  One of the more popular objections towards shape dynamics is the suspicion
that anomalies in the spatial Weyl symmetry will arise upon quantization. The
purpose of this short paper is to establish the tools required for an
investigation of the sort of anomalies that can possibly arise. The first step
is to adapt to our setting Barnich and Henneaux's formulation of gauge
cohomology in the Hamiltonian setting, which serve to decompose the anomaly
into a spatial component and time component. The spatial part of the anomaly,
i.e. the anomaly in the symmetry algebra itself ($[\Omega, \Omega]\propto
\hbar$ instead of vanishing) is given by a projection of the second ghost
cohomology of the Hamiltonian BRST differential associated to $\Omega$, modulo
spatial derivatives. The temporal part, $[\Omega, H]\propto\hbar$ is given by a
different projection of the first ghost cohomology and an extra piece arising
from a solution to a functional differential equation. Assuming locality of the
gauge cohomology groups involved, this part is always local. Assuming locality
for the gauge cohomology groups, using Barnich and Henneaux's results, the
classification of Weyl cohomology for higher ghost numbers performed by
Boulanger, and following the descent equations, we find a complete
characterizations of anomalies in 3+1 dimensions. The spatial part of the
anomaly and the first component of the temporal anomaly are always local given
these assumptions even in shape dynamics. The part emerging from the solution
of the functional differential equations explicitly involves the shape dynamics
Hamiltonian, and thus might be non-local. If one restricts this extra piece of
the temporal anomaly to be also local, then overall no \emph{local} Weyl
anomalies, either temporal or spatial, emerge in the 3+1 case.
