Suppose is a -graded connected algebra over a field concentrated in positive degrees, and suppose is a projective graded -module concentrated in degrees above some . Then is actually a free -module.
Under the same hypothesis over , show that if is a graded module and where is the quotient of by the ideal of positively graded elements, then .
Using the above, show that if is graded and , then is free. To illustrate, it follows projective graded modules over polynomial algebras are free.