  Given a contraction of a variety X to a base Y, we enhance the locus in Y
over which the contraction is not an isomorphism with a certain sheaf of
noncommutative rings D, under mild assumptions which hold in the case of (1)
crepant partial resolutions admitting a tilting bundle with trivial summand,
and (2) all contractions with fibre dimension at most one. In all cases, this
produces a global invariant. In the crepant setting, we then apply this to
study derived autoequivalences of X. We work generally, dropping many of the
usual restrictions, and so both extend and unify existing approaches. In full
generality we construct a new endofunctor of the derived category of X by
twisting over D, and then, under appropriate restrictions on singularities,
give conditions for when it is an autoequivalence. We show that these
conditions hold automatically when the non-isomorphism locus in Y has
codimension 3 or more, which covers determinantal flops, and we also control
the conditions when the non-isomorphism locus has codimension 2, which covers
3-fold divisor-to-curve contractions.
