  A theoretical analysis of higher-order corrections to D=11 supergravity is
given in a superspace framework. It is shown that any deformation of D=11
supergravity for which the lowest-dimensional component of the four-form $G_4$
vanishes is trivial. This implies that the equations of motion of D=11
supergravity are specified by an element of a certain spinorial cohomology
group and generalises previous results obtained using spinorial or pure spinor
cohomology to the fully non-linear theory. The first deformation of the theory
is given by an element of a different spinorial cohomology group with
coefficients which are local tensorial functions of the massless supergravity
fields. The four-form Bianchi Identities are solved, to first order and at
dimension $-{1/2}$, in the case that the lowest-dimensional component of $G_4$
is non-zero. Moreover, it is shown how one can calculate the first-order
correction to the dimension-zero torsion and thus to the supergravity equations
of motion given an explicit expression for this object in terms of the
supergravity fields. The version of the theory with both a four-form and a
seven-form is discussed in the presence of the five-brane anomaly-cancelling
term. It is shown that the supersymmetric completion of this term exists and it
is argued that it is the unique anomaly-cancelling invariant at this dimension
which is at least quartic in the fields. This implies that the first
deformation of the theory is completely determined by the anomaly term from
which one can, in principle, read off the corrections to all of the superspace
field strength tensors.
