TITLE: Are projective modules "graded projective"?
QUESTION [8 upvotes]: Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). Finally let $A$-mod be category of modules over $A$. 
So we have an oblivion functor $$ Obl: A^{\bullet}-{\rm mod} \rightarrow  A-{\rm mod}.$$
Consider $P^{\bullet} \in A^{\bullet}$-mod such that $Obl(P^{\bullet}) $ is projective in category of $A$-mod.
Question: Is $P^{\bullet}$ projective in category $A^{\bullet}$-mod ?

REPLY [10 votes]: This is true and easy to prove.

Let $M\stackrel{g}\to N$, $P\stackrel{f}\to N$ be graded homomorphisms, and $P\stackrel{h}\to M$ be a homomorphism such that $f=gh$. Then there is a graded homomorphism $P\stackrel{h'}\to M$ such that $f=gh'$.

For $x_n\in P_n$ we have $f(x_n)\in N_n$. From $h(x_n)=\sum y_m$ with $y_m\in M_m$ we get $f(x_n)=gh(x_n)=\sum g(y_m)$, and since $g(y_m)\in N_m$ it follows $g(y_m)=0$ for $m\ne n$ and $f(x_n)=g(y_n)$. Now define $h':P\to M$ by $h'(\sum x_n)=\sum y_n$, where $y_n$ in the unique element in $M_n$ with $f(x_n)=g(y_n)$.