group theory

# Contents

## Idea

The Baer sum is the natural addition operation on abelian group extensions as well on the extensions of $R$-modules, for fixed ring $R$.

For $G$ a group and $A$ an abelian group, the extensions of $G$ by $A$ are classified by the degree-2 group cohomology

${H}_{\mathrm{Grp}}^{2}\left(G,A\right)={H}^{2}\left(BG,A\right)=H\left(BG,{B}^{2}A\right)\phantom{\rule{thinmathspace}{0ex}}.$H^2_{Grp}(G,A) = H^2(\mathbf{B}G, A) = H(\mathbf{B}G, \mathbf{B}^2 A) \,.

On cocycles $BG\to {B}^{2}A$ there is a canonical addition operation coming from the additive structure of $A$, and the Baer sum is the corresponding operation on the extensions that these cocycles classify.

## Definition

Below are discussed several different equivalent ways to define the Baer sum

### On concrete cocycles

A cocycle in degree-2 group cohomology ${H}_{\mathrm{Grp}}^{2}\left(G,A\right)$ is a function

$c:G×G\to A$c : G \times G \to A

satisfying the cocycle property.

###### Definition

Given two coycles ${c}_{1},{c}_{2}:G×G\to A$ their sum is the composite

$\left({c}_{1}+{c}_{2}\right):G×G\stackrel{{\Delta }_{G×G}}{\to }\left(G×G\right)×\left(G×G\right)\stackrel{\left({c}_{1},{c}_{2}\right)}{\to }A×A\stackrel{+}{\to }A$(c_1 + c_2) : G \times G \stackrel{\Delta_{G \times G}}{\to} (G \times G) \times (G \times G) \stackrel{(c_1,c_2)}{\to} A \times A \stackrel{+}{\to} A

of

• the diagonal on $G×G$;

• the direct product $\left(f,g\right)$;

• the group operation $+:A×A\to A$.

Hence for all ${g}_{1},{g}_{2}\in G$ this sum is the function that sends

$\left({c}_{1}+{c}_{2}\right):\left({g}_{1},{g}_{2}\right)↦{c}_{1}\left({g}_{1},{g}_{2}\right)+{c}_{2}\left({g}_{1},{g}_{2}\right)$(c_1 + c_2) : (g_1, g_2) \mapsto c_1(g_1,g_2) + c_2(g_1, g_2)

### On abstract cocycles

As discussed at group cohomology, a cocycle $c:G×G\to A$ is equivalently a morphism of 2-groupoids from the delooping groupoid $BG$ of $G$ to the double-delooping 2-groupoid ${B}^{2}A$ of $A$:

${c}_{1},{c}_{2}:BG\to {B}^{2}A\phantom{\rule{thinmathspace}{0ex}}.$c_1,c_2 : \mathbf{B}G \to \mathbf{B}^2 A \,.

Since $A$ is an abelian group, $matbf{B}^{2}A$ is naturally an abelian 3-group, equipped with a group operation $+:\left({B}^{2}A\right)×\left({B}^{A}\right)\to {B}^{2}A$.

With respect to this the sum operation is

${c}_{1}+{c}_{2}:BG\stackrel{{\Delta }_{BG}}{\to }BG×BG\stackrel{\left({c}_{1},{c}_{2}\right)}{\to }{B}^{2}A×{B}^{2}A\stackrel{+}{\to }{B}^{2}A$c_1 + c_2 : \mathbf{B}G \stackrel{\Delta_{\mathbf{B}G}}{\to} \mathbf{B}G \times \mathbf{B}G \stackrel{(c_1,c_2)}{\to} \mathbf{B}^2 A \times \mathbf{B}^2 A \stackrel{+}{\to} \mathbf{B}^2 A

