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
- The extension of scalars
![f_{\ast} : C(k) \to C(L)]()
of a P-object is P.
- 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.
of a P-object is P.
