Relation between two twisted inverse image pseudofunctors in duality theory

Grothendieck duality theory assigns to essentially-finite-type maps f of noetherian schemes a pseudofunctor f× right-adjoint to Rf∗, and a pseudofunctor f ! agreeing with f× when f is proper, but equal to the usual inverse image f∗ when f is étale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by “compactly supported” versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes. Introduction The relation in the title is given by a canonical pseudofunctorial map ψ : (−)× → (−)! between “twisted inverse image” pseudofunctors with which Grothendieck duality theory is concerned. These pseudofunctors on the category E of essentially-finite-type separated maps of noetherian schemes take values in bounded-below derived categories of complexes with quasi-coherent homology, see 1.1 and 1.2. The map ψ, derived from the pseudofunctorial “fake unit map” id→ (−)! ◦R(−)∗ of Proposition 2.1, is specified in Corollary 2.1.4. A number of concrete examples appear in §3. For instance, if f is a map in E, then ψ(f) is an isomorphism if f is proper; but if f is, say, an open immersion, so that f ! is the usual inverse image functor f∗ whereas f× is right-adjoint to Rf∗ , then ψ(f) is usually quite far from being an isomorphism (see e.g., 3.1.2, 3.1.3 and 3.3). After some preliminaries are covered in §1, the definition of the pseudofunctorial map ψ is worked out at the beginning of §2. Its good behavior with respect to flat base change is given by Proposition 2.2. The rest of Section 2 shows that under suitable “compact support” conditions, various operations from duality theory take ψ to an isomorphism. To wit: Let Dqc(X) be the derived category of OX -complexes with quasi-coherent homology, and let RHom X (−,−) be the internal hom in the closed category Dqc(X) (§1.5). Proposition 2.3.2 says: 2010 Mathematics Subject Classification Primary: 14F05, 13D09. Secondary: 13D03