Countably many isomorphism classes of reductive groups over a field with countable Brauer and Witt groups Assume a field has a countable Brauer group and a countable Witt group. Are there countably many isomorphism classes of reductive groups over it?
 A: $\newcommand\kalg{k\textrm-\mathbf{alg}}\newcommand\kalgf{\kalg_\text f}$Welcome new contributor.  I am just writing my comment as an answer.  Let $k$ be a field and denote by $\kalg$ the category whose objects are $k$-algebras.  Denote by $\kalgf$ the full subcategory whose objects are finite type $k$-algebras.
Definition 1. Define $\mathcal{I}$ to be the subcategory of $\kalg$ whose objects are those finite type $k$-algebras that are integral domains with fraction field that is a separably generated extension of $k$ in which $k$ is separably closed.  For objects of $\mathcal{I}$, the $\mathcal{I}$-morphisms is the subset of $k$-algebra homomorphisms that are injective set maps such that the corresponding field extension is separably generated in which the domain field is separably closed.
Lemma 2. The category $\mathcal{I}$ is almost filtering.
Proof.
For every pair of finite type $k$-algebras $A$ and $B$, both $A$ and $B$ are flat over $k$ with geometrically integral fibers. Thus, the tensor product $A\otimes_k B$ is flat with geometrically integral fibers over $A$ and over $B$. Thus, by composition, $A\otimes_k B$ is an integral domain with geometrically integral fibers over $k$, i.e., it is an object of $\mathcal{I}$ and the natural $k$-algebra homomorphisms from $A$ and from $B$ to $A\otimes_k B$ are morphisms in $\mathcal{I}$.  Thus, every pair of objects of $\mathcal{I}$ admit maps to a common target object.
Similarly, for every pair of morphisms of $\mathcal{I}$, say
$$
u:A\to B, \ \ v:A \to C,
$$
by Grothendieck's Generic Freeness Theorem, there exists a nonzero element $a$ of $A$ such that after localizing all rings at the image of $a$, the induced ring homomorphisms are flat.  When $u$ and $v$ are flat, by the same argument as in the previous paragraph,
the tensor product ring $B\otimes_A C$ is flat with geometrically integral generic fibers over both $A$ and $B$.  Thus, $B\otimes_A C$ is an object of $\mathcal{I}$ and the natural $k$-algebra homomorphisms from $B$ and from $C$ to $B\otimes_A C$ are morphisms of $\mathcal{I}$.  Thus, every pair of morphisms are equalized by composing with morphisms to a common target.  (Please note, this is a weaker notion than in a filtering category, where every pair of morphisms with common source and target must be equalized by a single morphism.)  QED
Definition 3.  The tautological directed system of $k$-algebras indexed by $\mathcal{I}$ is the system that associates to every object $A$ of $\mathcal{I}$ the $k$-algebra $A$.
The category $\mathcal{I}$ is small.  Thus, there is a colimit ring of this directed system.  Denote the colimit by $K$.
Lemma 4. The $k$-algebra $K$ is a field, and for every object $A$ of $\mathcal{I}$, the natural $k$-algebra homomorphism $A\to K$ is an injective set map.
Proof.
For every nonzero element $a$ of $A$, the fraction ring $A[1/a]$ is also an object of $\mathcal{I}$ and the localization homomorphism,
$$
A\to A[1/a],
$$
is a morphism of $\mathcal{I}$.  Thus, the image in $K$ of the element $1/a$ of $A[1/a]$ is an inverse of the image in $K$ of the element $a$ of $A$.  QED
Lemma 5.  The subfield $k$ of $K$ is separably closed in $K$, and $K$ is a filtering colimit of its finitely generated $k$-subfields, each of which is separably generated over $k$.
Proof.
Since every subextension of a separably generated field extension is again separably generated, it follows that every finitely generated $k$-subfield of $K$ is separably generated over $k$ with $k$ separably closed in the $k$-subfield.  In particular, $k$ is separably closed in $K$.  Since $K$ is a filtered colimit of its finitely generated $k$-subfields, it is a filtered colimit of finitely generated $k$-subfields each of which is separably generated over $k$. QED
Lemma 6.  For every finitely generated $K$-algebra, there exists an object $A$ of $\mathcal{I}$ and a finitely generated, flat $A$-algebra whose base change by $A\to K$ is $K$-isomorphic to the finitely generated $K$-algebra.  The geometric generic fiber over $K$ is integral if and only if the geometric generic fiber over $\operatorname{Frac}(A)$ of the finitely generated $A$-algebra is integral.
