Suppose I have a faithfully flat cover of schemes $\phi:X\to Y$, and a sheaf $F$ on $Y$. I might be interested in so-called ``twisted forms for $F$." That is, sheaves $F'$ on $Y$ such that $\phi^\ast(F)\cong \phi^\ast(F')$. In the case that $\phi$ is a cover by some collection $\{U_i\}$, this is often stated as saying that $F$ and $F'$ are locally isomorphic on this cover. Okay, great. You'll have to trust me that this is something I might be interested in.

It is ``known" that if we compute $\check{H}^1(\phi,Aut(F))$, we can determine all twisted forms for $F$ (up to isomorphism class) along $\phi$. Or at least, that is what I believe people to be saying.

1) Is this the correct statement?

2) If so, can someone at least give me a rough explanation of why this is true, if not a complete proof?

3) Instead of starting out with a sheaf on $Y$, can I start off with an effective descent datum for $\phi$ and determine all twisted forms of that descent datum? In other words, can I start with a sheaf $F''$ on $X$ with effective descent data (which tells us that it comes from something over $Y$) and then ask for sheaves on $Y$ which pull back to (something isomorphic to) $F''$? How is their canonical descent datum (coming from pulling back) related to the descent datum on $F''$?

4) What can I do/say if $\phi$ is not faithfully flat? Can I still make these kinds of statements? What if I am interested in only finding twisted forms for one particular descent datum?

I know this is a ton of questions. Sorry if it's way too much.

Thanks!!

-Jon