Second cohomology group for trivial group action between additive groups of ring of integers localized at sets of primes

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The goal of this page is to discuss the second cohomology group for trivial group action H^2(G;A) where both G and A are additive groups of rings of integers localized at sets of primes. Explicitly, there are subsets \pi_1,\pi_2 of the set of all prime numbers such that G = \mathbb{Z}[\pi_1^{-1}] and A = \mathbb{Z}[\pi_2^{-1}].

Computation of the group

We use the formula for second cohomology group for trivial group action of abelian group in terms of Schur multiplier and abelianization (which is a special case of the formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization), namely the following short exact sequence:

0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0

where \bigwedge^2G = M(G) is the exterior square of G and also coincides with the Schur multiplier of G.

Note that since locally cyclic implies epabelian, the Schur multiplier of G is the trivial group, and all extensions are themselves abelian groups. In particular, \bigwedge^2G = 0, and we get:

H^2(G;A) \cong \operatorname{Ext}^1_{\mathbb{Z}}(G,A)

Thus, the computation of H^2(G;A) is equivalent to the computation of \operatorname{Ext}^1_{\mathbb{Z}}(G;A), and the elements of these classify the congruence classes of group extensions, all of which are abelian.

We now turn to an explicit description in terms of the sets \pi_1 and \pi_2. First, define \pi_3 as the set of all primes in \pi_1 that are not in \pi_2. Now:

  • Let R be the external direct product of the additive groups of p-adic integers for all p \in \pi_3. Note that if \pi_3 is empty, we take R as the trivial group.
  • Recall that there is a natural embedding of \mathbb{Z}[\pi_2^{-1}] inside each of the direct factors of R (we begin with the natural embedding of \mathbb{Z} inside \mathbb{Z}_{(p)}, the p-adics, then extend to an embedding of all of \mathbb{Z}[\pi_2^{-1}] using the fact that \mathbb{Z}_{(p)} is \pi_2-powered. Let S be the image of the diagonal embedding in R arising from each of these embeddings.
  • The group H^2(G;A) \cong \operatorname{Ext}^1_{\mathbb{Z}}(G,A) that we are trying to compute is R/S.

Some simple cases are described below:

Case Description of H^2(G;A) \cong \operatorname{Ext}^1_{\mathbb{Z}}(G,A)
\pi_1 \subseteq \pi_2, in other words, the base of the extension has all the powering that the extension group does, and possibly more. trivial group
\pi_1 = \{ p \}, \pi_2 = \{ \} quotient of additive group of p-adic integers by group of integers, i.e., the quotient \mathbb{Z}_{(p)}/\mathbb{Z}
\pi_1 is the set of all primes, \pi_2 = \{ \} quotient of profinite completion of the integers by the integers, i.e., the group \hat{\mathbb{Z}}/\mathbb{Z}.

Elements

Basically, we use the element of the p-adic group to describe what p^{th} root to adjoin at each stage.