2 Definable cohomology and A-forms of G

We carry on the notation and assumptions from the previous section; in particular, those on M, A, $\mathcal G$ , and G. We let $\operatorname {Aut}_{\operatorname {def}}(G)$ denote the set of definable (with parameters in M) group automorphisms of G. Note that this is a group—with respect to composition. Also note that the automorphism group $\mathcal G$ acts naturally on $\operatorname {Aut}_{\operatorname {def}}(G)$ ; namely, if $\sigma \in \mathcal G$ and $\phi \in \operatorname {Aut}_{\operatorname {def}}(G)$ , then $\sigma (\phi )$ is obtained by applying $\sigma $ to the graph of $\phi $ (a definable set), which yields another definable automorphism of G. Furthermore, one readily checks that for $\phi _1,\phi _2\in \operatorname {Aut}_{\operatorname {def}}(G)$ we have

(2.1) $$ \begin{align} \sigma(\phi_1\cdot \phi_2)=\sigma(\phi_1)\cdot\sigma(\phi_2). \end{align} $$

In other words, the action of $\mathcal G$ in $\operatorname {Aut}_{\operatorname {def}}(G)$ is by group automorphisms. Furthermore, equipping $\operatorname {Aut}_{\operatorname {def}}(G)$ with its discrete topology, elimination of imaginaries implies that this action is continuous.

Using the action of $\mathcal G$ on $\operatorname {Aut}_{\operatorname {def}}(G)$ , we define a $($ 1- $)$ cocycle from $\mathcal G$ to $\operatorname {Aut}_{\operatorname {def}}(G)$ to be a map

$$ \begin{align*}\Phi:\mathcal G\to \operatorname{Aut}_{\operatorname{def}}(G)\end{align*} $$

such that

$$ \begin{align*}\Phi(\sigma_1\cdot \sigma_2)=\Phi(\sigma_1)\cdot \sigma_1(\Phi(\sigma_2)).\end{align*} $$

The trivial group homomorphism from $\mathcal G$ to $\operatorname {Aut}_{\operatorname {def}}(G)$ is a cocycle, called the trivial cocycle. Two cocycles $\Phi $ and $\Psi $ from $\mathcal G$ to $\operatorname {Aut}_{\operatorname {def}}(G)$ are said to be cohomologous if there exist $\phi \in \operatorname {Aut}_{\operatorname {def}}(G)$ such that

$$ \begin{align*}\Psi(\sigma)=\phi^{-1}\cdot \Phi(\sigma)\cdot \sigma(\phi) \quad \textrm{ for all } \sigma\in \mathcal G.\end{align*} $$

Being cohomologous is an equivalence relation: transitivity follows by (2.1). We say that the cocycle $\Phi $ is definable if if there exists a tuple a from M and an A-definable (partial) function $h(x,y,z)$ such that

$$ \begin{align*}\Phi(\sigma)(-)=h(a,\sigma(a), -) \quad \textrm{ for all } \sigma\in \mathcal G.\end{align*} $$

We note that the trivial cocycle is definable, and if $\Phi $ is a cocycle that is cohomologous to a definable cocycle then $\Phi $ is itself definable. Furthermore, one readily verifies that definable cocycles are continuous. The set of definable cocycles is denoted by $Z^1_{\operatorname {def}}(\mathcal G,\operatorname {Aut}_{\operatorname {def}}(G))$ . The set of cohomology classes of $Z^1_{\operatorname {def}}(\mathcal G,\operatorname {Aut}_{\operatorname {def}}(G))$ (under the equivalence relation of being cohomologous) is the first definable cohomology of $\mathcal G$ with coefficients in $\operatorname {Aut}_{\operatorname {def}}(G)$ and is denoted by $H^1_{\operatorname {def}}(\mathcal G,\operatorname {Aut}_{\operatorname {def}}(G))$ .

Remark 2.1. As in the case of the definable cohomology $H^1_{\operatorname {def}}(\mathcal G,G)$ , the cohomology $H^1_{\operatorname {def}}(\mathcal G,\operatorname {Aut}_{\operatorname {def}}(G))$ has the structure of a pointed set, with the cohomology class of the trivial cocycle (also called the set of coboundaries from $\mathcal G$ to $\operatorname {Aut}_{\operatorname {def}}(G)$ ) being the distinguished element. Further, in the case when the group $\operatorname {Aut}_{\operatorname {def}}(G)$ is abelian, $H^1_{\operatorname {def}}(\mathcal G,\operatorname {Aut}_{\operatorname {def}}(G))$ has the structure of an abelian group. This follows from the fact that, in this case, the natural (commutative) group structure on the set of maps from $\mathcal G$ to $\operatorname {Aut}_{\operatorname {def}}(G)$ given by

$$ \begin{align*}\Psi_1* \Psi_1(\sigma)=\Psi_1(\sigma)\cdot \Psi_2(\sigma)\end{align*} $$

restricts to a (commutative) group structure on $Z^1_{\operatorname {def}}(\mathcal G,\operatorname {Aut}_{\operatorname {def}}(G))$ .

