# Finitely generated projective modules form exact category

I am looking for a proof of the fact that the category of f.g. projective modules over some commutative ring is exact. Actually, I would already be happy to know why this category is closed under pullbacks of split epis and under pushouts of split monos. The rest seems straightforward to me.

The pullback of a split epi $A\oplus B\to B$ along a map $f:C\to B$ is given by $\{(a,b,c)\in A\oplus B\oplus C: f(c)=b\}$. This is isomorphic to $A\oplus C$ by $(a,b,c)\mapsto (a,c)$ and $(a,c)\mapsto (a,f(c),c)$. This is a finite direct sum of finitely generated projectives. A pushout of $A\to A\oplus B$ along $g:A\to D$ is $A\oplus B\oplus D/R$, where the equivalence relation $R$ has $(a,b,d)R(0,0,0)$ if and only if $b=0$ and $d=-G(a)$. Sending $(a,b,d)\mapsto (b,g(a)+d)$ gives an isomorphism with $B\oplus D$.