  A central problem in liaison theory is to decide whether every arithmetically
Cohen-Macaulay subscheme of projective $n$-space can be linked by a finite
number of arithmetically Gorenstein schemes to a complete intersection. We show
that this can be indeed achieved if the given scheme is also generically
Gorenstein and we allow the links to take place in an $(n+1)$-dimensional
projective space. For example, this result applies to all reduced
arithmetically Cohen-Macaulay subschemes. We also show that every union of fat
points in projective 3-space can be linked in the same space to a union of
simple points in finitely many steps, and hence to a complete intersection in
projective 4-space.
