  Let $Y$ be an algebraic variety, $\mathcal{F}$ a locally free sheaf of
$\mathcal{O}_Y$-modules, and $\mathcal{R}(\mathcal{F})$ the
$\mathcal{O}_Y$-algebra $\operatorname{Sym}^\bullet \mathcal{F}$. In this paper
we study local properties of sheaves of
$\mathcal{O}_{\mathcal{R}(\mathcal{F})}$-ideals $\mathcal{I}$ such that
$\mathcal{R}(\mathcal{F}))/\mathcal{I}$ is an algebraic cover of $Y$. Following
the work of Miranda for triple covers, for $\mathcal{Q}$ a direct summand of
$\mathcal{R}(\mathcal{F})$, we say that a morphism $\Phi\colon
\mathcal{Q}\rightarrow\mathcal{R}(\mathcal{F})/\langle\mathcal{Q}\rangle$ is a
covering homomorphism if it induces such an ideal. As an application we study
in detail the case of Gorenstein covering maps of degree $6$ for which the
direct image of $\varphi_*\mathcal{O}_X$ admits an orthogonal decomposition.
These are deformation of $S_3$-Galois branch covers.
