  Let $\xi$ be a $\tau$-invariant contact structure on $N(W) =
\mathbb{R}_{\tau} \times W$ for a closed, $2n$-dimensional manifold $W$, so
that each $\{\tau\} \times W$ is a convex hypersurface. When $n=1$, Giroux's
criterion provides a simple means of determining exactly when $\xi$ is tight.
It is an open problem to find a generalization applicable for $n>1$. This
article solves an algebraic version of the problem, determining exactly when
$(N(W), \xi)$ has non-vanishing contact homology ($CH$) and computing $CH(N(W),
\xi)$ when it is non-zero. The result can be expressed in terms of homotopy
equivalence of augmentations of the chain level $CH$ algebra of the dividing
set or in terms of bilinearized homology theories, which we define for free,
commutative DGAs over $\mathbb{Q}$. Our proof relies on the development of
obstruction bundle gluing in the Kuranishi setting.