### On short exact sequences

In any category with products, for any object $C$ there is a diagonal morphism ${\Delta }_{C}:C\to C×C$; in a category with coproducts there is a codiagonal morphism ${\nabla }_{C}:C\coprod C\to C$ (addition in the case of modules). Every additive category is, in particular, a category with finite biproducts, so both morphisms are there. Short exact sequences in the category of $R$-modules, or in arbitrary abelian category $𝒜$, form an additive category (morphisms are commutative ladders of arrows) in which the biproduct $0\to {A}_{i}\to {\stackrel{^}{H}}_{i}\to {G}_{i}\to 0$ for $i=1,2$ is $0\to {A}_{1}\oplus {A}_{2}\to {H}_{1}\oplus {H}_{2}\to {G}_{1}\oplus {G}_{2}\to 0$.

Now if $0\to M\to N\to P\to 0$ is any extension, call it $E$, and $\gamma :{P}_{1}\to P$ a morphism, then there is a morphism $\Gamma \prime =\left({\mathrm{id}}_{M},{\beta }_{1},\gamma \right)$ from an extension ${E}_{1}$ of the form $0\to M\to {N}_{1}\to {P}_{1}\to 0$ to $E$, where the pair $\left({E}_{1},{\Gamma }_{1}\right)$ s unique up to isomorphism of extensions, and it is called $E\gamma$. In fact, the diagram

$\begin{array}{ccc}{N}_{1}& \to & {P}_{1}\\ ↓{\beta }_{1}& & ↓\gamma \\ N& \to & P\end{array}$\array{ N_1&\to &P_1\\ \downarrow\beta_1 && \downarrow\gamma\\ N&\to &P }

is a pullback diagram. Every morphism of abelian extensions $\left(\alpha ,\beta ,\gamma \right):E\to E\prime$ in a unique way decomposes as

$E\stackrel{\left(\alpha ,{\beta }_{a},\mathrm{id}\right)}{⟶}\alpha E\gamma \stackrel{\left(\mathrm{id},{\beta }_{1},\gamma \right)}{⟶}E\prime$E\stackrel{(\alpha,\beta_a,id)}\longrightarrow \alpha E\gamma \stackrel{(id,\beta_ 1,\gamma)}\longrightarrow E'

for some ${\beta }_{a}$ with ${\beta }_{1}$ as above. In short, the morphism of extensions factorizes through $E\gamma$.

Dually, for any morphism $\alpha :M\to {M}_{2}$, there is a morphism ${\Gamma }_{2}=\left(\alpha ,{\beta }_{2},{\mathrm{id}}_{P}\right)$ to an extension ${E}_{2}$ of the form $0\to {M}_{2}\to {N}_{2}\to P$; the pair $\left({E}_{2},{\Gamma }_{2}\right)$ is unique up to isomorphism of extensions and it is called $\alpha E$.

In fact, the diagram

$\begin{array}{ccc}M& \to & N\\ ↓\alpha & & ↓{\beta }_{2}\\ {M}_{2}& \to & {N}_{2}\end{array}$\array{ M&\to &N\\ \downarrow\alpha && \downarrow\beta_2\\ M_2&\to &N_2 }

is a pushout diagram. Every morphism of abelian extensions $\left(\alpha ,\beta ,\gamma \right):E\to E″$ in a unique way decomposes as

$E\stackrel{\left(\alpha ,{\beta }_{a},\mathrm{id}\right)}{⟶}\alpha E\stackrel{\left(\mathrm{id},{\beta }_{2},\gamma \right)}{⟶}E″$E\stackrel{(\alpha,\beta_a,id)}\longrightarrow \alpha E \stackrel{(id,\beta_ 2,\gamma)}\longrightarrow E''

for some ${\beta }_{a}$, with ${\beta }_{2}$ as above. In short, the morphism of extensions factorizes through $\alpha E$.

