Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I am reading Milne's Étale cohomology, III.4.

A twisted form of an object $Y$ (a scheme, a sheaf of modules, of algebras...) over a scheme $X$ is an object $Y'$ such that there exists a covering in some topology (let's say étale to fix the ideas), $(U_i \to X)$ such that $Y \times_X U_i \cong Y' \times_X U_i$ for all $i$. Then, the book says, any twisted form of $Y$ defines a cocycle in $\check{H}^1(X,\mathrm{Aut}(Y))$ as follows: let $f_i$ be the isomorpisms $Y \times_X U_i \to Y' \times_X U_i$, then the cocycle is given by $(\alpha_{ij}): Y \times_X U_i \times_X U_j \to Y \times_X U_i \times_X U_j$ where $\alpha_{ij} = f_i^{-1} \circ f_j$, which, in turn, constitutes the descent data that eventually allows to recover $Y'$. As far as I understand, the descent data does not have to be always effective in general, but in some cases it is. The book gives two examples when this is the case: Severi-Brauer varieties and vector bundles.

I would like to understand when the Cech cohomology classes are in bijective correspondnence with the (isomorphism classes of) twisted forms. So I have two questions:

  1. what are the general criteria that the descent data as described above is effective?

  2. what happens if the trivialising cover of a twisted form $Y'$ contains just one étale morphism $U_0 \to X$? It seems like the cocycle defined by $Y'$ is always trivial then ($\alpha_{00} = f_0^{-1} \circ f_0$), yet the form $Y'$ might be not isomorphic to $Y$.

share|improve this question

2 Answers 2

up vote 5 down vote accepted

For your question 2, note that the fiber product $U_0\times_XU_0$ can be non-trivial (unlike the case of an open sub-set $U_0\subset X$, where it would be $U_0$ again) with two different projections $p_1$ and $p_2$ onto $U_0$ giving two different structures to it as a $U_0$-scheme. The isomorphism $\alpha_{0,0}$ is an isomorphism between these two different $U_0$-structures on $Y\times_XU_0\times_XU_0$ and is also a non-trivial piece of information

As an example, if $X=Spec\;k$ and $U_0=Spec\;K$ where $K/k$ is a Galois extension with Galois group $G$, then we get an isomorphism $G\times U_0\rightarrow U_0\times_XU_0$ using the group action. The two structures of $G\times U_0$ as a $U_0$-scheme correspond to the two maps $(g,u)\mapsto u$ and $(g,u)\mapsto gu$. The isomorphism $\alpha_{0,0}$ satisfying the co-cycle condition now equates precisely to giving an action of $G$ on $Y'$ compatible with its action on $U_0$. This is Galois descent. See Serre's Local fields, for example.

EDIT: It occurs to me that I didn't directly talk about twisted objects. The idea is the same. Let us start with an object $Y'$ over $U_0$ equipped with an isomorphism $f_0:Y'\rightarrow Y\times_XU_0$. This isomorphism has to satisfy the co-cycle condition, which means the following:

We have two projections $p_1,p_2:U_0\times_XU_0\rightarrow U_0$. So we have two different ways to pull-back $Y'$ to over $U_0\times_XU_0$, giving us $p_1^*Y'$ and $p_2^*Y'$. The two pull-backs of $Y\times_XU_0$ under these projections are canonically isomorphic, since the two projections $p_1,p_2$, when composed with the map $U_0\rightarrow X$ agree with the structure map for $U_0\times_XU_0\rightarrow X$. So pulling back $f_0$ gives us isomorphisms $$p_1^*Y'\rightarrow Y\times_X(U_0\times_XU_0)\rightarrow p_2^*Y'.$$

This is your $\alpha_{0,0}$; if it satisfies the co-cycle condition--this amounts to the required compatibility between the pull-backs of $\alpha_{0,0}$ to $U_0\times_{X}U_0\times_{X}U_0$ under the three different projections to $U_0\times_XU_0$--then it gives you descent data for $Y'$. In the Galois setting, if you use $f_0$ to identify $Y'$ with $Y\times_XU_0$, then $\alpha_{0,0}$ is giving you a `twisted' action of $G$ on $Y\times_XU_0$.

This data is not always effective. I would highly recommend the chapter on descent in Bosch-Lutkebohmmert-Raynaud's `Neron Models' for an explanation of all these things.

share|improve this answer
    
Dear Keerthi, thank you for you answer. I will look at the chapter in "Neron models". Is it easy to answer from what general theorem does it follow that in the case of vector bundles or Severi-Brauer varieties the descent data is always effective? In the chapter of Milne's book I have mentioned, SGA1.I.VIII.7.8 is referenced, but so far I couldn't extract the precise statement that would imply the bijection between isomorphism types of twisted forms and cocycle classes. –  Dima Sustretov Apr 21 '11 at 19:19
    
Dear Dmitry--Yes, I believe BLR has a resume of effectivity results. For vector bundles, it's the fact that descent data for quasi-coherent sheaves are always effective. For Severi-Brauer varieties, it's the fact that varieties with an ample line bundle always have effective descent data. I don't have BLR at hand, so I can't give you references for the precise statements there. Once you have effectivity, the bijection you need follows formally. Again, I don't know of a good reference where this is laid out cleanly, though I can say that SGA1 is quite clearly written in general. –  Keerthi Madapusi Pera Apr 21 '11 at 20:43

If your $Y$ is an object over $X$ of a stack in the étale topology, then you can use the cocycles as descent data to get a twisted form (because by definition of a stack all descent data are effective).

Some examples other than the ones you mentioned are quasi-coherent sheaves and affine morphisms of schemes (i.e. quasi-coherent sheaves of algebras).

This is far from a necessary condition on $Y$, I don't know if there is a sharper characterization.

share|improve this answer

Your Answer

 
discard

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.