1
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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 $B\oplus D$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.