nLab
twisted form

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Typically, if one has a generalized topology 𝒯\mathcal{T} and one has an object MM “over” that topology (e.g. a sheaf, bundle or stack) one can ask about other objects over 𝒯\mathcal{T} which are locally isomorphic to MM. Here, all of these terms (e.g. topology, equivalent, over) depend on the situation at hand. However, we often say than an object MM' which is locally isomorphic to MM is a twisted form of MM. For some authors, the notion of being locally isomorphic is only for a chosen cover UXU\to X of an object XX. At least for the extent of this wiki, discussion of such twisted forms will always reference the cover. In other words, to say that MM' is a twisted form of MM over XX is to say that MM and MM' are locally isomorphic for all covers. However, to say that MM' is a twisted form of MM for UXU\to X would indicate that MM and MM' 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 KLK\to L, and an LL-vector space WW, there is exactly one KK-vector space VV such that WV KLW\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 ϕ:RS\phi:R\to S, we have an extension of scalars functor RS:Mod RMod S-\otimes_R S:Mod_R\to Mod_S. For a given RR-module MM, a twisted form of MM is another RR-module MM' such that M RSM\otimes_R S is isomorphic to M RSM'\otimes_R S. It turns out that if ϕ\phi is of effective descent for modules, then isomorphism classes of twisted forms of a given SS-module NM RSN\cong M\otimes_R S are in bijection with the set of descent data on NN! What’s more, this set can be computed using nonabelian cohomology.

Last revised on April 5, 2014 at 21:38:03. See the history of this page for a list of all contributions to it.