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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).〔(Springer EoM article )〕 (The term ''flat'' here comes from flat modules.) Strictly, there is no single definition of the flat topology, because, technically speaking, different finiteness conditions may be applied. == The big and small fppf sites == Let ''X'' be an affine scheme. We define an fppf cover of ''X'' to be a finite and jointly surjective family of morphisms :(''φ''a : ''X''a → ''X'') with each ''X''a affine and each ''φ''a flat, finitely presented, and quasi-finite. This generates a pretopology: for ''X'' arbitrary, we define an fppf cover of ''X'' to be a family :(''φ'a : ''X''a → ''X'') which is an fppf cover after base changing to an open affine subscheme of ''X''. This pretopology generates a topology called the ''fppf topology''. (This is not the same as the topology we would get if we started with arbitrary ''X'' and ''X''a and took covering families to be jointly surjective families of flat, finitely presented, and quasi-finite morphisms.) We write ''Fppf'' for the category of schemes with the fppf topology. The small fppf site of ''X'' is the category ''O''(''X''fppf) whose objects are schemes ''U'' with a fixed morphism ''U'' → ''X'' which is part of some covering family. (This does not imply that the morphism is flat, finitely presented, and quasi-finite.) The morphisms are morphisms of schemes compatible with the fixed maps to ''X''. The large fppf site of ''X'' is the category ''Fppf/X'', that is, the category of schemes with a fixed map to ''X'', considered with the fppf topology. "Fppf" is an abbreviation for "fidèlement plate de présentation finie", that is, "faithfully flat and of finite presentation". Every surjective family of flat and finitely presented morphisms is a covering family for this topology, hence the name. The definition of the fppf topology is generally given without the quasi-finiteness condition; its equivalence with the above definition follows from Corollary 17.16.2 in EGA IV4. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Flat topology」の詳細全文を読む スポンサード リンク
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