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## Forms and Galois cohomology

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

$\displaystyle C(k) \cong C(L)^G$

where $f : k \to L$ is a Galois extension, $G = \text{Aut}(L)$ is the Galois group of $L$ (thinking of $L$ as an object of the category of field extensions of $k$ at all times), $C(k)$ is a category of “objects over $k$,” and $C(l)$ is a category of “objects over $L$.”

In fact this description is probably only correct if $k \to L$ is a finite Galois extension; if $k \to L$ is infinite it should probably be modified by requiring that every function of $G$ that occurs (e.g. in the definition of homotopy fixed points) is continuous with respect to the natural profinite topology on $G$. To avoid this difficulty we’ll stick to the case that $k \to L$ is a finite extension.

Today we’ll recover from this abstract description the somewhat more concrete punchline that $k$-forms $c_k \in C(k)$ of an object $c_L \in C(L)$ can be classified by Galois cohomology $H^1(BG, \text{Aut}(c_L))$, and we’ll give some examples.

Basepoints

Way back when we defined a homotopy fixed point structure, we saw that it could be regarded as a generalization of a 1-cocycle, which it reduces to in the special case that the action of $G$ on a category $C$ can be strictified so that it becomes an action of $G$ on the automorphism group $\text{Aut}(c)$ of a single object $c$. This is possible in concrete examples; for example, if $c = L^n$ is the standard $n$-dimensional vector space over $L$, then $\text{Aut}(L^n) \cong GL_n(L)$, and the action of the Galois group on this is just componentwise. When is it possible abstractly?

Topologically, we’re starting from an action of a group $G$ on a space $X$ (in our examples, a groupoid), and we want to know when such an action gives rise to an action on the fundamental group $\pi_1(X, x)$. Naively, this is only possible if $x$ is fixed by the action of $G$. But this is not a homotopy-theoretic condition, and the homotopy-theoretic refinement is that $x$ should be a homotopy fixed point of $G$; then the action of $G$ on the unpointed space $X$ can be upgraded to an action of $G$ on a pointed space $(X, x)$, and taking the fundamental group is a functor on pointed spaces. In fact this defines an equivalence from pointed connected groupoids to groups, and so an equivalence from actions of $G$ on pointed connected groupoids to actions of $G$ on groups.

For our purposes, what this means is that in order to strictify the Galois action on $C(L)$ to an action on $\text{Aut}(c_L)$ for a particular object $c_L \in C(L)$, we need to pick a homotopy fixed point structure on $c_L$ to act as a basepoint; equivalently, we need to pick a $k$-form $c_k$. The corresponding homotopy fixed point structure is encoded by maps

$\displaystyle \alpha(g) : c_L \cong g_{\ast} c_L$

as usual, and we can now use these maps to coherently identify each $g_{\ast} c_L$ with $c_L$. The corresponding strictified action of $G$ on $\text{Aut}(c_L)$ is given by sending an automorphism $\varphi : c_L \to c_L$ to the automorphism

$\displaystyle c_L \xrightarrow{\alpha(g)} g_{\ast} c_L \xrightarrow{g_{\ast} \varphi} g_{\ast} c_L \xrightarrow{\alpha(g)^{-1}} c_L$

which we’ll write as $\varphi^g$ for simplicity.

Fixing this homotopy fixed point structure allows us to describe other homotopy fixed point structures $\beta(g)$ by describing their difference, which we’ll write as

$\gamma(g) : c_L \xrightarrow{\beta(g)} g_{\ast} c_L \xrightarrow{\alpha(g)^{-1}} c_L$.

After some simplification, we compute that $\gamma(-)$ satisfies the compatibility condition that

$\gamma(gh) = \gamma(g) \gamma(h)^g$

(where as usual we’ll need to interpret compositions in diagrammatic order for consistency), which is the usual definition of a 1-cocycle on $G$ with coefficients in $\text{Aut}(c)$ (with respect to the action defined above). Similarly we get that isomorphisms of homotopy fixed point data correspond to 1-cocycles being cohomologous. Hence:

Theorem: Suppose that $c_L \in C(L)$ has at least one $k$-form $c_k \in C(k)$. Using this $k$-form as a basepoint, isomorphism classes of $k$-forms on $c_L$ can be identified with elements of the Galois cohomology set $H^1(BG, \text{Aut}(c_L))$ with respect to the strictified action above.

Note that this set is naturally pointed by the trivial 1-cocycle, whereas the set of isomorphism classes of homotopy fixed points does not have a natural “trivial” object in it. This reinforces the need to pick a basepoint / $k$-form.

Note also that we haven’t yet provided a prescription for actually writing down a $k$-form $c_k$ given homotopy fixed point data on some $c_L$.

