  Convex codes were recently introduced as models for neural codes in the
brain. Any convex code $\C$ has an associated minimal embedding dimension
$d(\C)$, which is the minimal Euclidean space dimension such that the code can
be realized by a collection of convex open sets. In this work we import tools
from combinatorial commutative algebra in order to obtain better bounds on
$d(\C)$ from an associated simplicial complex $\Delta(\C)$. In particular, we
make a connection to minimal free resolutions of Stanley-Reisner ideals, and
observe that they contain topological information that provides stronger bounds
on $d(\C)$. This motivates us to define the Leray dimension $d_L(\C),$ and show
that it can be obtained from the Betti numbers of such a minimal free
resolution. We compare $d_L(\C)$ to two previously studied dimension bounds,
obtained from Helly's theorem and the simplicial homology of $\Delta(\C)$.
Finally, we show explicitly how $d_L(\C)$ can be computed algebraically, and
illustrate this with examples.
