Select any concept, construction junction, or annotated relation by pointer, touch, search, or keyboard.

Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Ext functor

Open this concept in the interactive graph Read the Markdown equivalent

Summary

The right-derived functors $\operatorname{Ext}^n_R(M,N)$ of $\operatorname{Hom}_R(M,-)$ , equivalently derived-category morphisms $M\to N[n]$ under standard hypotheses.

Record metadata

Canonical name: Ext functor
Concept ID: ext_functor
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Homological algebra
Domains: Homological algebra, Algebra, Category theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Derived functor

Carrier(s)

modules or objects of an abelian category

Data

graded groups $\operatorname{Ext}^n$

Axioms / constraints

derived-functor construction or equivalent Yoneda construction

Canonically induces

long exact Ext sequences
classification of extensions in degree one

Notes

Variance and the choice of which Hom argument is derived must be stated; module categories permit computation by projective or injective resolutions.

Concept sources

Wikipedia — Ext functor — Ext functor · encyclopedic reference · source ID wp-ext-functor
nLab — Ext — Ext · mathematical reference · source ID nlab-ext-functor
Stacks Project — Ext groups — Ext groups · mathematical reference · source ID stacks-ext-groups

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Derived category Ext functor

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: derived_category_to_ext
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: shifted derived morphisms

Authored explanation

In the derived category of an abelian category, $\operatorname{Ext}^n(A,B)$ is represented by morphisms $A\to B[n]$ .

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Ext functor — Ext functor · encyclopedic reference · source ID wp-ext-functor
nLab — Ext — Ext · mathematical reference · source ID nlab-ext-functor
Stacks Project — Ext groups — Ext groups · mathematical reference · source ID stacks-ext-groups

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

Atlas of Fundamental Concepts

Select a concept

Move selected concept

Stories & Views

Compare concepts

Guide to the Atlas