Let $A$ and $B$ be two rings with a ring homomorphism $f: A\to B$ such that $B$ has finite projective dimension over $A$. Is it true that any module which has finite projective dimension as $B$-module must also have finite projective dimension over $A$ (as $A$-module via restriction of scalars).
I was trying using “Ext” criterion but somewhere I was using that “ Ext” well behave with restriction of scalars. I don’t know any such result. Can anyone help me? Thanks in advance.
Given a short exact sequence of $A$-modules $0\to L\to M\to N\to 0$, we have $\mathrm{p.dim}_AN\leq\max\{\mathrm{p.dim}_AM,1+\mathrm{p.dim}_AL\}$. Using this, we see that if $d:=\mathrm{p.dim}_AB$ and $M$ is a $B$-module, then $\mathrm{p.dim}_AM\leq d+\mathrm{p.dim}_BM$.