  We study matrix factorizations of locally free coherent sheaves on a scheme.
For a scheme that is projective over an affine scheme, we show that
homomorphisms in the homotopy category of matrix factorizations may be computed
as the hypercohomology of a certain mapping complex. Using this explicit
description, we give another proof of Orlov's theorem that there is a fully
faithful embedding of the homotopy category of matrix factorizations into the
singularity category of the corresponding zero subscheme. We also give a
complete description of the image of this functor.