By an A-form of G we mean an A-definable group H which is definably (over M) isomorphic to G. Let H be an A-form of G. Note that, for any definable (group) isomorphism $f:G\to H$ and $\sigma \in \mathcal G$ , the map $\sigma (f):G\to H$ is again a definable isomorphism. Thus, we have a well-defined map $\Phi _{H,f}:\mathcal G\to \operatorname {Aut}_{\operatorname {def}}(G)$ given by

$$ \begin{align*}\Phi_{H,f}(\sigma)=f^{-1}\circ \sigma(f).\end{align*} $$

Furthermore, $\Phi _{H,f}$ is a definable cocycle. It is clearly definable and

$$ \begin{align*} \Phi_{H,f}(\sigma_1\cdot\sigma_2) &=f^{-1} \circ \sigma_1(\sigma_2(f)) \\ &=f^{-1}\circ \sigma_1(f\circ f^{-1}\circ \sigma_2(f)) \\ &=f^{-1}\circ\sigma_1(f)\circ\sigma_1(f^{-1}\circ\sigma_2(f)) \\ &=\Phi_{H,f}(\sigma_1)\cdot \sigma_1(\Phi_{H,f}(\sigma_2)). \end{align*} $$

Now, if $g:G\to H$ is another definable isomorphism, then $\Phi _{H,f}$ and $\Phi _{H,g}$ are cohomologous. Indeed,

$$ \begin{align*}\Phi_{H,g}(\sigma)=(f^{-1}g)^{-1}\Phi_{H,f}(\sigma)\sigma(f^{-1}g).\end{align*} $$

More generally, if N is another A-form of G which is (group) definably isomorphic to H over A, say witnessed by $\eta :N\to H$ , and $p:G\to N$ is a definable isomorphism, then $\Phi _{H,f}$ and $\Phi _{N,p}$ are cohomologous. Indeed

$$ \begin{align*}\Phi_{N,p}(\sigma)=(f^{-1}\, \eta\, p)^{-1}\Phi_{H,f}(\sigma)\sigma(f^{-1}\, \eta\, p).\end{align*} $$

Let $\mathfrak {F}_{\operatorname {def}}(G)$ denote the set of A-forms of G up to A-definable isomorphism. The above discussion yields a well-defined map

$$ \begin{align*}\mathfrak{F}_{\operatorname{def}}(G)\to H^1_{\operatorname{def}}(\mathcal G,\operatorname{Aut}_{\operatorname{def}}(G))\end{align*} $$

given by mapping $(H,f)$ to the cohomology class of $\Phi _{H,f}$ , where H an A-form of G and $f:G\to H$ a definable (over M) isomorphism. We now show that the displayed map is an isomorphism of pointed sets. The distinguished element in $\mathfrak {F}_{\operatorname {def}}(G)$ being G. Clearly, $\Phi _{G,\operatorname {Id}}$ is the trivial cocycle from $\mathcal G$ to $\operatorname {Aut}_{\operatorname {def}}(G)$ , hence $(G,\operatorname {Id})$ maps to the cohomology class of the trivial cocycle (i.e., the set of coboundaries). The remaining of this section is devoted to showing that this map is a bijection.

Let us show injectivity of $\mathfrak {F}_{\operatorname {def}}(G)\to H^1_{\operatorname {def}}(\mathcal G,\operatorname {Aut}_{\operatorname {def}}(G))$ . Take $(H,f)$ and $(N,p),$ where H and N are A-forms of G and $f:G\to H$ and $p:G\to N$ are definable isomorphisms. Assume $\Phi _{H,f}$ is cohomologous to $\Phi _{N,p}$ , then there exists $\phi \in \operatorname {Aut}_{\operatorname {def}}(G)$ such that for all $\sigma \in \mathcal G$

$$ \begin{align*}\Phi_{N,p}(\sigma)=\phi^{-1}\Phi_{H,f}(\sigma)\,\sigma(\phi).\end{align*} $$

In other words,

$$ \begin{align*}p^{-1}\sigma(p)=\phi^{-1}f^{-1}\sigma(f) \sigma(\phi),\end{align*} $$

rearranging we get

$$ \begin{align*}f\phi p^{-1}=\sigma(f\phi p^{-1}).\end{align*} $$

It follows, see Remark 2.2, that the isomorphism $f\phi p^{-1}:N\to H$ is A-definable. Thus N and H are identified in $\mathfrak {F}_{\operatorname {def}}(G)$ , showing the map is indeed injective.

We now show surjectivity. Let $\Phi \in Z^1_{\operatorname {def}}(\mathcal G,\operatorname {Aut}_{\operatorname {def}}(G))$ . Then there is tuple a from M and an A-definable (partial) function $h(x,y,z)$ such that $\Phi (\sigma )(-)=h(a,\sigma (a),-)$ for all $\sigma \in \mathcal G$ .

