Finite graded projectives are finite shifted-free summands

Let $q:E\to M$ be a degree-zero map in ${\mathrm{GrMod}}_0(A)$. Since the category is abelian, $q$ is an epimorphism if and only if its cokernel vanishes; by [L2] that cokernel is $M\mathrm{/}\mathrm{im}q$ with degreewise pieces $M_d\mathrm{/}\mathrm{im}{q}_d$. Hence $q$ is an epimorphism exactly when $q_d:E_d\to M_d$ is surjective for every $d$.

For every $r$ the shifted regular module $A\{r\}$ is projective in ${\mathrm{GrMod}}_0(A)$. Let $q:E\twoheadrightarrow M$ be a degree-zero epimorphism and $f:A\{r\}\to M$ degree-zero; put $m:=f(1_A)$, which lies in $M_r$ because $1_A\in (A\{r\}{)}_r$. By step 1.1 there is $e\in E_r$ with $q(e)=m$. Define $g:A\{r\}\to E$ by $g(a):=ae$ for $a\in (A\{r\}{)}_d=A_{d-r}$; this is well defined, $A$-linear, and $g(a)\in E_d$, so $g$ is degree-zero, and $q(g(a))=aq(e)=af(1_A)=f(a{1}_A)=f(a)$ for every $a$. Hence $g$ is a lift and [L3] makes $A\{r\}$ projective.

Let $P$ be finitely generated and $p_1,\dots ,p_n$ homogeneous generators of degrees $r_1,\dots ,r_n$; put $F:=A\{r_1\}\oplus \cdots \oplus A\{r_n\}$ and let $1_j:=1_A\in A\{r_j\}$ be the $j$-th basis vector of $F$, of degree $r_j$. By [L4] the assignment $1_j\mapsto p_j$ on the finite set $\{1_1,\dots ,1_n\}$ extends uniquely to an $A$-module homomorphism $\phi :F\to P$; it is degree-zero because $\phi \left(A_{d-r_j}{1}_j\right)=A_{d-r_j}{p}_j\subseteq P_d$, and it is surjective because the $p_j$ generate $P$. By step 1.1 applied to the cokernel description, $\phi$ is an epimorphism of ${\mathrm{GrMod}}_0(A)$.

Finite direct sums of projective objects of ${\mathrm{GrMod}}_0(A)$ are projective: if $P_1,\dots ,P_n$ are projective and $q:E\twoheadrightarrow M$, $f:P_1\oplus \cdots \oplus P_n\to M$ are given, the composites $f\circ {\mathrm{ȷ}}_j$ lift through $q$ by [L3], the universal property of the finite biproduct of [L2] assembles the $n$ lifts into $g$ with $g\circ {\mathrm{ȷ}}_j$ equal to the $j$-th lift, and then $qg=f$ because both sides agree on every summand. Only finitely many lifts are chosen, one for each $j$.

Assume now that $P$ is finite graded projective. With $\phi :F\to P$ the degree-zero epimorphism of step 2.2, projectivity of $P$ and [L3] give a degree-zero $\psi :P\to F$ with $\phi \psi =1_P$. Thus $P$ is a degree-zero direct summand of the finite direct sum $F$ of internal shifts $A\{r_j\}$.

Conversely, let $P$ be a degree-zero direct summand of a finite direct sum $F=A\{r_1\}\oplus \cdots \oplus A\{r_n\}$, so that there are degree-zero maps $i:P\to F$ and $p:F\to P$ with $pi=1_P$. The module $A\{r_j\}$ is finitely generated (by its generator $1_j$) and projective by step 2.1, so $F$ is projective by step 3.1 and finitely generated; hence $F$ is finite graded projective, and its direct summand $P$ is projective by [L3] and finitely generated because $P=p(F)$ is generated by the images of a finite generating set of $F$.

Steps 3.2 and 4.1 prove the two implications: $P$ is finite graded projective exactly when it is a degree-zero direct summand of a finite direct sum of internal shifts $A\{r_j\}$. The constructed data are a given finite homogeneous generating family, finitely many lifts indexed by that finite family, and one lift of the identity; no family indexed by an infinite set is selected, so the argument assumes no arbitrary-index choice.