Étale torsors and contracted products

Let $B$ be a scheme and let $J\to B$ be a commutative group scheme, meaning a $B$-scheme with multiplication, identity, and inverse morphisms satisfying the group axioms. On $B_{\acute{e}t}$ its section functor $U\mapsto {\mathrm{Hom}}_B(U,J)$ is a sheaf of abelian groups, with operations induced sectionwise. A right $J$-torsor is a sheaf $P$ of sets on $B_{\acute{e}t}$ with a right action of this group sheaf such that there is a declared étale cover $(U_i\to B)$ on which $P{\mathrm{\mid }}_{U_i}$ is equivariantly isomorphic to $J{\mathrm{\mid }}_{U_i}$. Here restriction to the slice site over $U_i$ is $P{\mathrm{\mid }}_{U_i}(V\to U_i)=P(V\to B)$; the composite $V\to B$ is étale. For two right torsors $P,Q$, define their contracted product $P{\times }^{J}Q$ as the associated sheaf of the orbit presheaf of $P\times Q$ under the anti-diagonal action $(p,q)\cdot g=(p\cdot g,q\cdot g^{-1})$; its right $J$-action is $[p,q]\cdot h=[p\cdot h,q]$. Write ${\mathrm{Sha}}_{\mathrm{e}\mathrm{t}}(J\mathrm{/}B)$ for the set of isomorphism classes of right $J$-torsors; contracted product makes this set an abelian group.

Presheaves and sheaves of sets on a site

Representable presheaves are sheaves on the small étale site

Every presheaf of sets on the small étale site has an associated sheaf

Jointly surjective étale families form a Grothendieck pretopology

The Stacks Project, Section 59.27

Milne, Lectures on Étale Cohomology, Chapter I, §§5–6 and §11

The definition is for sheaves of sets and does not assert that a torsor sheaf is represented by a scheme. It does not identify ${\mathrm{Sha}}_{\mathrm{e}\mathrm{t}}(J\mathrm{/}B)$ with derived sheaf cohomology $H_{\acute{e}t}^1(B,J)$ or with an arithmetic Tate–Shafarevich group.