Etale topology
The most important example of a Grothendieck topology (see Topologized category), making it possible to define cohomology and homotopy invariants for abstract algebraic varieties and schemes. Let be a scheme. The étale topology on
is the name for the category
of étale
-schemes the objects of which are étale morphisms (cf. Etale morphism)
and the morphisms of which are those of the
-schemes. Finite families
such that
are taken as coverings and so in
a topology is introduced.
A pre-sheaf of sets (groups, Abelian groups, etc.) on is defined as a contravariant functor
from the category
into that of sets (groups, etc.). A pre-sheaf
is called a sheaf if for any covering
a section
is determined by its restriction to
and if for any compatible collection of sections
there exists a unique section
such that
. Many standard concepts of sheaf theory carry over to étale sheaves (that is, sheaves on
). For example, if
is a morphism of schemes and
is an étale sheaf on
, then by putting
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one obtains the so-called direct image of
for the morphism
. The functor
adjoint to
on the left is called the inverse-image functor. In particular, the stalk of
at a geometric point
(where
is an algebraically closed field) is defined as the set
.
An important example of a sheaf on is
, representable by a certain
-scheme
; for it
. If
is a finite étale
-scheme, then the sheaf
is called locally constant. A sheaf
is said to be constructible if there exists a finite partition of
into locally closed subschemes
such that the restriction
is locally constant on every
.
See also Etale cohomology; Homotopy type of a topological category.
References
[1] | Yu.I. Manin, "Algebraic topology of algebraic varieties" Russian Math. Surveys , 20 : 5/6 (1965) pp. 183–192 Uspekhi Mat. Nauk , 20 : 6 (1965) pp. 3–12 |
[2] | J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) |
[3] | M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , Théorie des topos et cohomologie étale des schémas (SGA 4) , Lect. notes in math. , 269; 270; 305 , Springer (1972–1973) |
[4] | P. Deligne, "Cohomologie étale (SGA 4 1/2)" , Lect. notes in math. , 569 , Springer (1977) |
Etale topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Etale_topology&oldid=15079