Feeds:
Posts

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

$\displaystyle \eta(f, g) : f_{\ast} g_{\ast} \cong (fg)_{\ast}$

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

$\displaystyle \text{Mod}(L_1) \ni V \mapsto V \otimes_{L_1} L_2 \in \text{Mod}(L_2)$.

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

In words, this condition says that the category of objects over $k$ is equivalent to the category of objects over $L$ 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 $k = L^G$ for a Galois extension, the Galois descent condition just says that the 2-functor $C(-)$ 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 $L \mapsto C(L)$ as the functor of points of some sort of moduli space of objects $X$ such that maps $\text{Spec } L \to X$ correspond to objects over $L$ (that is, objects in $C(L)$).

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 $V(L)$ denotes the set of $L$-points of a variety over $k$, then the natural map $V(k) \to V(L)^G$ is an isomorphism.

The natural factorization

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

$\displaystyle \left( X \xrightarrow{f} Y \xrightarrow{g} Y \right) = \left( X \xrightarrow{f} Y \right)$

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

$\displaystyle \text{Hom}(X, Y^G) \cong \text{Hom}(X, Y)^G$.

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 $[X, Y]$ denotes the functor category between two categories $X, Y$ and $G$ acts on $Y$, then we have a natural equivalence of categories

$\displaystyle [X, Y^G] \cong [X, Y]^G$

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

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

$\displaystyle \left( k \xrightarrow{f} L \xrightarrow{g} L \right) = \left( k \xrightarrow{f} L \right)$

for all $g \in G$, and applying $C(-)$ gives natural isomorphisms

$\displaystyle \eta(f, g) : \left( C(k) \xrightarrow{f_{\ast}} C(L) \xrightarrow{g_{\ast}} C(L) \right) \cong \left( C(k) \xrightarrow{f_{\ast}} C(L) \right)$

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 $\text{Mod}(k) \to \text{Mod}(L)$ naturally factors through homotopy fixed points $\text{Mod}(L)^G$.

Forms

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

Definition: Fix an object $c_L \in C(L)$. An object $c_k \in C(k)$ is a $k$-form of $c_L$ if there is an isomorphism $f_{\ast} c_k \cong c_L$.

If we understand the objects in $C(L)$ well, we can hope to understand the objects in $C(k)$ by understanding the $k$-forms of objects in $C(L)$.

Example. Let $k = \mathbb{R}, L = \mathbb{C}$, and let $C(-)$ 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 $\mathfrak{g}_0$ of a complex Lie algebra $\mathfrak{g}$ is a real Lie algebra such that

$\mathfrak{g}_0 \otimes_{\mathbb{R}} \mathbb{C} \cong \mathfrak{g}$.

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

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

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