12:39 PM
Hello. I'm looking at Milnes Etale Cohomology, and I'm slightly confused about his notion of twisted forms. He defines a twisted form $Y'\to X$ of $Y\to X$ for, say, an etale cover $\{U_i\to X\}$ to be $Y'\to X$ such that there are isomorphisms $\varphi_i:Y\times_X U_i\to Y'\times_X U_i$ for every $i\in I$. This is fine. He then observes that this induces an automorphism $\varphi_j^{-1}\circ\varphi_i$ of $Y\times_X U_i\times_X U_j$ where (him and) I have abused notation very slightly by...
writing $\varphi_i$ for the induced morphism on $Y\times_X U_i\times_X U_j$. Then it's clear that such a twisted form induces a $1$-cocycle for the sheaf associated to the presheaf $Aut(Y):U\mapsto Aut_U(Y\times_X U)$. Now the way he writes this section, he makes it seem like this goes through the same for twisted forms of, say, $\mathcal{O}_X$-modules on $X$.
Wait, sorry, that last paragraph (now pseudo-redacted, i.e. still viable by history) isn't quite what I mean to write
What sheaf should play the part of $Aut(Y)$ in the case that we want to consider twisted forms of a sheaf of $\mathcal{O}_X$-modules on $X$?
I'm guessing we would ask for a twisted form $\mathcal{F}'$ of $\mathcal{F}$ that there is a family of isomorphisms $\varphi_i^*\mathcal{F}\cong \varphi_i^*\mathcal{F}'$.
Btw, the relevant page is 134