  In this paper we present an arbitrary-order fully discrete Stokes complex on
general polyhedral meshes. We enriche the fully discrete de Rham complex with
the addition of a full gradient operator defined on vector fields and fitting
into the complex. We show a complete set of results on the novelties of this
complex: exactness properties, uniform Poincar\'e inequalities and primal and
adjoint consistency. The Stokes complex is especially well suited for problem
involving Jacobian, divergence and curl, like the Stokes problem or
magnetohydrodynamic systems. The framework developed here eases the design and
analysis of scheme for such problems. Schemes built that way are nonconforming
and benefit from the exactness of the complex. We illustrate with the design
and study of a scheme to solve the Stokes equations and validate the
convergence rates with various numerical tests.
