The Schur multiplier and stem covers of Leibniz $n$-algebras

Given a free presentation $0\to\mathcal{R}\to\mathcal{F}\stackrel{\rho}\to\mathcal{G}\to0$ of a Leibniz $n$-algebra $\mathcal{G}$, the quotient $\frac{\mathcal{R}\cap[\mathcal{F},\overset{n}\dots,\mathcal{F}]}{[\mathcal{R},\mathcal{F},\overset{n-1}\dots,\mathcal{F}]}$ is known as the Schur multiplier of $\mathcal{G}$. In the article, we construct a four-term exact sequence relating the Schur multiplier of $\mathcal{G}$ and $\mathcal{G}/\mathcal{N}$, from which we derive some formulas concerning dimensions of the underlying vector spaces of the corresponding Schur multipliers. Additionally, this exact sequence is useful to characterize nilpotency of Leibniz $n$-algebras. Finally, we characterize stem covers of Leibniz $n$-algebras, showing their existence in case of finite dimension. We also analyze the interaction between stem covers of Leibniz $n$-algebras and the Schur multiplier.