In Chapter III,$\S 4$ of Milne's *Etale cohomology* a correspondence between twisted forms and Cech cohomology cocycles is described.


Fix some Grothendieck topology, say, etale, and let $A$ be a sheaf of algebras over a scheme $X$. A sheaf of algebras $A'$ is called a twisted form of $A$ if there exists a cover $\mathscr{U}=(U\_i \to X)$ such that $A \times\_X U\_i \cong A' \times\_X U\_i$. Milne then writes that to such an isomorphism one can associate a cocycle in $\check{H}^1(\mathscr{U}, \underline{Aut}(A))$ where $\underline{Aut}(A)$ is the sheaf associated to the presheaf of groups $Aut(U)=Aut\_U(A \times\_X U)$.


Why does one need to sheafify $Aut$? Isn't it true that $Aut$ is a sheaf already? It is clearly separated, because $A$ is separated, and I don't see why the second sheaf condition would fail, and I couldn't find a counterexample.



is the presheaf of automorphisms a sheaf?
You are absolutely right, it is a sheaf, you can glue local automorphisms.


