Flat cohomology

In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also plays a fundamental role in the theory of descent (faithfully flat descent). The term flat here comes from flat modules.

There are several slightly different flat topologies, the most common of which are the fppf topology and the fpqc topology. fppf stands for fidèlement plate de présentation finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation. fpqc stands for fidèlement plate et quasi-compacte, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat. In both categories, a covering family is defined be a family which is a cover on Zariski open subsets. In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover. These topologies are closely related to descent. The "pure" faithfully flat topology without any further finiteness conditions such as quasi compactness or finite presentation is not used much as is not subcanonical; in other words, representable functors need not be sheaves.

Unfortunately the terminology for flat topologies is not standardized. Some authors use the term "topology" for a pretopology, and there are several slightly different pretopologies sometimes called the fppf or fpqc (pre)topology, which sometimes give the same topology.

Flat cohomology was introduced by Grothendieck in about 1960.

Let X be an affine scheme. We define an fppf cover of X to be a finite and jointly surjective family of morphisms

with each X_a affine and each φ_aflat, finitely presented. This generates a pretopology: for X arbitrary, we define an fppf cover of X to be a family

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