Stating Galois descent

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 Image may be NSFW.
Clik here to view.. The gadgets we want to study assign to each separable extension Image may be NSFW.
Clik here to view. $k \to L$ a category Image may be NSFW.
Clik here to view. C(L) of “objects over Image may be NSFW.
Clik here to view.,” to each morphism Image may be NSFW.
Clik here to view. $f : L_1 \to L_2$ of extensions an “extension of scalars” functor Image may be NSFW.
Clik here to view. $f_{\ast} : C(L_1) \to C(L_2)$ , and to each composable pair Image may be NSFW.
Clik here to view. $L_1 \xrightarrow{f} L_2 \xrightarrow{g} L_3$ of morphisms of extensions a natural isomorphism

Image may be NSFW.
Clik here to view. $\displaystyle \eta(f, g) : f_{\ast} g_{\ast} \cong (fg)_{\ast}$

of functors Image may be NSFW.
Clik here to view. $C(L_1) \to C(L_3)$ (where again we’re taking compositions in diagrammatic order) satisfying the usual cocycle condition that the two natural isomorphisms Image may be NSFW.
Clik here to view. $f_{\ast} g_{\ast} h_{\ast} \cong (fgh)_{\ast}$ we can write down from this data agree. We’ll also want unit isomorphisms Image may be NSFW.
Clik here to view. $\varepsilon : \text{id}_{C(L)} \cong (\text{id}_L)_{\ast}$ satisfying the same compatibility as before. This is just spelling out the definition of a 2-functor from the category of separable extensions of Image may be NSFW.
Clik here to view. to the 2-category Image may be NSFW.
Clik here to view. $\text{Cat}$ , and in particular each Image may be NSFW.
Clik here to view. C(L) naturally acquires an action of Image may be NSFW.
Clik here to view. $\text{Aut}(L)$ (where we mean automorphisms of extensions of Image may be NSFW.
Clik here to view., hence if Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
Clik here to view.) for short. The basic example is the Galois prestack of vector spaces Image may be NSFW.
Clik here to view. $\text{Mod}(-)$ , which sends an extension Image may be NSFW.
Clik here to view. to the category Image may be NSFW.
Clik here to view. $\text{Mod}(L)$ of Image may be NSFW.
Clik here to view.-vector spaces and sends a morphism Image may be NSFW.
Clik here to view. $f : L_1 \to L_2$ to the extension of scalars functor

Image may be NSFW.
Clik here to view. $\displaystyle \text{Mod}(L_1) \ni V \mapsto V \otimes_{L_1} L_2 \in \text{Mod}(L_2)$ .

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 Image may be NSFW.
Clik here to view. $f : k \to L$ is an extension, then the functor Image may be NSFW.
Clik here to view. $f_{\ast} : C(k) \to C(L)$ naturally factors through the category Image may be NSFW.
Clik here to view. C(L)^G of homotopy fixed points for the action of Image may be NSFW.
Clik here to view. $G = \text{Aut}(L)$ on Image may be NSFW.
Clik here to view. C(L) . 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 Image may be NSFW.
Clik here to view. $k \to L$ the natural functor Image may be NSFW.
Clik here to view. $C(k) \to C(L)^G$ (where Image may be NSFW.
Clik here to view. $G = \text{Aut}(L) = \text{Gal}(L/k)$ ) is an equivalence of categories.

In words, this condition says that the category of objects over Image may be NSFW.
Clik here to view. is equivalent to the category of objects over Image may be NSFW.
Clik here to view. 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.

Intuitions

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 Image may be NSFW.
Clik here to view. k = L^G for a Galois extension, the Galois descent condition just says that the 2-functor Image may be NSFW.
Clik here to view. C(-) 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 Image may be NSFW.
Clik here to view. $L \mapsto C(L)$ as the functor of points of some sort of moduli space of objects Image may be NSFW.
Clik here to view. such that maps Image may be NSFW.
Clik here to view. $\text{Spec } L \to X$ correspond to objects over Image may be NSFW.
Clik here to view. (that is, objects in Image may be NSFW.
Clik here to view. C(L) ).

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 Image may be NSFW.
Clik here to view. V(L) denotes the set of Image may be NSFW.
Clik here to view.-points of a variety over Image may be NSFW.
Clik here to view., then the natural map Image may be NSFW.
Clik here to view. $V(k) \to V(L)^G$ is an isomorphism.

The natural factorization

In order to state the above condition we needed to know that Image may be NSFW.
Clik here to view. $f_{\ast} : C(k) \to C(L)$ naturally factors through the category Image may be NSFW.
Clik here to view. C(L)^G of homotopy fixed points. Let’s think about the analogous question one category level down: suppose Image may be NSFW.
Clik here to view. $f : X \to Y$ is a map of sets, and Image may be NSFW.
Clik here to view. is a group acting on Image may be NSFW.
Clik here to view.. When does Image may be NSFW.
Clik here to view. factor through the set Image may be NSFW.
Clik here to view. Y^G of fixed points? The answer is precisely when

Image may be NSFW.
Clik here to view. $\displaystyle \left( X \xrightarrow{f} Y \xrightarrow{g} Y \right) = \left( X \xrightarrow{f} Y \right)$

for all Image may be NSFW.
Clik here to view. $g \in G$ . This reflects the universal property of taking fixed points: it’s a certain limit, and so it’s preserved by functors of the form Image may be NSFW.
Clik here to view. $\text{Hom}(X, -)$ in the sense that we have a natural isomorphism

Image may be NSFW.
Clik here to view. $\displaystyle \text{Hom}(X, Y^G) \cong \text{Hom}(X, Y)^G$ .

