nLab twisted form

Idea

Typically, if one has a generalized topology $\mathcal{T}$ and one has an object $M$ “over” that topology (e.g. a sheaf, bundle or stack) one can ask about other objects over $\mathcal{T}$ which are locally isomorphic to $M$. Here, all of these terms (e.g. topology, equivalent, over) depend on the situation at hand. However, we often say than an object $M'$ which is locally isomorphic to $M$ is a twisted form of $M$. For some authors, the notion of being locally isomorphic is only for a chosen cover $U\to X$ of an object $X$. At least for the extent of this wiki, discussion of such twisted forms will always reference the cover. In other words, to say that $M'$ is a twisted form of $M$ over $X$ is to say that $M$ and $M'$ are locally isomorphic for all covers. However, to say that $M'$ is a twisted form of $M$ for $U\to X$ would indicate that $M$ and $M'$ are only isomorphic when pulled back along the given cover.

The notion of a twisted form is very general, and has manifestations in differential geometry, commutative algebra, category theory, algebraic geometry, and more recently in homotopy theory. However, one unifying property in all of these cases is that such forms should be classified by an appropriate cohomology. Typically, this will be a sort of sheaf cohomology or Čech cohomology with coefficients in a sheaf of automorphisms of the object of interest. Moreover, computations of such cohomologies can often be simplified by identifying them as Galois cohomology or Hopf-Galois cohomology?. In many cases, twisted forms are in bijection with some relevant notion of torsor for the automorphism object and when a specific cover is referenced, twisted forms are equivalent to descent data for the object along that cover.

History

Probably the most famous twisted form computation is Hilbert's Theorem 90. That theorem can be reinterpreted as saying that for a Galois extension of fields $K\to L$, and an $L$-vector space $W$, there is exactly one $K$-vector space $V$ such that $W\cong V\otimes_K L$. In that case, the relevant nonabelian cohomology has been reinterpreted to look like Galois cohomology. Serre discussed twisted forms at some length (though he just called them “forms”) in his book Corps Locaux.

Twisted forms in differential geometry

In differential geometry, one often twists differential forms by a line bundle; see differential form#twisted.

Twisted forms in commutative algebra

Given a homomorphism of commutative rings $\phi:R\to S$, we have an extension of scalars functor $-\otimes_R S:Mod_R\to Mod_S$. For a given $R$-module $M$, a twisted form of $M$ is another $R$-module $M'$ such that $M\otimes_R S$ is isomorphic to $M'\otimes_R S$. It turns out that if $\phi$ is of effective descent for modules, then isomorphism classes of twisted forms of a given $S$-module $N\cong M\otimes_R S$ are in bijection with the set of descent data on $N$! What’s more, this set can be computed using nonabelian cohomology.