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Projective-resolution definition of group cohomology

{c}

For $G=\langle x,y\mid xy=yx\rangle$, the trivial $\mathbb ZG$-module has the free resolution
$$
0\longrightarrow\mathbb ZG\xrightarrow{\binom{1-y}{x-1}}(\mathbb ZG)^2
\xrightarrow{(x-1\quad y-1)}\mathbb ZG\longrightarrow\mathbb Z\longrightarrow0.
$$


Koszul resolution for a rank-two free abelian group

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

Koszul resolution for a rank-two free abelian group

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

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Koszul resolution for a rank-two free abelian group

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

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