nLab acyclic object

Contents

Context

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

Definition

In homological algebra:

For F:π’œβ†’β„¬F : \mathcal{A} \to \mathcal{B} a left exact additive functor between abelian categories, an object Aβˆˆπ’œA \in \mathcal{A} is FF-acyclic if the right derived functor of FF has no cohomology on AA in positive degree

(p>0)β‡’R pFA=0. (p \gt 0) \Rightarrow R^p F A = 0 \,.

Properties

A resolution by FF-acyclic objects serves to compute the derived functor of FF. See at derived functor in homological algebra – Via acyclic resolutions

References

See most references on homological algebra.

Also:

Last revised on November 4, 2023 at 07:16:08. See the history of this page for a list of all contributions to it.