  Let $F$ be a non-Archimedean local field. This paper studies homological
properties of irreducible smooth representations restricted from
$\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. A main result shows that each
Bernstein component of an irreducible smooth representation of
$\mathrm{GL}_{n+1}(F)$ restricted to $\mathrm{GL}_n(F)$ is indecomposable. We
also classify all irreducible representations which are projective when
restricting from $\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. A main tool of
our study is a notion of left and right derivatives, extending some previous
work joint with Gordan Savin. As a by-product, we also determine the branching
law in the opposite direction.
