Deformations of Categories of Coherent Sheaves and Fourier-Mukai Transforms

In modern algebraic geometry, an algebraic variety is often studied by way of its category of coherent sheaves or derived category. Recent work by Toda has shown that infinitesimal deformations of the category of coherent sheaves can be described as twisted sheaves on a noncommutative deformation of the variety. This thesis generalizes Toda's work by creating a chain of inclusions from deformations of schemes to commutative deformations to deformations of the category of coherent sheaves. We define projections from coherent deformations to commutative deformations to scheme deformations and show that the fiber of the projection from commutative deformations to schemes is a gerbe.;We also prove that for two derived equivalent K3 surfaces in characteristic p and any scheme deformation of one of these, there is a scheme deformation of the other so that the two deformations are also derived equivalent.