Derived category of coherent sheaves

Summary

Let $X$ be a Noetherian scheme. Its derived category of coherent sheaves $D^b(X)$ is the bounded derived category of the abelian category of coherent sheaves on $X$. It is a triangulated category. For $X$ smooth and projective over a field, the derived functors of pushforward, pullback, tensor product and Hom act on these categories. It is the natural home of Serre and Grothendieck duality and of Fourier–Mukai transforms. Two varieties with equivalent derived categories share many invariants, so $D^b(X)$ is studied as an invariant of $X$ in its own right.