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

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

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 coycles $c_1, c_2 : G \times G \to A$ 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 $(f,g)$;

• 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, $\matbf{B}^2 A$ 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 $(E_1,\Gamma_1)$ s 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

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

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 $\Gamma_2 = (\alpha,\beta_2,id_P)$ 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 rhe 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)

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