  Factorization in gauge theories holds at the amplitude or amplitude-squared
level for states of given soft or collinear momenta. When performing
phase-space integrals over such states, one would generally like to avoid
putting in explicit cuts to separate soft from collinear momenta. Removing
these cuts induces an overcounting of the soft-collinear region and adds new
infrared-ultraviolet divergences in the collinear region. In this paper, we
first present a regulator-independent subtraction algorithm for removing
soft-collinear overlap at the amplitude level which may be useful in
pertubative QCD. We then discuss how both the soft-collinear and
infrared-ultraviolet overlap can be undone for certain observables in a way
which respects factorization. Our discussion clarifies some of the subtleties
in phase-space subtractions and includes a proof of the infrared finiteness of
a suitably subtracted jet function. These results complete the connection
between factorized QCD and Soft-Collinear Effective Theory.
