  Given a generic map between flagged vector bundles on a Cohen-Macaulay
variety, we construct maximal Cohen-Macaulay modules with linear resolutions
supported on the Schubert-type degeneracy loci. The linear resolution is
provided by the Schubert complex, which is the main tool introduced and studied
in this paper. These complexes extend the Schubert functors of Kra\'skiewicz
and Pragacz, and were motivated by the fact that Schur complexes resolve
maximal Cohen-Macaulay modules supported on determinantal varieties. The
resulting formula in K-theory provides a "linear approximation" of the
structure sheaf of the degeneracy locus, which can be used to recover a formula
due to Fulton.
