For a <simplicial abelian group> $A$, its <normalized chain complex of a simplicial abelian group> is
$$
N_nA=\bigcap_{i=1}^n\ker(d_i:A_n\to A_{n-1}),
\qquad \partial_n=d_0|_{N_nA}.
$$
The simplicial identities give $\partial^2=0$. Equivalently, $N_nA$ is the quotient of $A_n$ by the subgroup generated by degenerate simplices, with differential induced by $\sum_i(-1)^id_i$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 165 / 2 / i / Solution

{title2=$N_*A$}
{wiki=Moore_complex}

For a simplicial abelian group $A$, one convention for normalized chains is
$$
N_nA=\bigcap_{i=1}^n\ker(d_i:A_n\to A_{n-1}),
\qquad \partial=d_0.
$$
Equivalently, it is the quotient of the unnormalized chain complex by its degenerate subcomplex.


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 Incoming links: Normalized chain complex of a simplicial abelian group