Some homological properties of category $\mathcal{O}$ for Lie superalgebras

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\mathrm{\Delta }(\lambda )$ to be such that every non-zero homomorphism from another Verma supermodule to $\mathrm{\Delta }(\lambda )$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak{p}\mathfrak{e}(n)$ and, furthermore, to reduce the problem of description of ${\mathrm{E}\mathrm{x}\mathrm{t}}_{\mathcal{O}}^1(L(\mu ),\mathrm{\Delta }(\lambda ))$ for $\mathfrak{p}\mathfrak{e}(n)$ to the similar problem for the Lie algebra $\mathfrak{g}\mathfrak{l}(n)$. Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category ${\mathcal{O}}^{\mathfrak{p}}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\mathfrak{p}\mathfrak{e}(n)$ and the ortho-symplectic Lie superalgebra $\mathfrak{o}\mathfrak{s}\mathfrak{p}(2\mathrm{\mid }2n)$.