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 $X$ be a scheme. The étale topology on $X$ is the name for the category $X_{\text{et}}$ of étale $X$-schemes the objects of which are étale morphisms (cf. Etale morphism) $U \to X$ and the morphisms of which are those of the $X$-schemes. Finite families $\left({f_i:U_i\to U}\right)$ such that $U = \cup_i f_i[U_i]$ are taken as coverings and so in $X_{\text{et}}$ a topology is introduced.

A pre-sheaf of sets (groups, Abelian groups, etc.) on $X_{\text{et}}$ is defined as a contravariant functor $\mathcal{F}$ from the category $X_{\text{et}}$ into that of sets (groups, etc.). A pre-sheaf $\mathcal{F}$ is called a sheaf if for any covering $\left({f_i:U_i\to U}\right)$ a section $s \in \mathcal{F}(U)$ is determined by its restriction to $U_i$ and if for any compatible collection of sections $s_i \in \mathcal{F}(U_i)$ there exists a unique section $s \in \mathcal{F}(U)$ such that $F_i^*(s) = S_i$. Many standard concepts of sheaf theory carry over to étale sheaves (that is, sheaves on $X_{\text{et}}$). For example, if $f : X \to Y$ is a morphism of schemes and $\mathcal{F}$ is an étale sheaf on $X$, then by putting $$ (f * \mathcal{F})(V) = \mathcal{F}(X \times_Y V) $$ one obtains the so-called direct image $f*\mathcal{F}$ of $\mathcal{F}$ for the morphism $f$. The functor $f^*$ adjoint to $f*$ on the left is called the inverse-image functor. In particular, the stalk of $\mathcal{F}$ at a geometric point $\eta : \mathrm{Spec}(F) \to X$ (where $K$ is an algebraically closed field) is defined as the set $\mathcal{F}_\eta = \eta^*\mathcal{F}(\mathrm{Spec}(K))$.

An important example of a sheaf on $X_{\text{et}}$ is $\mathcal{F}_Z$, representable by a certain $X$-scheme $Z$; for it $\mathcal{F}_Z(U) = \mathrm{Hom}_X(U,Z)$. If $Z$ is a finite étale $X$-scheme, then the sheaf $\mathcal{F}_Z$ is called locally constant. A sheaf $F$ is said to be constructible if there exists a finite partition of $X$ into locally closed subschemes $X_i$ such that the restriction $\mathcal{F}|_{X_i}$ is locally constant on every $X_i$.

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)