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Projective-resolution definition of group cohomology

Koszul resolution for a rank-two free abelian group

If $G\cong\mathbb Z^2$, $I$ is the augmentation ideal, and $M=\mathbb ZG/I^2$, then
$$
H^2(G,M)\cong M/IM\cong\mathbb Z.
$$
The quotient map $M\to\mathbb ZG/I\cong\mathbb Z$ induces an isomorphism in second cohomology.


Second cohomology of a rank-two free abelian group with truncated group-ring coefficients

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 151 / 1 / Solution

Second cohomology of a rank-two free abelian group with truncated group-ring coefficients

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Second cohomology of a rank-two free abelian group with truncated group-ring coefficients

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