The real and complex numbers

Galois cohomology becomes particularly easy to describe in the special case that $k = \mathbb{R}, L = \mathbb{C}$ (or more generally any quadratic extension), which is already useful for many applications. Here $G = \mathbb{Z}_2$ is generated by a single nontrivial element $g$, namely complex conjugation. The action of $G$ on $\text{Aut}(c_L)$ will generally also have the interpretation of complex conjugation (e.g. on matrices), and so the data of a $1$-cocycle $\gamma \in Z^1(BG, \text{Aut}(c_L))$ amounts to (after trivializing $\gamma(e)$) the data of a single element $\gamma(g) \in \text{Aut}(c_L)$ such that

$\displaystyle \gamma(g^2) = \text{id}_{c_L} = \gamma(g) \gamma(g)^g$.

In other words, $\gamma(g)$ is an automorphism of $c_L$ whose inverse is its complex conjugate. Two such automorphisms $\gamma(g), \gamma'(g)$ are cohomologous as 1-cocycles iff there is some $f \in \text{Aut}(c_L)$ such that

$\displaystyle f \gamma(g) = \gamma'(g) f^g$.

Some examples

In all of the examples below we are claiming without proof that some Galois prestack is in fact a Galois stack.

Example. Let $C(-)$ be the Galois stack of vector spaces, and let $c_L = L^n \in C(L)$, with distinguished $k$-form $c_k = k^n \in C(k)$. The strictified Galois action on $\text{Aut}(c_L) \cong GL_n(L)$ is componentwise, and this will tell us what the Galois action is in many other examples involving vector spaces with extra structure. Since every $k$-form of $L^n$ must be an $n$-dimensional vector space over $k$ and hence must be isomorphic to $k^n$, we conclude that

$\displaystyle H^1(BG, GL_n(L)) = 0$.

This is a generalization of Hilbert’s Theorem 90, which it reduces to in the special case that $n = 1$.

When $k = \mathbb{R}, L = \mathbb{C}$ we learn the following: a 1-cocycle is a matrix $M \in GL_n(\mathbb{C})$ such that $M^{-1} = \overline{M}$, and the fact that every such 1-cocycle is cohomologous to zero means that every such matrix $M$ can be written in the form $N \overline{N}^{-1}$ for some $N \in GL_n(\mathbb{C})$. This recently came up on MathOverflow.

When $k = \mathbb{Q}, L = \mathbb{Q}(\sqrt{d})$, and $n = 1$ we learn the following: a 1-cocycle is an element $\alpha = a + b \sqrt{d} \in L^{\times}$ such that

$\alpha \overline{\alpha} = N(\alpha) = a^2 - b^2 d = 1$

(that is, an element of norm $1$), and the fact that every such 1-cocycle is cohomologous to zero means that every such element can be written in the form

$\displaystyle \beta \overline{\beta}^{-1} = \frac{p + q \sqrt{d}}{p - q \sqrt{d}} = \frac{p^2 + 2pq \sqrt{d} + q^2 d}{p^2 - q^2 d}$

for some $\beta \in L^{\times}$. This gives a parameterization

$\displaystyle a = \frac{p^2 + q^2 d}{p^2 - q^2 d}, b = \frac{2pq}{p^2 - q^2 d}$

of the set of rational solutions to the Diophantine equation $a^2 - b^2 d = 1$, and in particular when $d = -1$ we recover the usual parameterization of Pythagorean triples. (I learned this from Noam Elkies.)

Example. Let $C(-)$ be the Galois stack of commutative algebras, and consider $c_L = L^n \in C(L)$. Its automorphism group as an $L$-algebra is $S_n$ with trivial Galois action, so $k$-forms of $L^n$ are classified by

$\displaystyle H^1(BG, S_n)$.

Because the Galois action is trivial, this is the set of conjugacy classes of homomorphisms $G \to S_n$, or equivalently isomorphism classes of actions of $G$ on $n$-element sets. Such an isomorphism class is a disjoint union of transitive actions of $G$ on $d$-element sets, which by the Galois correspondence can be identified with finite separable extensions of $k$ of degree $d$, and in fact it turns out that $k$-forms of $L^n$ are precisely $k$-algebras of the form

$\prod L_i$

where each $L_i$ is a subextension of $L$ and $\sum \dim_k L_i = n$. So in this case we more or less get ordinary Galois theory back.

Example. Let $C(-)$ be the Galois stack of algebras, not necessarily commutative, and consider $c_L = M_n(L) \in C(L)$. Its automorphism group as an $L$-algebra is $PGL_n(L)$ with Galois action inherited from $GL_n(L)$, so $k$-forms of $M_n(L)$ are classified by

$H^1(BG, PGL_n(L))$.

This classification is related to the Brauer group of $k$, namely the part involving those central simple $k$-algebras which become isomorphic to $M_n(L)$ after extension by scalars to $L$. It is also related to Severi-Brauer varieties, which are $k$-forms of projective space.

