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Let $F$ be a non-Archimedean local field. This paper studies homologicalproperties of irreducible smooth representations restricted from$\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. A main result shows that eachBernstein component of an irreducible smooth representation of$\mathrm{GL}_{n+1}(F)$ restricted to $\mathrm{GL}_n(F)$ is indecomposable. Wealso classify all irreducible representations which are projective whenrestricting from $\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. A main tool ofour study is a notion of left and right derivatives, extending some previouswork joint with Gordan Savin. As a by-product, we also determine the branchinglaw in the opposite direction.

Homological branching law for $(\mathrm{GL}_{n+1}(F), \mathrm{GL}_n(F))$: projectivity and indecomposability