Lemma 2.3. Let a and h be as above. For any $b, c, d$ tuples from M realising $tp(a/A)$ we have that $h(b,c,-)\in \operatorname {Aut}_{\operatorname {def}}(G)$ . Furthermore,

$$ \begin{align*}h(b,c,-)=h(b,d,h(d,c,-)).\end{align*} $$

It follows that $h(b,b,-)=\operatorname {Id}_G$ and $h(b,c,-)^{-1}=h(c,b,-).$

Proof Let $\sigma ,\tau ,\eta \in \mathcal G$ be such that

$$ \begin{align*}\sigma(a)=b,\; \sigma\tau\eta(a)=c\; \textrm{ and }\; \sigma\tau(a)=d.\end{align*} $$

Then

$$ \begin{align*} h(b,c,-)&=h(\sigma(a), \sigma\tau\eta(a),-) \\ &=\sigma(\Phi(\tau\eta))(-) \\ &=\sigma(\Phi(\tau)\tau(\Phi(\eta)))(-) \\ &=h(\sigma(a),\sigma\tau(a), h(\sigma\tau(a),\sigma\tau\eta(a),-)) \\ &=h(b,d,h(d,c,-)). \end{align*} $$

All of the remaining properties follow from this, e.g., $h(b,c,-) \in \operatorname {Aut}_{\operatorname {def}}(G)$ as $h(b,c,-) = h(b,a,h(a,c,-))$ .

Now let Y be the set of realisations of $tp(a/A)$ in M. Then Y is an A-definable set. Set $Z=Y\times G$ . Define a relation R on Z given by

$$ \begin{align*}(b,d)\,R\, (c,e) \; \iff \; h(b,c,e)=d.\end{align*} $$

It follows, using Lemma 2.3, that R is an equivalence relation (which is of course A-definable). Let $H=Z/R$ . Then, by elimination of imaginaries, we identify the quotient H with an A-definable set. We now equip H with a A-definable group structure. Define, for $(b,d), (c,e)\in Z$ ,

$$ \begin{align*}(b,d)*(c,e)=(b,d\cdot h(b,c,e))\in Z.\end{align*} $$

Using Lemma 2.3, one readily checks that this binary operation $*$ on Z is R-invariant and thus induces a A-definable binary operation on H that we also denote by $*$ . We show that this is in fact a group structure on H. We prove this by showing that there is a definable (over M) bijection f between H and G such that the induced group structure on H is precisely $*$ . Note that this further shows that H is a A-form of G.

Let $f:Z\to G$ be defined by $f(b,d)=h(a,b,d)$ . Since $h(a,b,-)\in \operatorname {Aut}_{\operatorname {def}}(G)$ , the function f is clearly surjective. Furthermore, again using Lemma 2.3, one checks that

$$ \begin{align*}f(a,b,d)=f(a,c,e) \; \iff \; h(b,c,e)=d \; \iff\; (b,d) \, R\, (c,e).\end{align*} $$

Thus f induces a definable (over M) bijection from $H=Z/R$ to G, which we also denote by f. Now, using Lemma 2.3, the induced group structure on H via this bijection is

$$ \begin{align*} f^{-1}(f(b,d)\cdot f(c,e)) &=f^{-1}(h(a,b,d)\cdot h(a,c,e)) \\ &=f^{-1}(h(a,b,d)\cdot h(a,b,h(b,c,e))) \\ &=f^{-1}(h(a,b,d\cdot h(b,c,e)) \\ &=f^{-1}(f(b,d\cdot h(b,c,e))) \\ &=(b,d\cdot h(b,c,e)) \\ &=(b,d)*(c,e). \end{align*} $$

We have thus shown that $(H,*)$ is a A-form of G (witnessed by $f^{-1}:G\to H$ ). We claim that $(H,f^{-1})$ maps to $\Phi $ (namely $\Phi =\Phi _{H,f^{-1}}$ ). Indeed, for any $\sigma \in \mathcal G$ , using once again Lemma 2.3, we have

$$ \begin{align*}\Phi(\sigma)\sigma(f)(b,d)&=h(a,\sigma(a), \sigma(f)(b,d))=h(a,\sigma(a), h(\sigma(a), b,d))\\ &=h(a,b,d)=f(b,d),\end{align*} $$

and so

$$ \begin{align*}\Phi(\sigma)=f\cdot \sigma(f^{-1}),\end{align*} $$

this shows that $\Phi =\Phi _{H,f^{-1}}$ , as desired.

We have thus shown:

3 Exact sequences and twisting in definable cohomology

We carry forward the conventions and assumptions on M, A, $\mathcal G,$ and G from previous sections. Let X be an A-definable set in the structure M. Then $\mathcal G$ acts naturally on X. The 0th definable cohomology from $\mathcal G$ to X, denoted $H^0_{\operatorname {def}}(\mathcal G,X)$ , is the set of $\mathcal G$ -invariant points of X. By our assumptions (see Assumptions 1.1), this coincides with the points of X whose entries are all in $\operatorname {dcl}(A)$ . In the case when X has in addition the structure of an A-definable pointed set (i.e., has a distinguished point which is $\mathcal G$ -invariant and thus in $\operatorname {dcl}(A)$ ), then $H^0_{\operatorname {def}}(\mathcal G,X)$ inherits the structure of a pointed set sharing its distinguished element with X. Furthermore, when $X=G$ (an A-definable group), the 0th definable cohomology $H^0_{\operatorname {def}}(\mathcal G,G)$ is clearly a subgroup of G.

