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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
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.
Below are discussed several different equivalent ways to define the Baer sum
A cocycle in degree-2 group cohomology $H^2_{Grp}(G,A)$ is a function
satisfying the cocycle property.
Given two cocycles $c_1, c_2 : G \times G \to A$ their sum is the composite
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
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 $A$:
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
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
is a pullback diagram. Every morphism of abelian extensions $(\alpha,\beta,\gamma):E\to E'$ in a unique way decomposes as
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
is a pushout diagram. Every morphism of abelian extensions $(\alpha,\beta,\gamma):E\to E''$ in a unique way decomposes as
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
The first step forms the pullback of the short exact sequence along the diagonal on $G$:
The second forms the pushout along the addition map on $A$:
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.