The bounded derived category of coherent sheaves on a smooth projective variety replaces sheaf cohomology groups with the complexes that compute them, localizing chain complexes at quasi-isomorphisms so that derived functors, exact triangles and Serre duality become statements about a single triangulated category. Students learn how this category is described by exceptional and semiorthogonal decompositions, how integral transforms with a kernel on a product give every known exact functor between such categories, and how much of a variety (its points, line bundles, and pluricanonical ring) can be reconstructed from it. The theory explains when two different varieties are derived equivalent and which invariants, such as Hochschild homology and cohomology, survive that equivalence.
This arxiv lecture note by Andrei Caldararu provides an accessible introduction to derived categories of coherent sheaves on algebraic varieties. The notes cover foundational results including the de…