Descent for quasi-coherent sheaves on stacks

Abstract

Stacks were introduced in order to parametrize problems in algebraic geometry where the presence of automorphisms prevented representability by a scheme or even a sheaf (see Artin [1], Deligne–Mumford [3] and Giraud [5]). One early application was Deligne and Mumford’s use of stacks to prove the irreducibility of the space of curves of a given genus [3]. More recently stacks have also played an important role in algebraic topology. Complex oriented cohomology theories give rise to stacks over the moduli stack of formal groups and, in certain situations, stacks over the moduli stack of formal groups give rise to spectra (see Goerss [6], Goerss–Hopkins [8] and Rezk [21]) which play an important role in understanding the homotopy groups of spheres (see Goerss–Henn–Mahowald–Rezk [7] and Behrens [2]). One fundamental example is the spectrum of topological modular forms (see Hopkins [12]) which is associated to the moduli stack of elliptic curves.