A principal -bundle need not be Zariski locally trivial, e.g. see here, but it is at least etale-locally trivial. The idea is that every point admits an etale neighborhood over which has a right inverse.
A stronger claim is true if is affine: Every principal bundle is locally isotrivial. This means the etale neighborhood can be chosen to be finite over its image in (i.e. the number of points in the fiber is constant, see here), e.g. see this post.
An important proposition: Let be a principal -variety, and another -variety, then is a principal -variety. For this we need to be quasiprojective as well to ensure (*) Every finite subset of is contained in an affine open subset.
A -equivariant sheaf on a -variety is a sheaf together with a rule that assigns to every and open an isomorphism , and it should satisfies some associativity. Equivalently, we can say for every , there is a map satisfying associativity where is the action of on . However, this doesn’t take into account the topology of (in other words we just treat like a discrete group). A better definition is to view as part of in . Then the associativity axiom is where is multiplication of the first two coordinates, is the action of the second coordinates, and is the projection onto the last two coordinates. The idea is that , , and , so this boils down to the usual axiom. The Proposition is that is an abelian category.
There are certain subtlety relating to equivariant derived category. It is neither nor just copy the above definition but allowing to be a general object in . See section 6.4. We want the definition to satisfy the desideratum 6.1, including forgetful functor and six-functor formalism that is mutually compatible. See example 6.4.2 that uses the below proposition.
(Prop. 6.2.13) Every -equivariant perverse sheaf on a -homogeneous space is a shifted local system and there is an equivalence of categories -mod.
However, if is a principal -variety with geometric quotient , then we can define as . The key is Proposition 6.2.10 asserting induces an equivalence of categories between and . This is just smooth descent for perverse sheaves plus identifying descent datum with the definition of -equivariant sheaves using the Cartesian diagram for principal -varieties.
For general -varieties we use acyclic -resolutions. See Exercise 3.7.2 for the definition of -acyclic morphism. The key point is that for -acyclic smooth morphism the shifted pullback is fully faithful (easily seen from the adjunction). The motivation is that the cohomology of the fibers of can lead to ‘extra’ morphisms in or in the category of descent data so there is no smooth descent on the level of derived category. However, if is -acyclic, then for all , the unit and counit map and are both isomorphisms.
Lemma 6.1.20 tells us when is an -acyclic morphism in terms of . The idea is to reduce to the case of constant sheaf on and apply Exercise 3.2.1 (t-exactness of under good situations). From this we can produce lots of -ayclic resolutions and -resolutions more generally. See Example 6.1.21 and 6.1.22. For linear algebraic group this is suffice to conclude the existence of -acyclic -resolution of any -variety using Prop. 6.1.13 and Prop. 6.1.15.
Now the definition for as in Definition 6.4.4 appears very natural. Essentially for every -resolution we give an object and it should be compatible for different -resolutions in the sense that if is a morphism of -resolution then there is an isomorphism . Moreover, the isomorphism should compose.
Note that we always have the trivial -resolution which allows us to define the forgetful functor . It is not evidently a triangulated category with heart . See Berstein-Lunt’s book for detailed proof.
Reference: Achar’s book