There are the following isomorphisms of extensions: $\left(\alpha E\right)\gamma \cong \alpha \left(E\gamma \right)$, ${\mathrm{id}}_{M}E\cong E$, $E{\mathrm{id}}_{P}\cong P$, $\left(\alpha \prime \alpha \right)E\cong \alpha \prime \left(\alpha E\right)$, $\left(E\gamma \right)\gamma \prime \cong E\left(\gamma \gamma \prime \right)$.

The Baer’s sum of two extensions ${E}_{1},{E}_{2}$ of the form $0\to M\to {N}_{i}\to P\to 0$ (i.e. with the same $M$ and $P$) is given by ${E}_{1}+{E}_{2}={\nabla }_{M}\left({E}_{1}\oplus {E}_{2}\right){\Delta }_{P}$; this gives the structure of the abelian group on $\mathrm{Ext}\left(P,M\right)$ and $\mathrm{Ext}:{𝒜}^{\mathrm{op}}×𝒜\to \mathrm{Ab}$ is a biadditive (bi)functor. This is also related to the isomorphisms of extensions $\alpha \left({E}_{1}+{E}_{2}\right)\cong \alpha {E}_{1}+\alpha {E}_{2}$, $\left({\alpha }_{1}+{\alpha }_{2}\right)E\cong {\alpha }_{1}E+{\alpha }_{2}E$, $\left({E}_{1}+{E}_{2}\right)\gamma \cong {E}_{1}\gamma +{E}_{2}\gamma$, $E\left({\gamma }_{1}+{\gamma }_{2}\right)\cong E{\gamma }_{1}+E{\gamma }_{2}$.

In different notation, if $0\to A\to {\stackrel{^}{G}}_{i}\to G\to 0$ for $i=1,2$ are two short exact sequences of abelian groups, their Baer sum is

${\stackrel{^}{G}}_{1}+{\stackrel{^}{G}}_{2}≔{+}_{*}{\Delta }^{*}{\stackrel{^}{G}}_{1}×{\stackrel{^}{G}}_{2}$\hat G_1 + \hat G_2 \coloneqq +_* \Delta^* \hat G_1 \times \hat G_2

The first step forms the pullback of the short exact sequence along rhe diagonal on $G$:

$\begin{array}{ccc}A\oplus A& \to & A\oplus A\\ ↓& & ↓\\ {\Delta }^{*}\left({\stackrel{^}{G}}_{1}\oplus {\stackrel{^}{G}}_{2}\right)& \to & {\stackrel{^}{G}}_{1}\oplus {\stackrel{^}{G}}_{2}\\ ↓& & ↓\\ G& \stackrel{{\Delta }_{G}}{\to }& G\oplus G\end{array}$\array{ A \oplus A &\to& A \oplus A \\ \downarrow && \downarrow \\ \Delta^* (\hat G_1 \oplus \hat G_2) &\to& \hat G_1 \oplus \hat G_2 \\ \downarrow && \downarrow \\ G &\stackrel{\Delta_G}{\to}& G\oplus G }

The second forms the pushout along the addition map on $A$:

$\begin{array}{ccc}A\oplus A& \stackrel{+}{\to }& A\\ ↓& & ↓\\ {\Delta }^{*}\left({\stackrel{^}{G}}_{1}\oplus {\stackrel{^}{G}}_{2}\right)& \to & {+}_{*}{\Delta }^{*}\left({\stackrel{^}{G}}_{1}\oplus {\stackrel{^}{G}}_{2}\right)\\ ↓& & ↓\\ G& \to & G\end{array}$\array{ A \oplus A &\stackrel{+}{\to}& A \\ \downarrow && \downarrow \\ \Delta^* (\hat G_1 \oplus \hat G_2) &\to& +_* \Delta^*(\hat G_1 \oplus \hat G_2) \\ \downarrow && \downarrow \\ G &\to& G }

## References

• S. MacLane, Homology, 1963

• Patrick Morandi, Ext groups and Ext functors (pdf)

Revised on June 24, 2013 20:26:59 by Zoran Škoda (95.168.102.197)