nLab
free resolution

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The homological resolution of a chain complex by a complex of free modules.

Properties

Existence

Proposition

For RR a ring, every RR-module admits a free resolution.

Examples

Proposition

Assuming the axiom of choice, then every abelian group AA admits a free resolution of just length 2, hence with trivial syzygies, hence there exists a short exact sequence of abelian groups of the form

0F 2F 1A0 0 \to F_2 \longrightarrow F_1 \longrightarrow A \to 0

where both F 1F_1 and F 2F_2 are free abelian groups.

Proof

Let F 1=[A]F_1 = \mathbb{Z}[A] be the free abelian group on the underlying set of AA, and let F 1AF_1 \to A be the canonical morphism that sends a generator to itself (the adjunction counit of the free-forgetful adjunction).

Then let F 2F 1F_2 \to F_1 be the kernel of that map, hence F 2F_2 a subgroup of F 1F_1. Assuming the axiom of choice then every subgroup of a free abelian group is itself free abelian, hence F 2F_2 is free abelian (prop.).

Remark

Prop. 2 implies that all Ext-groups between abelian groups are concentrated in degrees 0 and 1.

(By the discussion at derived functor in homological algebra.)

Example

(Koszul complex is free resolution of quotient ring)

For RR a commutative ring and (x 1,,x d)(x_1, \cdots, x_d) a regular sequence of elements, then the Koszul complex K(x 1,,x d)K(x_1, \cdots, x_d) is a free resolutions of the quotient ring R/(x 1,,x d)R/(x_1, \cdots, x_d) by free RR-modules.

References

Lecture notes include for instance

around page 5 of

Revised on September 29, 2017 05:31:41 by Urs Schreiber (188.105.77.106)