How to prove the global dimension of the polynomial ring $F[x_1,...,x_n]$ is $n$?

Question

I am trying to prove that the global dimension of the polynomial ring $F[x_1,\dots,x_n]$, where $F$ is a field , is exactly $n$.

By Koszul complex, I know its global dimension is greater than or equal to $n$. But I don't know how to get it less than or equal to $n$. I have checked some books but couldn't understand well about their ideas of the proof. So I want to ask here for some guidance about the residue part of the proof.
Thank you for your assistance.

Answer 1

I suggest you look at Theorem 4.3.7 in Weibel's book, and then clarify your question on which part of the proof there you do not understand. Note the proof there does not use Koszul complexes explicitly, so you might find it easier to understand.