  We devise an algebraic procedure for the evaluation of Green's functions in
SU(N) Yang-Mills theory in the presence of a non-trivial background field. In
the ghost-free sector the dependence of the vertex functional on the background
is shown to be uniquely determined by the Slavnov-Taylor identities in terms of
a certain 1-PI correlator of the covariant derivatives of the ghost and the
anti-ghost fields. At non-vanishing background this amplitude is shown to
encode the quantum deformations to the tree-level background-quantum splitting.
The approach only relies on the functional identities of the model
(Slavnov-Taylor identities, $b$-equation, anti-ghost equation) and thus it is
valid beyond perturbation theory, and in particular in a lattice implementation
of the background field method. As an example of the formalism we analyze the
ghost two-point function and the Kugo-Ojima function in an instanton background
in SU(2) Yang-Mills theory, quantized in the background Landau gauge.
