Let $A=K[t_1,\dots, t_d]$ the ring of the polynomials in $d$ indeterminates. I know that a consequence of Serre's Theorem is that $gldim(A)=d$.
PROPOSITION 1. If $M$ is a finitely generated projective module over $A$, then $M$ is free.
PROPOSITION 2. If $M$ is a finitely generated projective module over $A$, then $M$ has a finite free resolution of the form $0\to F_k\to\dots\to F_1\to F_0\to M\to 0$ with $F_i$ free over $A$ for all $i=0,\dots, k$ and with $k\le d$.
I have some difficulties in showing this two proposition. Can anyone help me please?
Proposition 1 is the celebrated Quillen-Suslin theorem. The wikipedia page on the Quillen-Suslin Theorem gives links to the original papers and more information. Lang's Algebra contains an exposition of a later and shorter proof by Vaserstein, but even that is several pages long.
It is not an intended as an insult if I say that it is not surprising that you had some problems proving Proposition 1!
Proposition 2 is just the Hilbert Syzygy Theorem (or at least, the Hilbert Syzygy Theorem is the same statement without the requirement that $M$ is projective).