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Dual universal coefficient theorem for group cohomology

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Statement

For coefficients in an abelian group

Suppose G is a group and M is an abelian group. The dual universal coefficients theorem relates the homology groups for trivial group action of G on \mathbb{Z} and the cohomology groups for trivial group action of G on M as follows:

First, for any p\ge 0, there is a natural short exact sequence of abelian groups:

0 \to \operatorname{Ext}(H_{p-1}(G;\mathbb{Z}),M) \to H^p(G;M) \to \operatorname{Hom}(H_p(G;\mathbb{Z}),M) \to 0

Second, the sequence splits (not necessarily naturally), and we get:

H^p(G;M) \cong \operatorname{Hom}(H_p(G;\mathbb{Z}),M) \oplus \operatorname{Ext}(H_{p-1}(G;\mathbb{Z}),M)

For coefficients in the integers

This is the special case where M = \mathbb{Z}. In this case, we case:

H^p(G;\mathbb{Z}) \cong \operatorname{Hom}(H_p(G;\mathbb{Z}),\mathbb{Z}) \oplus \operatorname{Ext}(H_{p-1}(G;\mathbb{Z}),\mathbb{Z})

Typical case of finitely generated abelian groups

Suppose H_p(G;\mathbb{Z}) \cong \mathbb{Z}^{r_p} \oplus T_p for some finite group T_p and H_{p-1}(G;\mathbb{Z}) \cong \mathbb{Z}^{r_{p-1}} \oplus T_{p-1} for some finite group T_{p-1}. Suppose further that:

T_p \cong \mathbb{Z}/a_1\mathbb{Z} \oplus \dots \mathbb{Z}/a_s\mathbb{Z}

and

T_{p-1} \cong \mathbb{Z}/b_1\mathbb{Z} \oplus \dots \mathbb{Z}/b_t\mathbb{Z}

Then we have:

H^p(G;M) \cong M^{r_p} \oplus \operatorname{Hom}(T_p,M) \oplus \operatorname{Ext}(T_{p-1},M)

where we further have:

\operatorname{Hom}(T_p,M) \cong \bigoplus_{1 \le i \le s} \operatorname{Ann}_M(a_i)

where \operatorname{Ann}_M(a_i) = \{ x \in M \mid a_ix = 0 \}, i.e., the a_i-torsion of M.

Also:

\operatorname{Ext}(T_{p-1},M) \cong \bigoplus_{1 \le i \le t} M/b_iM

Thus, we get overall that:

H^p(G;M) \cong M^{r_p} \oplus  \bigoplus_{1 \le i \le s} \operatorname{Ann}_M(a_i) \oplus \bigoplus_{1 \le i \le t} M/b_iM

Finally, suppose:

M \cong \mathbb{Z}^w \oplus \mathbb{Z}/c_1\mathbb{Z} \oplus \dots \mathbb{Z}/c_u\mathbb{Z}

In this case, the expressions simplify further:

\operatorname{Hom}(T_p,M) \cong \bigoplus_{1 \le i \le s, 1 \le j \le u} \mathbb{Z}/\operatorname{gcd}(a_i,c_j)\mathbb{Z}

and:

\operatorname{Ext}(T_{p-1},M) \cong T_{p-1}^w \oplus \bigoplus_{1 \le i \le t, 1 \le j \le u} \mathbb{Z}/\operatorname{gcd}(b_i,c_j)\mathbb{Z}

Typical case of finitely generated abelian groups and coefficients in the integers

Suppose H_p(G;\mathbb{Z}) \cong \mathbb{Z}^{r_p} \oplus T_p for some finite group T_p and H_{p-1}(G;\mathbb{Z}) \cong \mathbb{Z}^{r_{p-1}} \oplus T_{p-1} for some finite group T_{p-1}. Then:

H^p(G;\mathbb{Z}) \cong \mathbb{Z}^{r_p} \oplus T_{p-1}

In other words, we pick the torsion-free part of H_p and the torsion part of H_{p-1} (roughly speaking).

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