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Ext (Definition)

For a ring $ R$, and $ R$-module $ A$, 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 $ B$ by $ A$ up to isomorphism of short exact sequences, where an extension of $ B$ by $ A$ 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
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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 MSC18G15 (Category theory; homological algebra :: Homological algebra :: Ext and Tor, generalizations, Künneth formula)

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