= Projective presentation
{title2=$0\to K\to P\to M\to0$}
A projective presentation of a <module> $M$ is an exact sequence
$$
P_1\xrightarrow{d_1}P_0\longrightarrow M\longrightarrow0
$$
with $P_0,P_1$ <projective modules>. The first map need not be <injective>. Its image is $K=\ker(P_0\to M)$, so it also yields a <short exact sequence> $0\to K\to P_0\to M\to0$; this kernel $K$ need not itself be a <projective module>.
For example, over $R=\mathbb Z/4\mathbb Z$, multiplication by two gives a projective presentation $R\xrightarrow{2}R\to\mathbb Z/2\mathbb Z\to0$. The kernel of the final quotient map has two elements and is isomorphic to $\mathbb Z/2\mathbb Z$. It is not a <projective module>: a splitting of $R\to\mathbb Z/2\mathbb Z$ would have to lift the element of order two to an odd residue, but every odd residue has additive order four. Continuing by resolving successive kernels gives a <projective resolution>.
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