Duality theory for Grothendieck categories

Non-noetherian Grothendieck Duality

For any separated map f : X → Y of quasi-compact quasiseparated schemes, Rf∗ : D + qc(X) → D (Y ) has a right adjoint f . If f is proper and pseudo-coherent (e.g., finitely-presented and flat) then Duality and tor-independent Base Change hold for f . Preface This is a research summary written early in 1991, concerning results obtained by the author during a stay at MSRI in Berkeley during 1989–...

Atomical Grothendieck Categories

Motivated by the study of Gabriel dimension of a Grothendieck category, we introduce the concept of atomical Grothendieck category, which has only two localizing subcategories, and we give a classification of this type of Grothendieck categories. 1. Introduction. Given a Grothendieck category Ꮽ, we can associate with it the lattice of all localizing categories of Ꮽ and denote it by Tors(Ꮽ). We ...

Lectures on Grothendieck Duality IV: A basic setup for duality

Let there be given, on a category C, a pair (*, *) of adjoint monoidal closed-category-valued pseudofunctors. Thus, to each object X ∈ C is associated a closed category DX , with unit object OX ; and to each C-map ψ : X → Y , adjoint monoidal functors DX ψ∗ ←−− −→ ψ∗ DY . There are also, as before, compatibilities—expressed by commutative diagrams—among adjunction, pseudofunctoriality, and mono...

Grothendieck Categories and Support Conditions

Grothendieck Categories and Support Conditions