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

Image may be NSFW.
Clik here to view. $\displaystyle C(k) \cong C(L)^G$

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

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

Today we’ll recover from this abstract description the somewhat more concrete punchline that Image may be NSFW.
Clik here to view.-forms Image may be NSFW.
Clik here to view. $c_k \in C(k)$ of an object Image may be NSFW.
Clik here to view. $c_L \in C(L)$ can be classified by Galois cohomology Image may be NSFW.
Clik here to view. $H^1(BG, \text{Aut}(c_L))$ , 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 Image may be NSFW.
Clik here to view. on a category Image may be NSFW.
Clik here to view. can be strictified so that it becomes an action of Image may be NSFW.
Clik here to view. on the automorphism group Image may be NSFW.
Clik here to view. $\text{Aut}(c)$ of a single object Image may be NSFW.
Clik here to view.. This is possible in concrete examples; for example, if Image may be NSFW.
Clik here to view. c = L^n is the standard Image may be NSFW.
Clik here to view.-dimensional vector space over Image may be NSFW.
Clik here to view., then Image may be NSFW.
Clik here to view. $\text{Aut}(L^n) \cong GL_n(L)$ , 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 Image may be NSFW.
Clik here to view. on a space Image may be NSFW.
Clik here to view. (in our examples, a groupoid), and we want to know when such an action gives rise to an action on the fundamental group Image may be NSFW.
Clik here to view. $\pi_1(X, x)$ . Naively, this is only possible if Image may be NSFW.
Clik here to view. is fixed by the action of Image may be NSFW.
Clik here to view.. But this is not a homotopy-theoretic condition, and the homotopy-theoretic refinement is that Image may be NSFW.
Clik here to view. should be a homotopy fixed point of Image may be NSFW.
Clik here to view.; then the action of Image may be NSFW.
Clik here to view. on the unpointed space Image may be NSFW.
Clik here to view. can be upgraded to an action of Image may be NSFW.
Clik here to view. on a pointed space Image may be NSFW.
Clik here to view. (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 Image may be NSFW.
Clik here to view. on pointed connected groupoids to actions of Image may be NSFW.
Clik here to view. on groups.

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

Image may be NSFW.
Clik here to view. $\displaystyle \alpha(g) : c_L \cong g_{\ast} c_L$

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

Image may be NSFW.
Clik here to view. $\displaystyle c_L \xrightarrow{\alpha(g)} g_{\ast} c_L \xrightarrow{g_{\ast} \varphi} g_{\ast} c_L \xrightarrow{\alpha(g)^{-1}} c_L$

which we’ll write as Image may be NSFW.
Clik here to view. $\varphi^g$ for simplicity.

Fixing this homotopy fixed point structure allows us to describe other homotopy fixed point structures Image may be NSFW.
Clik here to view. $\beta(g)$ by describing their difference, which we’ll write as

Image may be NSFW.
Clik here to view. $\gamma(g) : c_L \xrightarrow{\beta(g)} g_{\ast} c_L \xrightarrow{\alpha(g)^{-1}} c_L$ .

After some simplification, we compute that Image may be NSFW.
Clik here to view. $\gamma(-)$ satisfies the compatibility condition that

Image may be NSFW.
Clik here to view. $\gamma(gh) = \gamma(g) \gamma(h)^g$

(where as usual we’ll need to interpret compositions in diagrammatic order for consistency), which is the usual definition of a 1-cocycle on Image may be NSFW.
Clik here to view. with coefficients in Image may be NSFW.
Clik here to view. $\text{Aut}(c)$ (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 Image may be NSFW.
Clik here to view. $c_L \in C(L)$ has at least one Image may be NSFW.
Clik here to view.-form Image may be NSFW.
Clik here to view. $c_k \in C(k)$ . Using this Image may be NSFW.
Clik here to view.-form as a basepoint, isomorphism classes of Image may be NSFW.
Clik here to view.-forms on Image may be NSFW.
Clik here to view. c_L can be identified with elements of the Galois cohomology set Image may be NSFW.
Clik here to view. $H^1(BG, \text{Aut}(c_L))$ 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 / Image may be NSFW.
Clik here to view.-form.

Note also that we haven’t yet provided a prescription for actually writing down a Image may be NSFW.
Clik here to view.-form Image may be NSFW.
Clik here to view. c_k given homotopy fixed point data on some Image may be NSFW.
Clik here to view. c_L .

The real and complex numbers

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

Image may be NSFW.
Clik here to view. $\displaystyle \gamma(g^2) = \text{id}_{c_L} = \gamma(g) \gamma(g)^g$ .

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

Image may be NSFW.
Clik here to view. $\displaystyle f \gamma(g) = \gamma'(g) f^g$ .

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

Image may be NSFW.
Clik here to view. $\displaystyle H^1(BG, GL_n(L)) = 0$ .

This is a generalization of Hilbert’s Theorem 90, which it reduces to in the special case that Image may be NSFW.
Clik here to view. n = 1 .

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

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

Image may be NSFW.
Clik here to view. $\alpha \overline{\alpha} = N(\alpha) = a^2 - b^2 d = 1$

(that is, an element of norm Image may be NSFW.
Clik here to view.), and the fact that every such 1-cocycle is cohomologous to zero means that every such element can be written in the form

Image may be NSFW.
Clik here to view. $\displaystyle \beta \overline{\beta}^{-1} = \frac{p + q \sqrt{d}}{p - q \sqrt{d}} = \frac{p^2 + 2pq \sqrt{d} + q^2 d}{p^2 - q^2 d}$

for some Image may be NSFW.
Clik here to view. $\beta \in L^{\times}$ . This gives a parameterization

Image may be NSFW.
Clik here to view. $\displaystyle a = \frac{p^2 + q^2 d}{p^2 - q^2 d}, b = \frac{2pq}{p^2 - q^2 d}$

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

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

Image may be NSFW.
Clik here to view. $\displaystyle H^1(BG, S_n)$ .

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

Image may be NSFW.
Clik here to view. $\prod L_i$

where each Image may be NSFW.
Clik here to view. L_i is a subextension of Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. $\sum \dim_k L_i = n$ . So in this case we more or less get ordinary Galois theory back.

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

Image may be NSFW.
Clik here to view. H^1(BG, PGL_n(L)) .

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

To connect this to a previous computation, the short exact sequence Image may be NSFW.
Clik here to view. $1 \to L^{\times} \to GL_n(L) \to PGL_n(L) \to 1$ gives rise to a longish exact sequence part of which goes

Image may be NSFW.
Clik here to view. $H^1(BG, GL_n(L)) \to H^1(BG, PGL_n(L)) \to H^2(BG, L^{\times})$ .

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

The extension of scalars Image may be NSFW.
Clik here to view. $f_{\ast} : C(k) \to C(L)$ of a P-object is P.
The Image may be NSFW.
Clik here to view.-forms Image may be NSFW.
Clik here to view. of any P-object Image may be NSFW.
Clik here to view. $c_L \in C(L)$ are P.

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

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 Image may be NSFW.
Clik here to view. $k^n \otimes_k L \cong L^n$ , it is not preserved under taking Image may be NSFW.
Clik here to view.-forms, as we saw above.

Forms and Galois cohomology

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