Editing Grothendieck topology (section)

=== Topologies on the category of schemes ===
{{See also|List of topologies on the category of schemes}}
The category of [[scheme (mathematics)|scheme]]s, denoted ''Sch'', has a tremendous number of useful topologies.  A complete understanding of some questions may require examining a scheme using several different topologies.  All of these topologies have associated small and big sites.  The big site is formed by taking the entire category of schemes and their morphisms, together with the covering sieves specified by the topology.  The small site over a given scheme is formed by only taking the objects and morphisms that are part of a cover of the given scheme.

The most elementary of these is the [[Zariski topology]].  Let ''X'' be a scheme.  ''X'' has an underlying topological space, and this topological space determines a Grothendieck topology.  The Zariski topology on ''Sch'' is generated by the pretopology whose covering families are jointly surjective families of scheme-theoretic open immersions.  The covering sieves ''S'' for ''Zar'' are characterized by the following two properties:
*For all ''Y'' and every morphism ''f'' : ''Y'' → ''X'' in ''S''(''Y''), there exists a ''V'' and a ''g'' : ''V'' → ''X'' such that ''g'' is an open immersion, ''g'' is in ''S''(''V''), and ''f'' factors through ''g''.
*If ''W'' is the union of all the sets ''f''(''Y''), where ''f'' : ''Y'' → ''X'' is in ''S''(''Y''), then ''W'' = ''X''.
Despite their outward similarities, the topology on ''Zar'' is ''not'' the restriction of the topology on ''Spc''!  This is because there are morphisms of schemes that are topologically open immersions but that are not scheme-theoretic open immersions.  For example, let ''A'' be a non-[[reduced (ring theory)|reduced]] ring and let ''N'' be its ideal of nilpotents.  The quotient map ''A'' → ''A/N'' induces a map Spec ''A/N'' → Spec ''A'', which is the identity on underlying topological spaces.  To be a scheme-theoretic open immersion it must also induce an isomorphism on structure sheaves, which this map does not do.  In fact, this map is a closed immersion.

The [[étale topology]] is finer than the Zariski topology.  It was the first Grothendieck topology to be closely studied.  Its covering families are jointly surjective families of étale morphisms.  It is finer than the [[Nisnevich topology]], but neither finer nor coarser than the [[cdh topology|''cdh'']] and l&prime; topologies.

There are two [[flat topology|flat topologies]], the ''fppf'' topology and the ''fpqc'' topology.  ''fppf'' stands for ''{{lang|fr|fidèlement plate de présentation finie}}'', and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat, of finite presentation, and is quasi-finite.  ''fpqc'' stands for ''{{lang|fr|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 to be a family that is a cover on Zariski open subsets.<ref>[[Séminaire de Géométrie Algébrique du Bois Marie|SGA]] III<sub>1</sub>, IV 6.3.</ref>  In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover.<ref>SGA III<sub>1</sub>, IV 6.3, Proposition 6.3.1(v).</ref>  These topologies are closely related to [[descent (category theory)|descent]].  The ''fpqc'' topology is finer than all the topologies mentioned above, and it is very close to the canonical topology.

Grothendieck introduced [[crystalline cohomology]] to study the ''p''-torsion part of the cohomology of characteristic ''p'' varieties.  In the ''crystalline topology'', which is the basis of this theory, the underlying category has objects given by infinitesimal thickenings together with [[divided power structure]]s.  Crystalline sites are examples of sites with no [[final object]].