Projective module

A module $P$ is projective when every map from $P$ through the target of a surjection lifts through that surjection.

For a nonzero module, the projective dimension ${\mathrm{pd}}_{R}M$ is the shortest length of a projective resolution of the module $M$, or infinity if no finite projective resolution exists. Equivalently, it is the supremum of degrees with nonzero ${\mathrm{Ext}}_R^n(M,N)$ as $N$ ranges over all modules. This equivalence follows by dimension shifting and the characterization of projective modules by vanishing ${\mathrm{Ext}}^1$.

The global dimension of a ring is the supremum of the projective dimensions of all its modules, allowing infinity. It measures the longest projective resolution needed by any module. For a field it is zero, since every module is a vector space and hence free.

If global dimension is zero, every module is a projective module. Each short exact sequence splits because its quotient is projective. Conversely, splitting every short exact sequence makes each module a direct summand of a free module and therefore a projective module. Splitting also makes every module an injective module: for $A\hookrightarrow B$ choose a retraction $B\to A$ and compose it with any prescribed map out of $A$.

Lift the image of each basis element independently through the given surjection, then extend the chosen lifts linearly.

A projective cover of a module $M$ is an essential surjection $P\twoheadrightarrow M$ from a projective module. For a finite-dimensional algebra, every finite-dimensional module has one; the projective cover of a simple module is indecomposable and has that simple module as its head.

The head, or top, of a finite-length module $M$ is its largest semisimple quotient. It is $M/J(M)$, where $J(M)$ is the radical of a module.