# Tardily projective? You’re free then

Suppose $A$ is a $\mathbb Z$-graded connected algebra over a field $k$ concentrated in positive degrees, and suppose $P$ is a projective graded $A$-module concentrated in degrees above some $m\in \mathbb Z$ . Then $P$ is actually a free $A$-module.

Under the same hypothesis over $A$, show that if $M$ is a graded module and $M\otimes_A k=0$ where $k$ is the quotient of $A$ by the ideal of positively graded elements, then $M=0$.

Using the above, show that if $M$ is graded and $\mathrm{Tor}_A^1(M,k)=0$, then $M$ is free. To illustrate, it follows projective graded modules over polynomial algebras are free.