Proof.
Every finitely generated algebra over $K$ is a quotient of a  polynomial $K$-algebra in finitely many variables by a finitely generated ideal that has finitely many generators with finitely many coefficients. Each of these coefficients in the image of an element of an object of $\mathcal{I}$.  Since $\mathcal{I}$ is almost filtering, it follows that the finitely generated $K$-algebra is the base change of a finitely generated $A$-algebra for an object $A$ of $\mathcal{I}$. Again using Grothendieck's Generic Freeness Theorem, we may assume that the finitely generated algebra over $A$ is flat.
Since generic smoothness of the generic fiber can be checked after arbitrary field extension of the fraction field of $A$, e.g., after base change to $K$, the original algebra over $K$ has geometrically integral generic fiber if and only if the finitely generated algebra over $A$ has geometrically integral generic fiber.  QED
Proposition 7.  Every finitely generated $K$-algebra that is an integral domain with a geometrically integral fiber has a $K$-point, i.e., $K$ is a pseudo-algebraically closed field.
Proof.
With notation as in the previous proof, if the generic fiber is geometrically integral, then the morphism from $A$ to the flat, finitely generated $A$-algebra is an object of $\mathcal{I}$.  Thus, the original algebra over the colimit ring has a $K$-point.
QED
Altogether this proves that every field $k$ is contained in a PAC field $K$ in which $k$ is separably closed and such that $K$ is a filtering colimit of its finitely generated $k$-subfields, each of which is separably generated over $k$.  Since $K$ is a PAC field, it has trivial Brauer group and the Witt group is cyclic of order $2$.
Finally, let $\ell$ be a prime that is different from the characteristic of $k$, assume that $k$ contains a primitive root of unity of order $\ell$, and assume that $k$ contains uncountably many cyclic Galois extensions $L/k$ of degree $\ell$.  By the hypothesis, each of these is a Kummer extension that is characterized by the cyclic kernel of the induced map of $\ell$-torsion groups,
$$
k^\times/(k^\times)^\ell \to L^\times/(L^\times)^\ell.
$$
Since $k$ is separably closed in $K$, also the $K$-algebra $L\otimes_k K$ is a field extension of prime degree $\ell$, so it is a Kummer extension, and the kernel of the map from $k^\times/(k^\times)^\ell$ to the corresponding group is still the same cyclic group.  Thus, these countably many Kummer extensions with distinct cyclic kernel subgroups of $k^\times/(k^\times)^\ell$ give distinct extensions $L\otimes_k K$, since they still have the same kernel.
For each such extension, consider the Weil restriction of the multiplicative group.  The intersection of $k^\times$ with the $\ell$-power subgroup of the $K$-points of this group recovers the cyclic kernel.  Thus, as a group of multiplicative type together with a morphism from the multiplicative group over $L$, these reductive groups are distinct (since they recover the field extension $L/k$).  But for any two groups of multiplicative type, there are at most countably infinitely many morphisms of algebraic groups between these.  Thus, if there are uncountably many cyclic Kummer extensions of $k$ of degree $\ell$, it follows that these corresponding Weil restrictions of the multiplicative groups give uncountably many pairwise non-isomorphic algebraic groups over $K$ of multiplicative type.
Now for every uncountable, algebraically closed field $\kappa$, for every prime $\ell$ different from the characteristic, the purely transcendental extension $k=\kappa(t)$ is field $k$ as above that has uncountably many cyclic Kummer extensions of degree $\ell$, e.g., the cyclic extensions adjoining a degree-$\ell$ root of $t-a$ as $a$ ranges over the uncountably many elements of $\kappa$.  Applying the construction above to the PAC field $K$, it follows that $K$ has trivial Brauer group, it has Witt group that is cyclic of order $2$, yet it has uncountably many pairwise-nonisomorphic algebraic groups of multiplicative type of dimension $\ell$.
