If X is an algebraic variety being acted upon by a linearly reductive group G, Mumford's Geometric Invariant Theory tells us how to take the quotient X/G.  However, this quotient depends on a choice of a G-equivariant line bundle on X.  I will discuss a comparison theorem developed in joint work with M. Ballard and L. Katzarkov, about how the derived category of the quotient is affected by these choices.