If X = Spec(R), following customary usage and depending on the context,
we also use the notation Hf1ppf (R, G) instead of Hf1ppf (X, G). Similarly for
the étale and Zariski sites.
If G is in addition affine, by faithfully flat descent all of our sheaf torsors
are representable. They are thus torsors in the usual sense. For a G-torsor
E we denote by E G the twisted form of G by inner automorphisms; it is an
affine group scheme over X. Furthermore, if G is smooth all torsors are lo1 (X, G) = H 1
cally trivial for the étale topology. In particular, Hét
f ppf (X, G).
These assumptions on G hold in most of the situations that arise in our
work. Also, as we mentioned in the introduction, by default our topology
1 (X, G) we will write H 1 (X, G).
will be étale so that instead of Hét
Given an X–group G and a morphisms Y → X of schemes, we let GY
denote the Y –group G ×X Y obtained by base change. For convenience, we
will denote H 1 (Y, GY ) by H 1 (Y, G).
Assuming G affine of finite type, a maximal torus T of G is a subgroup
X-scheme T of G such that T ×X κ(x) is a maximal κ(x)–torus of G ×X κ(x)
for each point x ∈ X [SGA3, XII.1]. Here κ(x) denotes an algebraic closure
of κ(x). A G–torsor E is toral if the twisted group scheme E G admits a
maximal X-torus. We denote by Hf1ppf,toral (X, G) the subset of Hf1ppf (X, G)
consisting of (isomorphism) classes of toral X–torsors under G.

### Language ###
English

### Assessment ###
The text defines when a topology is used by default, which could be a straightforward information request.

### Query ###
What is the default topology used in algebraic geometry?
What is the default topology used in algebraic geometry?