Now consider an A-definable map $f:X\to X'$ of A-definable sets. Since f preserves $\mathcal G$ -invariants, it induces a map in cohomology

$$ \begin{align*}f^0:H^0_{\operatorname{def}}(\mathcal G,X)\to H^0_{\operatorname{def}}(\mathcal G,X').\end{align*} $$

In the case that X and $X'$ have in addition the structure of A-definable pointed sets (the distinguished elements being $\mathcal G$ -invariant) and f is a homomorphism of pointed sets (i.e., maps the distinguished element of X to that of $X'$ ), the above map $f^0$ is a homomorphism of pointed sets. Furthermore, when $X=G$ and $X'=G'$ are A-definable groups and f is a group homomorphism, the map

$$ \begin{align*}f^0:H^0_{\operatorname{def}}(\mathcal G,G)\to H^0_{\operatorname{def}}(\mathcal G,G')\end{align*} $$

is a group homomorphism.

Recall, from the introduction, that $Z^1_{\operatorname {def}}(\mathcal G,G)$ and $H^1_{\operatorname {def}}(\mathcal G,G)$ denote the sets of definable cocycles and cohomology from $\mathcal G$ to G, respectively. Let $f:G\to G'$ be an A-definable group homomorphism (between the A-definable groups G and $G'$ ). For any definable 1-cocycle $\Phi \in Z^1_{\operatorname {def}}(\mathcal G,G)$ , the composition $f\circ \Phi :\mathcal G\to G'$ is again a definable 1-cocycle. Furthermore, if $\Psi $ is cohomologous to $\Phi $ , then $f\circ \Phi $ is cohomologous to $f\circ \Psi $ as elements of $Z^1_{\operatorname {def}}(\mathcal G,G')$ . Indeed, suppose $b\in G$ is such that $\Phi (\sigma )=b^{-1}\Psi (\sigma )\sigma (b)$ for all $\sigma \in \mathcal G$ , then one easily verifies

$$ \begin{align*}(f\circ \Psi)(\sigma)=(f(b))^{-1}\, (f\circ \Psi)(\sigma)\, \sigma(f(b)).\end{align*} $$

Thus, the assignment $\Psi \mapsto (f\circ \Phi )$ defines a map from $Z^1_{\operatorname {def}}(\mathcal G,G)$ to $Z^1_{\operatorname {def}}(\mathcal G,G')$ that induces a map in cohomology

$$ \begin{align*}f^1:H^1_{\operatorname{def}}(\mathcal G,G)\to H^1_{\operatorname{def}}(\mathcal G,G').\end{align*} $$

Moreover, $f^1$ is a morphism of pointed sets (meaning it preserves distinguished elements), and when G and $G'$ are abelian then it is a group homomorphism.

3.1 Exact sequence in cohomology

We now fix an arbitrary (not necessarily normal) A-definable subgroup N of G. This yields the natural short exact sequence of A-definable pointed sets

(3.1) $$ \begin{align} 1\to N \xrightarrow{\iota} G \xrightarrow{\pi} G/N\to 1. \end{align} $$

Note that, since N is not necessarily normal in G, the left-coset space $G/N$ is not necessarily a group. It is however an A-definable (left) homogeneous space for G (i.e., $G/N$ is an A-definable set equipped with an A-definable transitive left G-action—in this case the natural G-action on the left). While the inclusion map $\iota $ is a group homomorphism, the projection map $\pi $ is generally just a morphism of A-definable homogenous G-spaces. Furthermore, $G/N$ has a natural structure of an A-definable pointed set, with distinguished element being the coset N, and so $\pi $ is also a homomorphism of A-definable pointed sets. When N is normal in G, the above sequence is a normal short exact sequence of A-definable groups.

Let $a\in H^0_{\operatorname {def}}(\mathcal G,G/N)$ , here we think of $G/N$ as a pointed set and so the 0th cohomology (which the same as the $\mathcal G$ -invariants of $G/N$ ) is also a pointed set as explained above. Then, a is of the form $\pi (b)$ for some $b\in G$ . For any $\sigma \in \mathcal G$ we have

$$ \begin{align*}\pi(\sigma(b))=\sigma(\pi(b))=\sigma(a)=a=\pi(b)\end{align*} $$

and so $b^{-1}\cdot \sigma (b)\in N$ . Thus the assignment $\sigma \mapsto b^{-1}\cdot \sigma (b)$ defines a map from $\mathcal G$ to N. Moreover, this map clearly belongs to $Z^1_{\operatorname {def}}(\mathcal G,N)$ ; and if $b'$ is another element of G with $\pi (b')=a$ , then $b'=bc$ for some $c\in N$ and hence

$$ \begin{align*}(bc)^{-1}\cdot \sigma(bc)=c^{-1}(b^{-1}\cdot \sigma(b))\sigma(c).\end{align*} $$

This shows that the assignment $\sigma \mapsto (b')^{-1}\cdot \sigma (b')$ is cohomologous to the previous one. Therefore, the cohomology class of this assignment is uniquely determined by a, and this class is denoted by $\delta (a)\in H^1_{\operatorname {def}}(\mathcal G,N)$ . We have now a morphism

$$ \begin{align*}\delta : H^0_{\operatorname{def}}(\mathcal G,G/N)\to H^1_{\operatorname{def}}(\mathcal G,N)\end{align*} $$

of pointed sets (and of groups when G is abelian) called the connecting homomorphism.

