexes that are not necessarily bounded. There are also applications to the theory developed by Avramov, Iyengar, Lipman and Nayak on the connection between Grothendieck duality and Hochschild homology and cohomology but, to keep this paper from becoming even longer, these are being relegated to separate articles. See for example (Neeman in K-Theory—Proceedings of the International Colloquium, Mumbai, 2016, Hindustan Book Agency, New Delhi, pp 91–126, 2018).