Mathematics

Given a scheme Y and a subscheme Z \subset Y we construct
a sequence of triangulated categories D_n(Z,Y), such that
D_0(Z,Y) is equivalent to the derived category of the blowup
X of Y in Z, and for n > 0 there is a semiorthogonal
decomposition D_n(Z,Y) with one component equivalent
to the derived category of X and n copies of the derived
category of Z. These categories are called the higher blowup
categories. Each higher blowup category comes with a pair of adjoint
functors \pi_*:D_n(Z,Y) \to D(Y) and \pi^*:D(Y) \to D_n(Z,Y)
(generalizing the pushforward and the pullback functors for the blowup
morphism \pi:X \to Y), and for n >> 0 the composition \pi_*\circ \pi^*
is isomorphic to identity. This is a joint work in progress with Dima Kaledin.