# Analisis Matematico Volumen I (Spanish Version) Analisis by Pedro Pi Calleja, Cesar A. Trejo Julio Rey Pastor

By Pedro Pi Calleja, Cesar A. Trejo Julio Rey Pastor

Read Online or Download Analisis Matematico Volumen I (Spanish Version) Analisis Algebraico. Teoria De Ecuaciones. Calculo Infinitesimal De Una Variable. PDF

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Extra resources for Analisis Matematico Volumen I (Spanish Version) Analisis Algebraico. Teoria De Ecuaciones. Calculo Infinitesimal De Una Variable.

Example text

Cohomology of profinite groups Let G be a profinite group and let A ∈ DMod(G). For each natural number n we consider an R-module H n (G, A), the nth cohomology group of G with coefficients in A. 6. Here, instead, we mention some of their fundamental properties (cf. 2 in [5]), which in fact characterize them: Introduction to Profinite Groups 221 (a) H n (G, A) are functors in the variable A; (b) H 0 (G, A) = Hom[[RG]] (R, A) = {a | a ∈ A, ga = a, ∀g ∈ G} = AG on R trivially); (G acts (c) H n (G, Q) = 0 for every discrete injective [[RG]]-module Q and n ≥ 1; (d) For each short exact sequence 0 −→ A1 −→ A2 −→ A3 −→ 0 in DMod(G), there exist ‘connecting homomorphisms’ δ : H n (G, A3 ) −→ H n+1 (G, A1 ) for all n ≥ 0, such that the sequence δ 0 → H 0 (G, A1 ) → H 0 (G, A2 ) → H 0 (G, A3 ) → H 1 (G, A1 ) → H 1 (G, A2 ) → · · · is exact; and (e) For every commutative diagram G 0 0 G A1  GA α A1  GA G β 2 G 0 G 0 A3 2 G  γ A3 in DMod(G) with exact rows, the following diagram commutes for every n≥0 H n (G, A3 ) δ G H n+1 (G, A1 ) H n (G,γ)  H n (G, A3 ) δ G  H n+1 (G,α) H n+1 (G, A1 ) .

Open subgroups of free pro-C groups are free pro-C. More precisely, let F be a free pro-C group on a profinite pointed space (X, ∗) and let H be an open subgroup of F . Let Φ be the free abstract group on Y = X − {∗} and let T be a Schreier transversal for H ∩ Φ in Φ. Define B = {tx(tx)−1 | (t, x) ∈ T × X}. Then 1 ∈ B, B is a profinite space and H is a free pro-C on the pointed space (B, 1). In [5] one can find two different proofs of this theorem. The first one (cf. 6) depends on the corresponding result for abstract free groups.

Example. A free pro-C group is C-projective. 17. Let C be a variety of finite groups and let G be a pro-C group. (a) If G is C-projective, then it is isomorphic to a closed subgroup of a free pro-C group. (b) Assume in addition that the variety C is extension closed. Then G is Cprojective if and only if G is a closed subgroup of a free pro-C group. Introduction to Profinite Groups 229 Proof. 12, there exists a free pro-C group F and a continuous epimorphism α : F −→ G. Since G is C-projective, there exists a continuous homomorphism σ : G −→ F such that ασ = idG .