nLab
Baer sum

Contents

Idea

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

For GG a group and AA an abelian group, the extensions of GG by AA are classified by the degree-2 group cohomology

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

On cocycles BGB 2A\mathbf{B}G \to \mathbf{B}^2 A there is a canonical addition operation coming from the additive structure of AA, 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 Grp 2(G,A)H^2_{Grp}(G,A) is a function

c:G×GA c : G \times G \to A

satisfying the cocycle property.

Definition

Given two coycles c 1,c 2:G×GAc_1, c_2 : G \times G \to A their sum is the composite

(c 1+c 2):G×GΔ G×G(G×G)×(G×G)(c 1,c 2)A×A+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×GG\times G;

  • the direct product (f,g)(f,g);

  • the group operation +:A×AA+ \colon A \times A \to A.

Hence for all g 1,g 2Gg_1, g_2 \in G this sum is the function that sends

(c 1+c 2):(g 1,g 2)c 1(g 1,g 2)+c 2(g 1,g 2) (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×GAc \colon G \times G \to A is equivalently a morphism of 2-groupoids from the delooping groupoid BG\mathbf{B}G of GG to the double-delooping 2-groupoid B 2A\mathbf{B}^2 A of AA:

c 1,c 2:BGB 2A. c_1,c_2 : \mathbf{B}G \to \mathbf{B}^2 A \,.

Since AA is an abelian group, matbfB 2A\matbf{B}^2 A is naturally an abelian 3-group, equipped with a group operation +:(B 2A)×(B A)B 2A+ \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:BGΔ BGBG×BG(c 1,c 2)B 2A×B 2A+B 2A 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 CC there is a diagonal morphism Δ C:CC×C\Delta_C:C\to C\times C; in a category with coproducts there is a codiagonal morphism C:CCC\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 RR-modules, or in arbitrary abelian category 𝒜\mathcal{A}, form an additive category (morphisms are commutative ladders of arrows) in which the biproduct 0A iH^ iG i00 \to A_i \to \hat H_{i} \to G_i \to 0 for i=1,2i = 1,2 is 0A 1A 2H 1H 2G 1G 200\to A_1\oplus A_2 \to H_1\oplus H_2\to G_1\oplus G_2\to 0.

Now if 0MNP00\to M\to N\to P\to 0 is any extension, call it EE, and γ:P 1P\gamma:P_1\to P a morphism, then there is a morphism Γ=(id M,β 1,γ)\Gamma' = (id_M,\beta_1,\gamma) from an extension E 1E_1 of the form 0MN 1P 100\to M\to N_1\to P_1\to 0 to EE, where the pair (E 1,Γ 1)(E_1,\Gamma_1) s unique up to isomorphism of extensions, and it is called EγE\gamma. In fact, the diagram

N 1 P 1 β 1 γ N P\array{ N_1&\to &P_1\\ \downarrow\beta_1 && \downarrow\gamma\\ N&\to &P }

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

E(α,β a,id)αEγ(id,β 1,γ)E E\stackrel{(\alpha,\beta_a,id)}\longrightarrow \alpha E\gamma \stackrel{(id,\beta_ 1,\gamma)}\longrightarrow E'

for some β a\beta_a with β 1\beta_1 as above. In short, the morphism of extensions factorizes through EγE\gamma.

Dually, for any morphism α:MM 2\alpha:M\to M_2, there is a morphism Γ 2=(α,β 2,id P)\Gamma_2 = (\alpha,\beta_2,id_P) to an extension E 2E_2 of the form 0M 2N 2P0\to M_2\to N_2\to P; the pair (E 2,Γ 2)(E_2,\Gamma_2) is unique up to isomorphism of extensions and it is called αE\alpha E.

In fact, the diagram

M N α β 2 M 2 N 2\array{ M&\to &N\\ \downarrow\alpha && \downarrow\beta_2\\ M_2&\to &N_2 }

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

E(α,β a,id)αE(id,β 2,γ)E E\stackrel{(\alpha,\beta_a,id)}\longrightarrow \alpha E \stackrel{(id,\beta_ 2,\gamma)}\longrightarrow E''

for some β a\beta_a, with β 2\beta_2 as above. In short, the morphism of extensions factorizes through αE\alpha E.

There are the following isomorphisms of extensions: (αE)γα(Eγ)(\alpha E)\gamma\cong \alpha (E\gamma), id MEEid_M E \cong E, Eid PPE id_P \cong P, (αα)Eα(αE)(\alpha'\alpha)E\cong\alpha' (\alpha E), (Eγ)γE(γγ)(E\gamma)\gamma' \cong E(\gamma\gamma').

The Baer’s sum of two extensions E 1,E 2E_1,E_2 of the form 0MN iP00\to M\to N_i\to P\to 0 (i.e. with the same MM and PP) is given by E 1+E 2= M(E 1E 2)Δ PE_1+E_2 = \nabla_M (E_1\oplus E_2) \Delta_P; this gives the structure of the abelian group on Ext(P,M)Ext(P,M) and Ext:𝒜 op×𝒜AbExt:\mathcal{A}^{op}\times\mathcal{A}\to Ab is a biadditive (bi)functor. This is also related to the isomorphisms of extensions α(E 1+E 2)αE 1+αE 2\alpha (E_1+E_2)\cong \alpha E_1+\alpha E_2, (α 1+α 2)Eα 1E+α 2E(\alpha_1+\alpha_2) E \cong \alpha_1 E+ \alpha_2 E, (E 1+E 2)γE 1γ+E 2γ(E_1+E_2)\gamma \cong E_1\gamma + E_2\gamma, E(γ 1+γ 2)Eγ 1+Eγ 2E(\gamma_1+\gamma_2)\cong E\gamma_1 + E\gamma_2.

In different notation, if 0AG^ iG00 \to A \to \hat G_{i} \to G \to 0 for i=1,2i = 1,2 are two short exact sequences of abelian groups, their Baer sum is

G^ 1+G^ 2+ *Δ *G^ 1×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 GG:

AA AA Δ *(G^ 1G^ 2) G^ 1G^ 2 G Δ G GG \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 AA:

AA + A Δ *(G^ 1G^ 2) + *Δ *(G^ 1G^ 2) G G \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)