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 Image may be NSFW.
Clik here to view. [X, Y] denotes the functor category between two categories Image may be NSFW.
Clik here to view. X, Y and Image may be NSFW.
Clik here to view. acts on Image may be NSFW.
Clik here to view., then we have a natural equivalence of categories

Image may be NSFW.
Clik here to view. $\displaystyle [X, Y^G] \cong [X, Y]^G$

where Image may be NSFW.
Clik here to view. (-)^G refers to taking homotopy fixed points on both the LHS and the RHS. In words, this equivalence says that the category of functors Image may be NSFW.
Clik here to view. $X \to Y^G$ is equivalent to the category of functors Image may be NSFW.
Clik here to view. $X \to Y$ equipped with homotopy fixed point structure for the induced action of Image may be NSFW.
Clik here to view..

Now we just need to observe that the extension of scalars functor Image may be NSFW.
Clik here to view. $f_{\ast} : C(k) \to C(L)$ is always equipped with homotopy fixed point structure for the action of Image may be NSFW.
Clik here to view. $G = \text{Aut}(L)$ . This follows from functoriality: because Image may be NSFW.
Clik here to view. is, by definition, fixed by the action of Image may be NSFW.
Clik here to view., we have

Image may be NSFW.
Clik here to view. $\displaystyle \left( k \xrightarrow{f} L \xrightarrow{g} L \right) = \left( k \xrightarrow{f} L \right)$

for all Image may be NSFW.
Clik here to view. $g \in G$ , and applying Image may be NSFW.
Clik here to view. C(-) gives natural isomorphisms

Image may be NSFW.
Clik here to view. $\displaystyle \eta(f, g) : \left( C(k) \xrightarrow{f_{\ast}} C(L) \xrightarrow{g_{\ast}} C(L) \right) \cong \left( C(k) \xrightarrow{f_{\ast}} C(L) \right)$

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 Image may be NSFW.
Clik here to view. $\text{Mod}(k) \to \text{Mod}(L)$ naturally factors through homotopy fixed points Image may be NSFW.
Clik here to view. $\text{Mod}(L)^G$ .

Forms

We can extract somewhat more concrete information from this abstract version of Galois descent as follows.

Definition: Fix an object Image may be NSFW.
Clik here to view. $c_L \in C(L)$ . An object Image may be NSFW.
Clik here to view. $c_k \in C(k)$ is a Image may be NSFW.
Clik here to view.-form of Image may be NSFW.
Clik here to view. c_L if there is an isomorphism Image may be NSFW.
Clik here to view. $f_{\ast} c_k \cong c_L$ .

If we understand the objects in Image may be NSFW.
Clik here to view. C(L) well, we can hope to understand the objects in Image may be NSFW.
Clik here to view. C(k) by understanding the Image may be NSFW.
Clik here to view.-forms of objects in Image may be NSFW.
Clik here to view. C(L) .

Example. Let Image may be NSFW.
Clik here to view. $k = \mathbb{R}, L = \mathbb{C}$ , and let Image may be NSFW.
Clik here to view. C(-) 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 Image may be NSFW.
Clik here to view. $\mathfrak{g}_0$ of a complex Lie algebra Image may be NSFW.
Clik here to view. $\mathfrak{g}$ is a real Lie algebra such that

Image may be NSFW.
Clik here to view. $\mathfrak{g}_0 \otimes_{\mathbb{R}} \mathbb{C} \cong \mathfrak{g}$ .

These can be quite varied. For example, the complex special orthogonal Lie algebra Image may be NSFW.
Clik here to view. $\mathfrak{so}_n(\mathbb{C})$ has real forms the indefinite special orthogonal Lie algebras Image may be NSFW.
Clik here to view. $\mathfrak{so}(p, q)$ for any nonnegative integers Image may be NSFW.
Clik here to view. p, q such that Image may be NSFW.
Clik here to view. p + q = n .

If Galois descent holds, then the extension of scalars functor Image may be NSFW.
Clik here to view. $f_{\ast} : C(k) \cong C(L)^G \to C(L)$ can be described simply as the functor that takes an object of Image may be NSFW.
Clik here to view. C(L) equipped with homotopy fixed point structure and forgets that structure. Hence:

Observation: If Image may be NSFW.
Clik here to view. C(-) is a Galois stack, then isomorphism classes of Image may be NSFW.
Clik here to view.-forms of an object Image may be NSFW.
Clik here to view. $c_L \in C(L)$ can be identified with isomorphism classes of homotopy fixed point structures on Image may be NSFW.
Clik here to view. c_L .

Stating Galois descent

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