Derived categories of projectivizations and flops

We prove a generalization of Orlov's projectivization formula for the derived category Dcohb(P(E)), where E does not need to be vector bundle; Instead, is coherent sheaf which locally admits two-step resolutions. As special case, this also gives generalized universal hyperplane section formula. applications, (i) we obtain blowup along codimension two Cohen-Macaulay subschemes, (ii) new “flop-flop=twist” results large class flops obtained by crepant resolutions degeneracy loci. another consequence, perverse schober on C. (iii) give applications above symmetric powers curves and Θ-flops, following Toda [79].