The reason I ask this question is I found two different statements about Hilbert's syzygy theorem from Jacobson's Basic Algebras 2nd and Wikipedia. Please have a look at the following pictures. The first one is from the former and the second one is from the latter.
In this two statements I found the modules required are different. One is called graded and one is called finitely generated. But the results are the same. So I want to ask are these two kinds of modules equivalent? Or there is some connection between them. Someone has told me that there are two editions about this theorem. If this is the case, then which one is more general or widely used?
At the same time, I found Quillen-Suslin theorem from wikipedia, which states that every finitely generated projective module over a polynomial ring is free. And from Jacobson's book mentioned above there is a corollary
Here I consider the corollary as Any projective graded R-module for $R=F[x_1,x_2,...,x_m]$ is free. I don't know whether I understand it in a correct way or not. Then are these two statements about projective modules to be free the same?
Thank you for your assistance.