Universal Extension In Abelian Categories
Let \(K\) be a field, \(\mathcal{C}\) a \(K\)-linear abelian category with finite dimensional ext groups.
What we call a universal extension of object \(V\) by \(W\) is either
an extension of \(V\otimes_K \mathrm{Ext}^1(V,W)\) by \(W\)
i.e. an element of \(\mathrm{Ext}^1(V\otimes_K \mathrm{Ext}^1(V,W),W)\)
or an extension of \(V\) by \(\mathrm{Ext}^1(V,W)^*\otimes_K W\)
i.e. an element of \(\mathrm{Ext}^1(V,W\otimes_K \mathrm{Ext}^1(V,W)^*)\)
all of which corresponds to the identity morphism in \(\mathrm{End}_K(\mathrm{Ext}^1(V,W))\),
\[ \mathrm{Ext}^1(V\otimes_K \mathrm{Ext}^1(V,W),W) \cong \mathrm{Ext}^1(V,\mathrm{Ext}^1(V,W)^*\otimes_K W) \cong \mathrm{End}_K(\mathrm{Ext}^1(V,W)). \]
Denote the image of identity in the first and second group \(E_1(V,W)\) and \(E_2(V,W)\) respectively.
Universal Property
The universal extension of \(V\) by \(W\) is then the universal object in \(\mathcal{C}\) such that for any extension
\[ 0 \to W \to X \to V \to 0 \]
there are commutative diagrams
and
Construction of Universal Extension
We can think of the extension
\[0\to W\to X\to V\to 0\]
as a sub-object of
\[0\to W\to E_1(V,W)\to V\otimes_K \mathrm{Ext}^1(V,W)\to 0\]
and also a quotient object of
\[0\to W\otimes_K \mathrm{Ext}^1(V,W)^*\to E_2(V,W)\to V\to 0.\]
In this way, we are naturally lead to consider taking a basis \([X_1],\dots,[X_n]\) of \(\mathrm{Ext}^1(V,W)\) and try to span all of \(\mathrm{Ext}^1(V,W)\otimes_K V\). This gives us the following diagram
where \(\alpha: \bigoplus W\otimes [X_i] \to \bigoplus X_i\otimes [X_i]\) is the sum of the inclusions,
\[ K = \ker\left(\bigoplus_{i=1}^n W\otimes[X_i]\to W\right). \]
and the rows and columns are all exact sequences by the \(9\)-lemma. This tells us that we can construct \(E_1(V,W)\) as
\[ E_1(V,W) = \frac {\bigoplus_{i=1}^n X_i\otimes [X_i]} {\alpha\ker\left(\bigoplus_{i=1}^n W\otimes[X_i]\to W\right)}. \]
Dual Construction
It is easy to imagine what we should do to construct \(E_2(V,W)\), first we draw a similar diagram
where
\[ C = \frac {\bigoplus_{i=1}^n V\otimes [X_i]} {V}. \]
Therefore we have
\[E_2(V,W) = \ker\left(\bigoplus_{i=1}^n X_i\otimes [X_i]\to \frac{\bigoplus_{i=1}^n V\otimes [X_i]}{V}\right).\]