We are now in the position to state (and prove) how exactness transfers when passing to the sequence in cohomology.

Theorem 3.1. Suppose N is an A-definable subgroup of $G ($ not necessarily normal $)$ . Then, the sequence in cohomology induced from (3.1)

$$ \begin{align*}1\to H^{0}_{\operatorname{def}}(\mathcal G,N)\xrightarrow{\iota^0}H^0_{\operatorname{def}}(\mathcal G,G)\xrightarrow{\pi^0} H^0_{\operatorname{def}}(\mathcal G,G/N)\xrightarrow{\delta}H^1_{\operatorname{def}}(\mathcal G,N)\xrightarrow{\iota^1} H^{1}_{\operatorname{def}}(\mathcal G,G)\end{align*} $$

is exact $($ as a sequence of pointed sets $)$ . Furthermore, if N is normal in G, then the sequence remains exact when extended by

$$ \begin{align*}H^{1}_{\operatorname{def}}(\mathcal G,G)\xrightarrow{\pi^1}H^1_{\operatorname{def}}(\mathcal G,G/N).\end{align*} $$

Proof It is easy to check that the sequence is exact at $H^{0}_{\operatorname {def}}(\mathcal G,N)$ and at $H^0_{\operatorname {def}}(\mathcal G,G)$ . We now prove it is exact at $H^0_{\operatorname {def}}(\mathcal G,G/N)$ . Let $a\in H^0_{\operatorname {def}}(\mathcal G,G/N)$ and $b\in G$ such that $\pi (b)=a$ . Then, $\delta (a)=1$ if and only if there is $c\in N$ such that

$$ \begin{align*}b^{-1}\sigma(b)=c^{-1}\sigma(c), \quad \text{ for all }\sigma\in \mathcal G;\end{align*} $$

that is, $bc^{-1}\in H^0_{\operatorname {def}}(\mathcal G,G)$ . Thus, $\delta (a)=1$ if and only there is $d\in H^0_{\operatorname {def}}(\mathcal G,G)$ such that $\pi ^0(d)=a$ .

We now check the sequence in exact at $H^1_{\operatorname {def}}(\mathcal G,N)$ . Let $\eta \in H^1_{\operatorname {def}}(\mathcal G,N)$ and $\Phi $ a cocycle in $Z^1_{\operatorname {def}}(\mathcal G,N)$ with cohomology class $\eta $ . Then, $\iota ^1(\eta )=1$ if and only there is $b\in G$ such that $\Phi (\sigma )=b^{-1}\sigma (b)$ for all $\sigma \in \mathcal G$ . The latter implies

$$ \begin{align*}\pi(b)^{-1}\sigma(\pi(b))=\pi(b^{-1}\sigma(b))=\pi(\Phi(\sigma))=1\end{align*} $$

and so $\pi (b)\in H^0_{\operatorname {def}}(\mathcal G,G/N)$ . Clearly, $\delta (\pi (b))=\eta $ , and exactness at $H^1_{\operatorname {def}}(\mathcal G,N)$ follows.

Finally, assuming N is normal in G, we check exactness at $H^1_{\operatorname {def}}(\mathcal G,G)$ . Let $\Phi \in Z^1_{\operatorname {def}}(\mathcal G,N)$ . Then, since $im(\iota )=ker(\pi )$ , we get that $\pi \circ \iota \circ \Phi $ is the trivial cocycle from $\mathcal G$ to $G/N$ . It follows that $im(\iota ^1)\subseteq ker(\pi ^1)$ . Now, let $\Psi \in Z^1_{\operatorname {def}}(\mathcal G,G)$ be such that $\pi \circ \Psi $ is a coboundary from $\mathcal G$ to $G/N$ . This means that there is $d\in G/N$ such that

$$ \begin{align*}\pi(\Psi(\sigma))=d^{-1}\cdot \sigma(d), \quad \text{ for all } \sigma\in \mathcal G.\end{align*} $$

Let $a\in G$ such that $\pi (a)=d$ . Then we can rearrange the above to

$$ \begin{align*}\pi(a\cdot \Psi(\sigma)\cdot \sigma(a)^{-1})=1.\end{align*} $$

Since $\iota $ is injective and $im(\iota )=ker(\pi )$ , there is a unique $c\in N$ such that $\iota (c)=a\cdot \Psi (\sigma )\cdot \sigma (a)^{-1}$ . If we set $\Lambda :\mathcal G\to N$ to be $\Lambda (\sigma )=c$ , where c is the unique c just found, then one can readily check that $\Lambda $ is a definable cocycle from $\mathcal G$ to N. Moreover, $\iota \circ \Lambda $ is cohomologous to $\Psi $ as witnessed by $a \in G$ . This proves $ker(\pi ^1)\subseteq im(\iota ^1)$ , as desired.

