Finite-dimensional graded algebras have graded projective covers

Choose a finite $k$-basis of $X$. Each basis vector has finitely many homogeneous components, and the collection of these components is a finite homogeneous generating family $x_1,\dots ,x_m$. If $X=0$, take $m=0$.

Write $d_j=\mathrm{deg}(x_j)$ and set $E:={⨁}_{j=1}^{m}A\{d_j\}$. The map $ϵ\text{\hspace{0.17em}⁣}:E\to X$ given on the $j$th shifted generator by $1_j\mapsto x_j$ and extended by $a{1}_j\mapsto a{x}_j$ is degree-zero by [L1], and is surjective by step 1.1. It is an epimorphism in ${\mathrm{GrMod}}_0(A)$ by [L2]. The empty case gives $E=X=0$.

The family of graded direct summands $Q$ of $E$ for which $ϵ{\mathrm{\mid }}_Q\text{\hspace{0.17em}⁣}:Q\to X$ is surjective is nonempty because it contains $E$. Their dimensions lie in the finite set $\{0,1,\dots ,{\dim }_{k}E\}$, so choose such a $P$ of minimum dimension and put $p:=ϵ{\mathrm{\mid }}_P$. By [L3], $P$ is finite graded projective. It is finite-dimensional because it is a submodule of the finite-dimensional $E$.

Let $K=\ker p$ and let $N\le P$ be a graded submodule with $N+K=P$. Then $p{\mathrm{\mid }}_N\text{\hspace{0.17em}⁣}:N\to X$ is surjective, hence an epimorphism by [L2]. Since $P$ is graded projective, [L4] lifts $p$ through this map to a degree-zero $h\text{\hspace{0.17em}⁣}:P\to N$ with $(p{\mathrm{\mid }}_N)h=p$. After inclusion $N\hookrightarrow P$, this gives a degree-zero endomorphism $f$ of $P$ with $pf=p$, and therefore $p{f}^n=p$ for every $n\ge 1$.

Apply [L7] to $f$: for some $n\ge 1$, $P=\ker (f^n)\oplus \mathrm{im}(f^n)$ by graded submodules. Since $p{f}^n=p$, the restriction of $p$ to $\mathrm{im}(f^n)$ is still surjective. This image is a graded direct summand of $P$, hence of $E$, so it is one of the candidates in step 3.1. Minimality gives ${\dim }_{k}P\le {\dim }_k\mathrm{im}(f^n)$; the reverse inequality follows from $\mathrm{im}(f^n)\subseteq P$, so $\mathrm{im}(f^n)=P$. As $\mathrm{im}(f^n)\subseteq \mathrm{im}(f)\subseteq N$, it follows that $N=P$. Thus $K$ is superfluous among graded submodules and, by the category-relative definition in the Statement (using the terminology of [L6]), $p$ is a finite graded projective cover.

Let $p\text{\hspace{0.17em}⁣}:P\to X$ and $q\text{\hspace{0.17em}⁣}:Q\to X$ be finite graded projective covers. By projectivity, there are degree-zero maps $f\text{\hspace{0.17em}⁣}:P\to Q$ and $g\text{\hspace{0.17em}⁣}:Q\to P$ with $qf=p$ and $pg=q$. Then $p(gf)=p$, so $P=gf(P)+\ker p$. The image $gf(P)$ is graded by [L2]; superfluity of $\ker p$ gives $gf(P)=P$. Similarly, $fg(Q)=Q$.

By [L5] the sources $P,Q$ are finitely generated by homogeneous elements; since $A$ is finite-dimensional, both are finite-dimensional over $k$. The surjective endomorphisms $gf$ and $fg$ from step 6.1 are therefore bijective. It follows that $f$ is injective from injectivity of $gf$ and surjective from surjectivity of $fg$. The inverse is $A$-linear; it preserves degrees because a homogeneous element's preimage can have no nonzero components in other degrees under an injective degree-zero map. Hence $f$ is a degree-zero isomorphism and $qf=p$, proving uniqueness over $X$.