PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
Ext (Definition)

For a ring $R$ , and $R$ -module $A$ , we have a covariant functor $\Hom A-R$ . The functions $\Ext^n_R(A,-)$ are defined to be the right derived functors of $\Hom A-R: \Ext^n_R(A,-)=R^n\Hom A-R$ .

$\Ext$ gets its name from the existence of a natural bijection between elements of $\Ext^1_R(A,B)$ and extensions of $B$ by $A$ up to isomorphism of short exact sequences, where an extension of $B$ by $A$ is an exact sequence $$0\to B\to C\to A\to 0.$$ For example, $$\Ext^1_\Z(\Z/n\Z,\Z)\cong\Z/n\Z,$$ with $0$ corresponding to the trivial extension $0\to\Z\to\Z\oplus\Z/n\Z\to 0$ , and $m\neq 0$ corresponding to

$\displaystyle \xymatrix{0 \ar [r] & \mathbb{Z}\ar [r]^n & \mathbb{Z}\ar [r]^m & \mathbb{Z}/n\mathbb{Z}\ar [r]&0}.$

A more modern interpretation of the $\Ext$ functors was given by S. Mac Lane, namely that there is a correspondence between $\Ext^n(A,B)$ with equivalence classes of exact sequences

$\displaystyle 0\rightarrow B\rightarrow C_n\rightarrow C_{n-1}\rightarrow\cdots\rightarrow C_1\rightarrow A\rightarrow 0$    




"Ext" is owned by mathcam. [ full author list (2) | owner history (1) ]
(view preamble | get metadata)

View style:

Log in to rate this entry.
(view current ratings)

Cross-references: equivalence classes, interpretation, exact sequence, short exact sequences, isomorphism, bijection, derived functors, right, functions, covariant functor, ring
There are 2 references to this entry.

This is version 6 of Ext, born on 2003-08-14, modified 2007-01-11.
Object id is 4588, canonical name is Ext.
Accessed 2849 times total.

Classification:
AMS MSC18G15 (Category theory; homological algebra :: Homological algebra :: Ext and Tor, generalizations, Künneth formula)

Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)