# nLab Baer sum

Contents

### Context

#### Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

# 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

$H^2_{Grp}(G,A) = H^2(\mathbf{B}G, A) = H(\mathbf{B}G, \mathbf{B}^2 A) \,.$

On cocycles $\mathbf{B}G \to \mathbf{B}^2 A$ 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 $H^2_{Grp}(G,A)$ is a function

$c : G \times G \to A$

satisfying the cocycle property.

###### Definition

Given two cocycles $c_1, c_2 : G \times G \to A$ their sum is the composite

$(c_1 + c_2) : G \times G \stackrel{\Delta_{G \times G}}{\to} (G \times G) \times (G \times G) \stackrel{(c_1,c_2)}{\to} A \times A \stackrel{+}{\to} A$

of

• the diagonal on $G\times G$;

• the direct product $(c_1,c_2)$;

• the group operation $+ \colon A \times A \to A$.

Hence for all $g_1, g_2 \in G$ this sum is the function that sends

$(c_1 + c_2) : (g_1, g_2) \mapsto c_1(g_1,g_2) + c_2(g_1, g_2)$

### On abstract cocycles

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

$c_1,c_2 : \mathbf{B}G \to \mathbf{B}^2 A \,.$

Since $A$ is an abelian group, $\mathbf{B}^2 A$ is naturally an abelian 3-group, equipped with a group operation $+ \colon (\mathbf{B}^2 A) \times (\mathbf{B}^A)\to \mathbf{B}^2 A$.

With respect to this the sum operation is

$c_1 + c_2 : \mathbf{B}G \stackrel{\Delta_{\mathbf{B}G}}{\to} \mathbf{B}G \times \mathbf{B}G \stackrel{(c_1,c_2)}{\to} \mathbf{B}^2 A \times \mathbf{B}^2 A \stackrel{+}{\to} \mathbf{B}^2 A$

### On short exact sequences

In any category with products, for any object $C$ there is a diagonal morphism $\Delta_C:C\to C\times C$; 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 $\mathcal{A}$, form an additive category (morphisms are commutative ladders of arrows) in which the biproduct $0 \to A_i \to \hat H_{i} \to G_i \to 0$ 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 $\Gamma' = (id_M,\beta_1,\gamma)$ from an extension $E_1$ of the form $0\to M\to N_1\to P_1\to 0$ to $E$, where the pair $(E_1,\Gamma_1)$ is unique up to isomorphism of extensions, and it is called $E\gamma$. In fact, the diagram

$\array{ N_1&\to &P_1\\ \downarrow\beta_1 && \downarrow\gamma\\ N&\to &P }$

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

$E\stackrel{(\alpha,\beta_a,id)}\longrightarrow E\gamma \stackrel{(id,\beta_ 1,\gamma)}\longrightarrow E'$

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 $\Gamma_2 = (\alpha,\beta_2,id_P)$ to an extension $E_2$ of the form $0\to M_2\to N_2\to P$; the pair $(E_2,\Gamma_2)$ is unique up to isomorphism of extensions and it is called $\alpha E$.

In fact, the diagram

$\array{ M&\to &N\\ \downarrow\alpha && \downarrow\beta_2\\ M_2&\to &N_2 }$

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

$E\stackrel{(\alpha,\beta_a,id)}\longrightarrow \alpha E \stackrel{(id,\beta_ 2,\gamma)}\longrightarrow E''$

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: $(\alpha E)\gamma\cong \alpha (E\gamma)$, $id_M E \cong E$, $E id_P \cong P$, $(\alpha'\alpha)E\cong\alpha' (\alpha E)$, $(E\gamma)\gamma' \cong E(\gamma\gamma')$.

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 $E_1+E_2 = \nabla_M (E_1\oplus E_2) \Delta_P$; this gives the structure of the abelian group on $Ext(P,M)$ and $Ext:\mathcal{A}^{op}\times\mathcal{A}\to Ab$ is a biadditive (bi)functor. This is also related to the isomorphisms of extensions $\alpha (E_1+E_2)\cong \alpha E_1+\alpha E_2$, $(\alpha_1+\alpha_2) E \cong \alpha_1 E+ \alpha_2 E$, $(E_1+E_2)\gamma \cong E_1\gamma + E_2\gamma$, $E(\gamma_1+\gamma_2)\cong E\gamma_1 + E\gamma_2$.

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

$\hat G_1 + \hat G_2 \coloneqq +_* \Delta^* \hat G_1 \times \hat G_2$

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

$\array{ A \oplus A &\to& A \oplus A \\ \downarrow && \downarrow \\ \Delta^* (\hat G_1 \oplus \hat G_2) &\to& \hat G_1 \oplus \hat G_2 \\ \downarrow && \downarrow \\ G &\stackrel{\Delta_G}{\to}& G\oplus G }$

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

$\array{ A \oplus A &\stackrel{+}{\to}& A \\ \downarrow && \downarrow \\ \Delta^* (\hat G_1 \oplus \hat G_2) &\to& +_* \Delta^*(\hat G_1 \oplus \hat G_2) \\ \downarrow && \downarrow \\ G &\to& G }$

## References

• S. MacLane, Homology, 1963

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

Last revised on January 3, 2023 at 00:17:48. See the history of this page for a list of all contributions to it.