PlanetMath: Ext

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Ext

(Definition)

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

$\mathrm{Ext}$ gets its name from the existence of a natural bijection between elements of $\mathrm{Ext}^1_R(A,B)$ and extensions of by up to isomorphism of short exact sequences, where an extension of by is an exact sequence

$\displaystyle 0\to B\to C\to A\to 0.$

For example,

$\displaystyle \mathrm{Ext}^1_\mathbb{Z}(\mathbb{Z}/n\mathbb{Z},\mathbb{Z})\cong\mathbb{Z}/n\mathbb{Z},$

with 0 corresponding to the trivial extension $0\to\mathbb{Z}\to\mathbb{Z}\oplus\mathbb{Z}/n\mathbb{Z}\to 0$ , 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 $\mathrm{Ext}$ functors was given by S. Mac Lane, namely that there is a correspondence between $\mathrm{Ext}^n(A,B)$ 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) ]

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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 2382 times total.

Classification:

AMS MSC:

18G15 (Category theory; homological algebra :: Homological algebra :: Ext and Tor, generalizations, Künneth formula)

Pending Errata and Addenda

None.[ View all 4 ]

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