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.
The usage of the term stack above is meant to activate two intuitions. On the one hand, the way we stated Galois descent is meant to look like a sheaf condition. Since for a Galois extension, the Galois descent condition just says that the 2-functor sends certain limits to certain homotopy limits (or 2-limits, depending on taste).
On the other hand, stacks are supposed to be generalizations of schemes, and we can also think of the assignment as the functor of points of some sort of moduli space of objects such that maps correspond to objects over (that is, objects in ).
In addition, as mentioned in the first post in this series on Galois descent, this condition should also be regarded as a categorification of the observation that if denotes the set of -points of a variety over , then the natural map is an isomorphism.
The natural factorization
In order to state the above condition we needed to know that naturally factors through the category of homotopy fixed points. Let’s think about the analogous question one category level down: suppose is a map of sets, and is a group acting on . When does factor through the set of fixed points? The answer is precisely when
for all . This reflects the universal property of taking fixed points: it’s a certain limit, and so it’s preserved by functors of the form in the sense that we have a natural isomorphism
It shouldn’t be surprising that taking homotopy fixed points has an analogous universal property. In fact, it’s not hard to check that if denotes the functor category between two categories and acts on , then we have a natural equivalence of categories
where refers to taking homotopy fixed points on both the LHS and the RHS. In words, this equivalence says that the category of functors is equivalent to the category of functors equipped with homotopy fixed point structure for the induced action of .
Now we just need to observe that the extension of scalars functor is always equipped with homotopy fixed point structure for the action of . This follows from functoriality: because is, by definition, fixed by the action of , we have
for all , and applying gives natural isomorphisms
satisfying the appropriate compatibilities to give homotopy fixed point data. This is the abstract version of the concrete argument we gave previously that the extension of scalars functor naturally factors through homotopy fixed points .
We can extract somewhat more concrete information from this abstract version of Galois descent as follows.
Definition: Fix an object . An object is a -form of if there is an isomorphism .
If we understand the objects in well, we can hope to understand the objects in by understanding the -forms of objects in .
Example. Let , and let be the Galois prestack of semisimple Lie algebras. This turns out to be a Galois stack; that is, semisimple Lie algebras satisfy descent. The classification of complex semisimple Lie algebras reduces the classification of real semisimple Lie algebras to the classification of real forms of complex semisimple Lie algebras; explicitly, a real form of a complex Lie algebra is a real Lie algebra such that
These can be quite varied. For example, the complex special orthogonal Lie algebra has real forms the indefinite special orthogonal Lie algebras for any nonnegative integers such that .
If Galois descent holds, then the extension of scalars functor can be described simply as the functor that takes an object of equipped with homotopy fixed point structure and forgets that structure. Hence:
Observation: If is a Galois stack, then isomorphism classes of -forms of an object can be identified with isomorphism classes of homotopy fixed point structures on .