Descent Theory
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Before starting any discussion about étale topology or étale cohomology, we discuss the notion of a descent. Roughly speaking, a descent asks for uniqueness/recoverability of gluing along the pullback (as a notion of restriction), which generalizes the gluing condition on a topology. We introduce descent theory over Zariski topology, with respect to fpqc and/or fppf morphisms, which includes descent of quasi-coherent sheaves, descent of properties of morphisms, descent of schemes, as well as an argument that allows “passage to limit” through a directed system. With the categorical formulation of descent data, we allow a generalization of descent theory over étale topology and/or with respect to a covering, or over general fibered categories over sites. [Notes]