3.2 Twisting definable cohomology

Recall that when G is abelian, the cohomology sequence in Theorem 3.1 is an exact sequence of groups and group homomorphism, and so fibres are just translates of the kernels. However, in the general situation (when G is not necessarily abelian), the exact sequence is just a sequence of pointed sets and thus carry less information than in the abelian case. Indeed, the kernel of a morphism of pointed sets does not generally yield any information about the other fibres.

As in algebraic Galois cohomology [Reference Platonov and Rapinchuk8, Section 1.3], this can be overcome using a method introduced by Serre [Reference Serre12, Section 1.5] called twisting. We now discuss how a special case (which suffices for our purposes) of this method transfers to our general definable model-theoretic context. Let $\Phi \in Z^1_{\operatorname {def}}(\mathcal G,G)$ and define a new (twisted) action of $\mathcal G$ on G by

$$ \begin{align*}\sigma * g = C_{\Phi(\sigma)}(\sigma(g)) \ \ (\ \ =\Phi(\sigma)\sigma(g)\Phi(\sigma)^{-1} \ \ )\end{align*} $$

where, for any $h \in G$ , $C_h:G \to G$ denotes conjugation by h; namely, $C_h(g) = hgh^{-1}$ for all $g \in G$ . We call this the induced action of $\mathcal G$ on G twisted via conjugation by $\Phi $ . When clarity is needed we denote this action by saying that $\mathcal G_\Phi $ acts on G. One readily checks that this is indeed a group action and in fact $\mathcal G_\Phi $ acts on G by group automorphisms. Furthermore, since $\Phi :\mathcal G\to G$ is continuous, this action is continuous.

With respect to this (non-natural) action of $\mathcal G_\Phi $ on G, we can define the notion of definable (1-)cocycles, denoted $Z^1_{\operatorname {def}}(\mathcal G_\Phi ,G)$ , as those maps $\Psi :\mathcal G\to G$ such that

$$ \begin{align*}\Psi(\sigma_1\sigma_2)=\Psi(\sigma_1)\cdot \sigma_1 *(\Psi(\sigma_2)).\end{align*} $$

As before, such cocycle is said to be definable if there is tuple a from M and an A-definable function $h(x,y)$ such that

$$ \begin{align*}\Psi(\sigma)=h(a,\sigma(a)), \quad \text{ for all } \sigma\in G.\end{align*} $$

We remark that definable cocycles from $\mathcal G_{\Phi }$ to G are continuous (see [Reference Pillay5] for more discussion of continuity and definability of cocycles). Two cocycles $\Psi $ and $\Lambda $ in $Z^1_{\operatorname {def}}(\mathcal G_\Phi ,G)$ are cohomologous if there is $b\in G$ such that

$$ \begin{align*}\Psi(\sigma)=b^{-1}\cdot \Lambda(\sigma)\cdot (\sigma * b), \quad \text{ for all }\sigma\in \mathcal G.\end{align*} $$

This again yields an equivalence relation. The set of equivalence classes is the first definable cohomology set $H^{1}_{\operatorname {def}}(\mathcal G_\Phi ,G)$ twisted by $\Phi $ . This has a natural structure of a pointed set.

Remark 3.2. Note that when G is abelian the twisted action of $\mathcal G_\Phi $ on G is just the natural action of $\mathcal G$ on G and so the cohomologies $H^{1}_{\operatorname {def}}(\mathcal G_\Phi ,G)$ and $H^1_{\operatorname {def}}(\mathcal G,G)$ coincide.

As with the 0th definable cohomology set $H^0_{\operatorname {def}}(\mathcal G,G)$ , the 0th definable twisted cohomology $H^0_{\operatorname {def}}(\mathcal G_\Phi ,G)$ is defined as the set of $\mathcal G_\Phi $ -invariant points of G, which is a subgroup of G (as $\mathcal G_\Phi $ acts on G by group automorphisms).

Now let N be an A-definable normal subgroup of G. We can then restrict the action of $\mathcal G_\Phi $ on G to N (since N is normal in G). This yields a continuous (twisted) action by group automorphisms of $\mathcal G_\Phi $ on N. In a similar way as above, we can construct the sets of twisted definable cocycles and cohomology $Z^1_{\operatorname {def}}(\mathcal G_\Phi ,N)$ and $H^1_{\operatorname {def}}(\mathcal G_\Phi ,N)$ , respectively. Furthermore, letting $\pi :G\to G/N$ be the natural projection, we can twist the natural action of $\mathcal G$ on $G/N$ by the cocycle $\pi \circ \Phi \in Z^1_{\operatorname {def}}(\mathcal G,G/N)$ . Then one can construct the twisted definable cocycles and cohomology sets $Z^1_{\operatorname {def}}(\mathcal G_{\pi \Phi },G/N)$ and $H^1_{\operatorname {def}}(\mathcal G_{\pi \Phi },G/N)$ . For convenience, and it should not cause confusion, we denote $\mathcal G_{\pi \Phi }$ (acting on $G/N$ ) simply by $\mathcal G_\Phi $ and the set of definable cocycles and cohomology by $Z^1_{\operatorname {def}}(\mathcal G_{\Phi },G/N)$ and $H^1_{\operatorname {def}}(\mathcal G_{\Phi },G/N).$

