296:"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 pretopology can also be given with an extra quasi-finiteness condition; it follows from Corollary 17.16.2 in EGA IV
94:, 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
101:
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.
98:. The "pure" faithfully flat topology without any further finiteness conditions such as quasi compactness or finite presentation is not used much as it is not subcanonical; in other words, representable functors need not be sheaves.
463:"Fpqc" is an abbreviation for "fidèlement plate quasi-compacte", that is, "faithfully flat and quasi-compact". Every surjective family of flat and quasi-compact morphisms is a covering family for this topology, hence the name.
523:
at this point, which is a discrete valuation ring whose spectrum has one closed point and one open (generic) point. We glue these spectra together by identifying their open points to get a scheme
272:
which is part of some covering family. (This does not imply that the morphism is flat, finitely presented.) The morphisms are morphisms of schemes compatible with the fixed maps to
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While such groups have a number of applications, they are not in general easy to compute, except in cases where they reduce to other theories, such as the
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726:
681:
82:, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation.
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The following example shows why the "faithfully flat topology" without any finiteness conditions does not behave well. Suppose
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which is part of some covering family. The morphisms are morphisms of schemes compatible with the fixed maps to
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The procedure for defining the cohomology groups is the standard one: cohomology is defined as the sequence of
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and took covering families to be jointly surjective families of flat, finitely presented morphisms.) We write
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754:, online book by James Milne, explains at the level of flat cohomology duality theorems originating in the
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552:, and these maps are the same on intersections. However they cannot be combined to give a map from
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548:) which are open in the faithfully flat topology, and each of these sets has a natural map to
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and took covering families to be jointly surjective families of flat morphisms.) We write
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There are several slightly different flat topologies, the most common of which are the
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381:} which is an fpqc cover after base changing to an open affine subscheme of
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which is an fppf cover after base changing to an open affine subscheme of
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Flat cohomology was introduced by
Grothendieck in about 1960.
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and J. S. Milne, "Duality in the flat cohomology of curves",
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to be a finite and jointly surjective family of morphisms {
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to be a finite and jointly surjective family of morphisms
654:, Documents Mathématiques (Paris) , vol. 3, Paris:
456:, that is, the category of schemes with a fixed map to
289:, that is, the category of schemes with a fixed map to
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is the affine line over an algebraically closed field
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for the category of schemes with the fpqc topology.
385:. This pretopology generates a topology called the
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for the category of schemes with the fppf topology.
47:; it also plays a fundamental role in the theory of
218:. This pretopology generates a topology called the
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8:
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359:arbitrary, we define an fpqc cover of
184:arbitrary, we define an fppf cover of
78:fidèlement plate de présentation finie
738:, Volume 35, Number 1, December, 1976
560:, because the underlying spaces of
460:, considered with the fpqc topology.
355:. This generates a pretopology: For
293:, considered with the fppf topology.
43:. It is used to define the theory of
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51:(faithfully flat descent). The term
312:be an affine scheme. We define an
300:that this gives the same topology.
597:"Form of an (algebraic) structure"
90:fidèlement plate et quasi-compacte
25:
708:Éléments de géométrie algébrique
636:, IV 6.3, Proposition 6.3.1(v).
514:we can consider the local ring
721:, Princeton University Press,
656:Société Mathématique de France
527:. There is a natural map from
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650:; Raynaud, Michèle (2003) ,
539:is covered by the sets Spec(
427:) whose objects are schemes
304:The big and small fpqc sites
260:) whose objects are schemes
109:The big and small fppf sites
750:Arithmetic Duality Theorems
602:Encyclopedia of Mathematics
568:have different topologies.
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475:of the functor taking the
736:Inventiones Mathematicae
506:. For each closed point
648:Grothendieck, Alexander
481:sheaf of abelian groups
431:with a fixed morphism
264:with a fixed morphism
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37:Grothendieck topology
176:. This generates a
756:Tate–Poitou duality
674:2002math......6203G
535:. The affine line
445:large fpqc site of
409:small fpqc site of
278:large fppf site of
242:small fppf site of
775:Algebraic geometry
658:, p. XI.4.8,
174:finitely presented
41:algebraic geometry
760:Galois cohomology
727:978-0-691-08238-7
683:978-2-85629-141-2
16:(Redirected from
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719:Étale Cohomology
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345:affine and each
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178:pretopology
86:stands for
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29:mathematics
769:Categories
717:. (1980),
701:References
314:fpqc cover
155:with each
123:fppf cover
623:, IV 6.3.
607:EMS Press
572:See also
477:sections
370: :
327: :
199: :
140: :
66:and the
39:used in
692:2017446
670:Bibcode
632:SGA III
619:SGA III
609:, 2001
494:Example
443:. The
276:. The
96:descent
49:descent
725:
690:
680:
454:Fpqc/X
287:Fppf/X
180:: for
117:be an
31:, the
752:(PDF)
660:arXiv
584:Notes
479:of a
35:is a
723:ISBN
678:ISBN
564:and
425:fpqc
407:The
402:Fpqc
393:and
353:flat
308:Let
258:fppf
240:The
235:Fppf
226:and
170:flat
113:Let
84:fpqc
72:fppf
53:flat
758:of
556:to
531:to
510:of
316:of
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27:In
771::
688:MR
686:,
676:,
668:,
605:,
599:,
490:.
483:.
435:→
377:→
334:→
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672::
662::
645:*
634:1
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