Finite graded projective modules

Fix a graded $k$-algebra $A$ and let $P$ be a graded left $A$-module (Associative graded algebras, bimodules, and internal shifts). The category ${\mathrm{GrMod}}_0(A)$ of graded left $A$-modules and degree-zero maps is abelian (Graded modules with degree-zero maps form an abelian category).

Graded projective. $P$ is graded projective when it is a projective object of ${\mathrm{GrMod}}_0(A)$ (Projective object): for every degree-zero epimorphism $q:E\twoheadrightarrow M$ and every degree-zero $f:P\to M$ there exists a degree-zero $\tilde{f}:P\to E$ with $q\tilde{f}=f$. Thus projectivity is tested only against degree-zero epimorphisms and degree-zero maps, and the lift need not be unique.

Finitely generated. $P$ is finitely generated as a graded left $A$-module, or generated by finitely many homogeneous elements, when for some $n\ge 0$ there are homogeneous elements $p_1,\dots ,p_n\in P$ with

$P=A{p}_1+\cdots +A{p}_n\mathrm{.}$

By Generated submodule, cyclic and finitely generated modules, module basis and free module this is the same notion as finite generation of the underlying $A$-module: a finite homogeneous family is a finite family, and conversely, if $P=⟨S{⟩}_A$ for a finite set $S$, then writing each $s={\sum }_e{s}_e$ as its finite sum of nonzero homogeneous components gives, for every $x={\sum }_{s\in S}{a}_{s}s\in P$,

$x={\sum }_{s\in S}{a}_s{\sum }_e{s}_e={\sum }_{s,e}{a}_s{s}_e,$

a finite $A$-linear combination of the homogeneous elements $s_e\in P$; so the finitely many components of the members of $S$ generate $P$. In particular the notion does not depend on the chosen finite generating set.

Finite graded projective. $P$ is finite graded projective when it is graded projective and generated by finitely many homogeneous elements, that is, when it is graded projective and finitely generated as an $A$-module. The family may be empty: $n=0$ gives $P=A{p}_1+\cdots +A{p}_n=0$, so the zero module is generated by the empty homogeneous family, and the corresponding finite direct sum of shifts in the characterization below is the empty direct sum. Its finite shifted-free characterization is the next result (Finite graded projectives are finite shifted-free summands), and it uses this same notion of finite generation throughout: a finite homogeneous generating family there is exactly a family as above. No choice principle is used in the equivalence above, which only rewrites a given finite expression.