Quot schemes of locally free quotients of a given coherent sheaf, introduced by Grothendieck, are generalizations of projectivizations and Grassmannian bundles, and are closely related to degeneracy loci of maps between vector bundles. In this talk, we will discuss the structure of the derived categories of these Quot schemes in the case when the coherent sheaf has homological dimension . This framework not only allows us to relax the regularity conditions on various known formulae -- such as the ones for blowups, Cayley's trick, standard flips, projectivizations, and Grassmannain flips, but it also leads us to many new phenomena such as virtual flips, and blowup formulae for blowups along determinantal subschemes of codimension . We will illustrate the idea of proof in concrete cases, and if time allowed, we will also discuss the applications to the case of moduli of linear series on curves, and Brill-Noether theory for moduli of stable objects in K3 categories.