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im δ n ker δ n 1 ZG n + 1 δ n ZG n δ n 1 ZG n 1 coker δ n im δ n 1 \array{ && im \delta_n &&\to&& ker \delta_{n-1} \\ & \nearrow && \searrow && \swarrow \\ ZG_{n+1} &&\stackrel{\delta_n}{\to}&& ZG_n &&\stackrel{\delta_{n-1}}{\to}&& ZG_{n-1} \\ & && \swarrow && \searrow && \nearrow \\ && coker \delta_n &&\stackrel{}{\to}&& im \delta_{n-1} } ω ( v , w ) = g ( J ( v ) , w ) AAAAA g ( v , w ) = ω ( v , J ( v ) ) \omega(v,w) \;=\; g(J(v),w) \phantom{AAAAA} g(v,w) \;=\; \omega(v,J(v)) \,.
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