Consider the natural short exact sequence of A-definable groups (recall that we are assuming that N is normal in G)

$$ \begin{align*}1\to N \xrightarrow{\iota} G \xrightarrow{\pi} G/N\to 1.\end{align*} $$

With the respect to the above twisted actions of $\mathcal G_\Phi $ on N, G, and $G/N$ , one readily checks that for $\sigma \in \mathcal G$ , $n\in N$ , and $g\in G$ we have

$$ \begin{align*}\iota(\sigma*n)=\sigma*\iota(n) \quad \text{ and }\quad \pi(\sigma*g)=\sigma*\pi(g).\end{align*} $$

It follows from these equalities, exactly as we did in the previous sub-section, that we can induce morphisms of pointed sets in cohomology

$$ \begin{align*}\iota^i_\Phi:H^i_{\operatorname{def}}(\mathcal G_\Phi,N)\to H^i_{\operatorname{def}}(\mathcal G_\Phi,G) \quad \text{ and }\quad \pi^i_\Phi:H^i_{\operatorname{def}}(\mathcal G_\Phi,G)\to H^i_{\operatorname{def}}(\mathcal G_\Phi,G/N)\end{align*} $$

for $i=0,1$ . Furthermore, we get a connecting morphism (or pointed sets)

$$ \begin{align*}\delta_\Phi:H^0(\mathcal G_\Phi,G/N)\to H^1_{\operatorname{def}}(\mathcal G_\Phi,N),\end{align*} $$

and obtain, by adapting the argument in the proof of Theorem 3.1, the (twisted) exact sequence

$$ \begin{align*} 1\to H^{0}_{\operatorname{def}}(\mathcal G_\Phi,N)\xrightarrow{\iota^0_\Phi}H^0_{\operatorname{def}}(\mathcal G_\Phi,G) & \xrightarrow{\pi^0_\Phi} H^0_{\operatorname{def}}(\mathcal G_\Phi,G/N) \\ & \xrightarrow{\delta}H^1_{\operatorname{def}_\Phi}(\mathcal G_\Phi,N)\xrightarrow{\iota^1_\Phi} H^{1}_{\operatorname{def}}(\mathcal G_\Phi,G)\xrightarrow{\pi^1_\Phi}H^1_{\operatorname{def}}(\mathcal G_\Phi,G/N) \end{align*} $$

of pointed sets.

While one is not generally able to multiply cocycles in $Z^1_{\operatorname {def}}(\mathcal G,G)$ , unless G is abelian say, by going to the twisted context this is possible. We make this precise in the following result which also shows how the cohomologies $H^1_{\operatorname {def}}(\mathcal G,G)$ and $H^1_{\operatorname {def}}(\mathcal G_\Phi ,G)$ are related (in fact in bijection).

Proposition 3.3. Let $f_\Phi :Z^1_{\operatorname {def}}(\mathcal G_\Phi ,G)\to Z^1_{\operatorname {def}}(\mathcal G,G)$ be the map defined by $f_\Phi (\Psi )=\Psi \cdot \Phi $ ; namely, for $\sigma \in G$ , $f_\Phi (\Psi )(\sigma )=\Psi (\sigma )\cdot \Phi (\sigma )$ . Then, $f_\Phi $ is a bijection. Furthermore, it induces a bijection in cohomology $F_\Phi :H^1_{\operatorname {def}}(\mathcal G_\Phi ,G)\to H^1_{\operatorname {def}}(\mathcal G,G)$ which maps the distinguished element to the cohomology class of $\Phi $ .

Proof We first prove that indeed $f_\Phi (\Psi )\in Z^{1}_{\operatorname {def}}(\mathcal G,G)$ for every $\Psi \in Z^1_{\operatorname {def}}(\mathcal G_\Phi ,G)$ .

Indeed, it is clearly definable (since $\Phi $ and $\Psi $ are definable) and for $\sigma _1,\sigma _2\in \mathcal G$ we have

$$ \begin{align*} f_\Phi(\Psi)(\sigma_1\sigma_2) &= \Psi(\sigma_1\sigma_2) \cdot \Phi(\sigma_1\sigma_2)\\ &= \Psi(\sigma_1) \cdot \sigma_1*(\Psi(\sigma_2)) \cdot \Phi(\sigma_1) \cdot \sigma_1(\Phi(\sigma_2)) \\ &= \Psi(\sigma_1) \cdot \Phi(\sigma_1)\cdot \sigma_1(\Psi(\sigma_2))\cdot \Phi(\sigma_1)^{-1}\cdot \Phi(\sigma_1)\cdot \sigma_1(\Phi(\sigma_2)) \\ &= f_\Phi(\Psi)(\sigma_1)\cdot \sigma_1(f_\Phi(\Psi)(\sigma_2)). \end{align*} $$

