nLab rank of a coherent sheaf

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

As the concept of coherent sheaf is a generalization of that of (complex, algebraic vector bundle) vector bundle, the concept of rank of a coherent sheaf is the corresponding generalization of that of rank of a vector bundle.

Definition

e.g. (Huybrechts-Lehn 96, def. 1.2.2)

References

This appeared in the Grothendieck’s FGA in order to treat the moduli problems leading to Quot schemes. See Nitsure’s contribution at FGA explained for a modern remake.

Last revised on October 3, 2018 at 14:20:46. See the history of this page for a list of all contributions to it.