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



Why does the first Cech cohomology classify twisted forms?
In general the statement is slightly different. What you typically have is a presheaf $F$ of groupoids (i.e. for every element $X$ you can have $QCoh(X)$ the groupoid of quasicoherent sheaves and isomorphisms, but maybe you are interested in the groupoid of algebras, Azumaya algebras, Hopf algebras etc..) and an object $A$ of $F(Y)$.


Then a twisted form of $A$ is an object $B$ of $F(Y)$ which becomes isomorphic to $A$ when they are pulled back to $X$. In this context you have is a map of pointed sets from the set of isomorphism classes of twisted forms of $A$ to $\check{H}^1(X,Aut(A))$, given by $B\mapsto Iso(A,B)$ where $Iso(A,B)$ is seen as an $Aut(A)$-torsor via precomposition. It is clear that if this torsor is trivial (that is it has a section on $Y$) then $B$ is isomorphic to $A$. Your question can be rephrased to asking conditions under which this map is an isomorphism.


The only case in which I know of a general theorem of this type is when the presheaf of groupoids is obtained by $QCoh$ via some sort of algebraic structure. At example call a *signature* $S$ a collection of pairs $(n\_i,m\_i)\_{i\in I}$. Then the ''algebraic structures of signature $S$'' are the quasicoherent sheaves $F$ with a collection of maps $\Phi\_i:F^{\otimes n\_i} \to F^{\otimes m\_i}$. Then for any signature $S$ the groupoid of ''algebraic structures of signature $S$'' and isomorphisms is such that the map of the previous paragraph is an isomorphism.


You can find a complete discussion when $X\to Y$ is a field extension in the book by Gille and Szamuely *Central simple algebras and Galois cohomology*. From that discussion and the general descent theorem is not hard to get the general case.


Sorry I pressed the submit button by mistake, hence the incomplete answer.


