nLab coherent cohomology

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Contents

Idea

Given a (Noetherian) ringed topos (𝒳,𝒪 X)(\mathcal{X}, \mathcal{O}_X), then a chain complex V V_\bullet of modules over the structure sheaf is said to have (quasi-)coherent cohomology if all its chain homology groups are (quasi-)coherent sheaves (coherent objects).

Properties

Over a (finite-dimensional) Noetherian scheme XX the derived category of quasi-coherent sheaves is canonically equivalent to that of sheaves with quasicoherent cohomology.

The coherent version of the statement is (SGA 6, Exp. II, Corollaire 2.2.2.1) while the quasi-coherent version is (SGA 6, Exp. II, Proposition 3.7, b)). A review appears also as (Orlov 03, prop. 1.3.2).

See also the discussion at triangulated categories of sheaves.

References

Last revised on May 30, 2014 at 09:30:18. See the history of this page for a list of all contributions to it.