Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|cite|improve this question

1 Answer 1

up vote 4 down vote accepted

You are absolutely right, it is a sheaf, you can glue local automorphisms.

share|cite|improve this answer
Thank you for confirming my suspicions. I was left confused after reading this passage from Milne's book. –  Dima Sustretov Nov 11 '11 at 21:53

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.