w x In 8 , K. Morita introduced a useful notion of duality between categories of modules, usually called ‘‘Morita duality.’’ He proved that every duality is given by contravariant hom functors defined by a bimodule which is an injective cogenerator for both categories of modules. On the other hand, the equivalences between categories of comodules over a coalgebra w x were characterized by M. Ta...

In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is Hecke algebra a reductive $p$-adic group, to case it finite central extension group. The most important being that concentrated in single degree for irreducible representations and gives rise Schneider--Stuhler Ext groups (a Serre like property). A...

Let f : X → Y be a Cohen-Macaulay map of finite type between Noetherian schemes, and g : Y ′ → Y a base change map, with Y ′ Noetherian. Let f ′ : X → Y ′ be the base change of f under g and g : X → X the base change of g under f . We show that there is a canonical isomorphism θ g : g ∗ ωf ≃ ωf ′ , where ωf and ωf ′ are the relative dualizing sheaves. The map θ g f is easily described when f is...

Grothendieck proved that if f : X −→ Y is a proper morphism ofnice schemes, then Rf∗ has a right adjoint, which is given as tensor productwith the relative canonical bundle. The original proof was by patching localdata. Deligne proved the existence of the adjoint by a global argument, andVerdier showed that this global adjoint may be computed locally.In this article we show ...

These notes are an account of a series lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic Diophantine Aspects', in July 2018, Imperial College London. The goal these is to see how motives may be used enhance cohomological methods, giving natural ways prove independence $\ell$ results for traces zeta-functions, constructions characteristic classes (a...

Abstract We characterize cyclic algebras over the associative and framed little 2-disks operad in any symmetric monoidal bicategory. The cyclicity is appropriately treated a coherent way, that up to isomorphism. When bicategory specified be certain of linear categories subject finiteness conditions, we prove algebras, respectively, are equivalent pivotal Grothendieck–Verdier ribbon categories, ...

In what follows, “the book” refers to Grothendieck duality and base change in its original form. The online version incorporates some of the minor changes below into the text (e.g., typos), but the more detailed discussions of certain points are not included in the on-line revision of the text (as this would have seriously disrupted the page numbering). All theorems, equations, page numbers, et...