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.

Tardily projective? You’re free then

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