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.