My MaBloWriMo 2015 run met an untimely end on the 18th, when LaTeX stopped working on WordPress for me; I could no longer see any of the LaTex I was writing. It’s still not working for me in Chrome, but it’s now working for me in another browser, so hopefully I’ll get some posts up soon.
Yesterday we gave a brief and abstract description of Galois descent, the punchline of which was that Galois descent could abstractly be described as a natural equivalence
where is a Galois extension, is the Galois group of (thinking of as an object of the category of field extensions of at all times), is a category of “objects over ,” and is a category of “objects over .”
In fact this description is probably only correct if is a finite Galois extension; if is infinite it should probably be modified by requiring that every function of that occurs (e.g. in the definition of homotopy fixed points) is continuous with respect to the natural profinite topology on . To avoid this difficulty we’ll stick to the case that is a finite extension.
Today we’ll recover from this abstract description the somewhat more concrete punchline that -forms of an object can be classified by Galois cohomology , and we’ll give some examples.
After a relaxing and enjoyable break, we’re finally in a position to state what it means for structures to satisfy Galois descent.
Fix a field . The gadgets we want to study assign to each separable extension a category of “objects over ,” to each morphism of extensions an “extension of scalars” functor , and to each composable pair of morphisms of extensions a natural isomorphism
of functors (where again we’re taking compositions in diagrammatic order) satisfying the usual cocycle condition that the two natural isomorphisms we can write down from this data agree. We’ll also want unit isomorphisms satisfying the same compatibility as before. This is just spelling out the definition of a 2-functor from the category of separable extensions of to the 2-category , and in particular each naturally acquires an action of (where we mean automorphisms of extensions of , hence if is Galois this is the Galois group) in precisely the sense we described earlier.
We’ll call such an object a Galois prestack (of categories, over ) for short. The basic example is the Galois prestack of vector spaces , which sends an extension to the category of -vector spaces and sends a morphism to the extension of scalars functor
Every example we consider will in some sense be an elaboration on this example in that it will ultimately be built out of vector spaces with extra structure, e.g. the Galois prestacks of commutative algebras, associative algebras, Lie algebras, and even schemes. In these examples, fields are not really the natural level of generality, and to make contact with algebraic geometry we should replace them with commutative rings, but for now we’ll ignore this.
In order to state the definition, we need to know that if is an extension, then the functor naturally factors through the category of homotopy fixed points for the action of on . We’ll elaborate on why this is in a moment.
Definition: A Galois prestack satisfies Galois descent, or is a Galois stack, if for every Galois extension the natural functor (where ) is an equivalence of categories.
In words, this condition says that the category of objects over is equivalent to the category of objects over equipped with homotopy fixed point structure for the action of the Galois group (or Galois descent data).
(Edit, 11/18/15:) This definition is slightly incorrect in the case of infinite Galois extensions; see the next post and its comments for some discussion.
Two weeks ago we proved the following formula. Let be a finitely generated subgroup and let be the number of subgroups of of index . Then
This identity reflects, in a way we made precise in the previous post, the decomposition of a finite -set (the terms on the LHS) into a disjoint union of transitive -sets (the terms on the RHS).
Noam Zeilberger commented on the previous post that he had seen results like this for more specific groups in the literature; in particular, Samuel Vidal describes a version of this analysis for , the modular group. In this post we’ll use the above formula to compute the number of subgroups of index in using a computer algebra system that can manipulate power series. We’ll also say something about how to visualize these subgroups.
Yesterday we described how a (finite-dimensional) projective representation of a group functorially gives rise to a -linear action of on such that the Schur class classifies this action.
Today we’ll go in the other direction. Given an action of on explicitly described by a 2-cocycle , we’ll recover the category of -projective representations, or equivalently the category of modules over the twisted group algebra , by taking the homotopy fixed points of this action. We’ll end with another puzzle.
Today we’ll resolve half the puzzle of why the cohomology group appears both when classifying projective representations of a group over a field and when classifying -linear actions of on the category of -vector spaces by describing a functor from the former to the latter.
(There is a second half that goes in the other direction.)
Three days ago we stated the following puzzle: we can compute that isomorphism classes of -linear actions of a group on the category of vector spaces over a field correspond to elements of the cohomology group
This is the same group that appears in the classification of projective representations of over , and we asked whether this was a coincidence.
Before answering the puzzle, in this post we’ll provide some relevant background information on projective representations.