 Projective presentation

A projective presentation of a module $M$ is an exact sequencewith $P_0,P_1$ 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 $0\to K\to P_0\to M\to 0$; 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 $R\stackrel{2}{\to }R\to \mathbb{Z}/2\mathbb{Z}\to 0$. 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.