Author: Eiko

Tags: connection, homological algebra

Time: 2024-10-31 06:23:09 - 2025-01-15 17:10:09 (UTC)

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 ${\mathrm{Ext}}^1(V,W{)}^{\ast }{\otimes }_{K}W$

  i.e. an element of ${\mathrm{Ext}}^1(V,W{\otimes }_K{\mathrm{Ext}}^1(V,W{)}^{\ast })$

all of which corresponds to the identity morphism in ${\mathrm{End}}_K({\mathrm{Ext}}^1(V,W))$,

$${\mathrm{Ext}}^1(V{\otimes }_K{\mathrm{Ext}}^1(V,W),W)\cong {\mathrm{Ext}}^1(V,{\mathrm{Ext}}^1(V,W{)}^{\ast }{\otimes }_{K}W)\cong {\mathrm{End}}_K({\mathrm{Ext}}^1(V,W)).$$

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

$$0\to W{\otimes }_K{\mathrm{Ext}}^1(V,W{)}^{\ast }\to E_2(V,W)\to V\to 0.$$

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 $\alpha :⨁W\otimes [X_i]\to ⨁X_i\otimes [X_i]$ is the sum of the inclusions,

$$K=\ker (\underset{i=1}{\overset{n}{⨁}}W\otimes [X_i]\to W).$$

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

$$E_1(V,W)=\frac{\underset{i=1}{\overset{n}{⨁}}{X}_i\otimes [X_i]}{\alpha \ker (\underset{i=1}{\overset{n}{⨁}}W\otimes [X_i]\to W)}.$$

Dual Construction

It is easy to imagine what we should do to construct $E_2(V,W)$, first we draw a similar diagram

where

$$C=\frac{\underset{i=1}{\overset{n}{⨁}}V\otimes [X_i]}{V}.$$

Therefore we have

$$E_2(V,W)=\ker (\underset{i=1}{\overset{n}{⨁}}{X}_i\otimes [X_i]\to \frac{\underset{i=1}{\overset{n}{⨁}}V\otimes [X_i]}{V}).$$