To connect this to a previous computation, the short exact sequence $1 \to L^{\times} \to GL_n(L) \to PGL_n(L) \to 1$ gives rise to a longish exact sequence part of which goes

$H^1(BG, GL_n(L)) \to H^1(BG, PGL_n(L)) \to H^2(BG, L^{\times})$.

Since $H^1(BG, GL_n(L))$ vanishes by Hilbert’s theorem 90, by exactness we conclude that if the relative Brauer group $H^2(BG, L^{\times})$ vanishes, then so does $H^1(BG, PGL_n(L))$; equivalently, in this case the only $k$-form of $M_n(L)$ is $M_n(k)$.

Good properties

One last comment. Galois descent gives us a reason to single out certain properties P that some objects satisfying Galois descent (such as algebras or commutative algebras) can have as particularly good: namely, those properties which also satisfy Galois descent. This means that

1. The extension of scalars $f_{\ast} : C(k) \to C(L)$ of a P-object is P.
2. The $k$-forms $c_k$ of any P-object $c_L \in C(L)$ are P.

For example, for algebras, being semisimple is not a good property in this sense: the extension of scalars $A \otimes_k L$ of a semisimple $k$-algebra $A$ can fail to be semisimple (e.g. if $A$ is an extension of $k$ which is not separable). The good version of this property is being separable, which is equivalent to being “geometrically semisimple” in the sense that $A \otimes_k L$ is semisimple for all field extensions $k \to L$.

Similarly, for commutative algebras, being isomorphic to a finite product of copies of the ground field is not a good property in this sense: although it is preserved under extension of scalars, since $k^n \otimes_k L \cong L^n$, it is not preserved under taking $k$-forms, as we saw above.

### 9 Responses

1. All LaTeX is and has been compiling fine for me.

2. Am I the only one getting red text on a yellow background saying “Latex pah not specified”?

• I’ve been seeing this for a few days now, and it’s the reason I stopped posting.

3. […] 11/18/15:) This definition is slightly incorrect in the case of infinite Galois extensions; see the next post and it comments for some […]

4. Your guess about infinite Galois extensions is correct. For instance, consider the stack of modules (or quasicoherent sheaves, if you like): it is an fpqc-stack, so it certainly satisfies the (effective) Galois descent condition. But if you unwind what fpqc descent means for a field extension (which is always fpqc!) you will see that it only reduces to homotopy fixed points for a group action in the case of a finite Galois extension.

In a bit more detail: let $L \mid k$ be a field extension and consider the simplicial kernel of $\text{Spec} L \to \text{Spec} k$. By fpqc descent, it has an effective quotient (in the category of fpqc sheaves), namely the representable sheaf on $\text{Spec} k$, and moreover, $\text{Qcoh}(-)$ sends this colimit diagram to a bilimit diagram. Since we are dealing with objects in a 2-category, we can truncate these (co)simplicial diagrams above degree 2. When $L \mid k$ is a finite Galois extension, the objects in the simplicial kernel are just disjoint unions of copies of $\text{Spec} L$. It is instructive to work out what the face operators are in terms of this identification: at the bottom, one is the diagonal embedding $L \to L \times \cdots \times L$, and the other is $x \mapsto (\sigma_1 (x), \ldots, \sigma_d (x))$, where $\sigma_1, \ldots, \sigma_d$ is some enumeration of the Galois group.

• (If you could fix the LaTeX there that would be much appreciated.)

• I tried, but WordPress’s LaTeX is freaking out on me right now…

• For anyone reading Zhen’s comment, and not particularly enamored by the words ‘simplicial kernel’, let me just state another way of seeing it (which I assume both Zhen and Qiaochu know well). Namely, fpqc descent along $U\to V$ involves, in particular, $U\times_V U$. Now, $\text{Spec}(L)\to\text{Spec}(k)$ is fpqc for any $L/k$ a field extension. What makes the finite case different is that, there, $\displaystyle L\otimes_k L$ is just $\displaystyle \prod_{\sigma\in\text{Gal}(L/k)}L$ with the map being $\ell\otimes \ell'\mapsto (\ell\sigma(\ell'))$.

Thus, fpqc descent works for any extension, but explicitly being able to relate it to the Galois group is a failed endeavor for non-finite Galois extensions. This also suggestions how one might replace this notion by studying stacks on the subcategory of $\mathsf{F\acute{e}t}(X)$ consisting of Galois objects, for other $X$.

Again, so Zhen doesn’t yell at me, this is the exact same thing he wrote (I assume) in different language. 🙂

• By the way, for descent conditions needed for infinite Galois extensions, at least in the case of quasi-projective varieties, see Corollary 16.25 here: http://www.jmilne.org/math/CourseNotes/AG16.pdf

This gets used all the time in, say, Shimura varieties, but is still semi-unsatisfying since it only works for quasi-projectives.