Finitely generated projective modules form exact category

Question

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.

Answer 1

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 $B\oplus D$.