One easily checks that $f_\Phi $ has inverse $f^{-1}_\Phi :Z^1_{\operatorname {def}}(\mathcal G,G)\to Z^{1}_{\operatorname {def}}(\mathcal G_\Phi ,G)$ given by

$$ \begin{align*}f_\Phi^{-1}(\Lambda)(\sigma)=\Lambda(\sigma)\cdot (\Phi(\sigma))^{-1}.\end{align*} $$

Now, to pass to cohomology, let $\Psi ,\Lambda \in Z^1_{\operatorname {def}}(\mathcal G_\Phi ,G)$ be cohomologous witnessed by $b\in G$ ; namely,

$$ \begin{align*}\Lambda(\sigma)=b^{-1}\cdot\Psi(\sigma) \cdot \sigma* b, \quad \text{ for all }\sigma \in \mathcal G.\end{align*} $$

Multiplying on the right by $\Phi (\sigma )$ we get that

$$ \begin{align*} f_\Phi(\Lambda)(\sigma) &= \Lambda(\sigma)\cdot \Phi(\sigma)\\ &= b^{-1}\cdot \Psi(\sigma)\cdot \sigma * b \cdot \Phi(\sigma)\\ &= b^{-1}\cdot \Psi(\sigma)\cdot \Phi(\sigma)\cdot \sigma(b)\cdot \Phi(\sigma)^{-1} \cdot \Phi(\sigma)\\ &= b^{-1}\cdot f_\Phi(\Psi)(\sigma)\cdot \sigma(b). \end{align*} $$

Thus, $f_\Phi (\Lambda )$ and $f_\Phi (\Psi )$ are cohomologous as elements in $Z^1_{\operatorname {def}}(\mathcal G,G)$ , and so $f_\Phi $ induces a well-defined map $F_\Phi :H^1_{\operatorname {def}}(\mathcal G_\Phi ,G)\to H^{1}_{\operatorname {def}}(\mathcal G,G)$ . Let $1$ be the trivial cocycle in $Z^1_{\operatorname {def}}(\mathcal G_\Phi ,G)$ , then, by definition, $F_\Phi $ maps the distinguished element of $H^1_{\operatorname {def}}(\mathcal G_\Phi ,G)$ to the cohomology class of

$$ \begin{align*}f_\Phi(1)=\Phi,\end{align*} $$

as desired. Finally, one can similarly check that the inverse function $f^{-1}_\Phi $ induces a well-defined map in cohomology

$$ \begin{align*}F_\Phi^{-1}:H^1_{\operatorname{def}}(\mathcal G,G)\to H^1_{\operatorname{def}}(\mathcal G_\Phi,G)\end{align*} $$

which an easy computation shows that it is in fact the inverse for $F_\Phi $ .

Let $\Psi \in Z^1_{\operatorname {def}}(\mathcal G_{\Phi }, G)$ and $\sigma \in \mathcal G$ . As $\pi : G \to G/N$ is a homomorphism, we have

$$ \begin{align*} f_{\pi\Phi}(\pi\circ \Psi)(\sigma) &= \pi \circ \Psi(\sigma)\cdot \pi \circ \Phi(\sigma)\\ &= \pi(\Psi(\sigma)\cdot \Phi(\sigma))\\ &= \pi \circ f_{\Phi}(\Psi)(\sigma), \end{align*} $$

which shows that the following diagram commutes:

This induces a commutative diagram in cohomology

which, together with Proposition 3.3 applied to $f_\Phi $ and $f_{\pi \Phi }$ , shows that the bijection

$$ \begin{align*}F_\Phi:H^1_{\operatorname{def}}(\mathcal G_\Phi,G)\to H^1_{\operatorname{def}}(\mathcal G,G)\end{align*} $$

restricts to a bijection between the kernel of $\pi ^1_\Phi $ and the fibre $(\pi ^1)^{-1}(\pi ^1(\mu )),$ where $\mu $ is the cohomology class of $\Phi $ . Thus, we have proved the following.

Corollary 3.4. With the above notation and $\mu $ denoting the cohomology class of $\Phi $ , there is a natural bijection between $ker(\pi ^1_\Phi )$ and the fibre

$$ \begin{align*}\mathfrak P(\mu):=(\pi^1)^{-1}(\pi^1(\mu)).\end{align*} $$

This demonstrates how, using the method of twisting, one can describe the fibres of $\pi ^1$ in the original (non-twisted) cohomology sequence.

Remark 3.5. We note that if $\Psi $ is a cocycle in $Z^1_{\operatorname {def}}(\mathcal G,G)$ which is cohomologous to $\Phi $ , witnessed by $b\in G$ (i.e., $\Phi (\sigma )=b^{-1}\Psi (\sigma )\sigma (b)$ ), then the inner automorphism of G given by conjugation $C_b:G\to G$ commutes with the twisted actions of $G_\Phi $ and $G_\Psi $ on G. More precisely,

$$ \begin{align*}C_b(\sigma*_\Phi g)=\sigma*_{\Psi}C_b(g), \quad \text{ for all }\sigma\in \mathcal G, g\in G.\end{align*} $$

Let $\Xi \in Z^1_{\operatorname {def}}(\mathcal G_{\Phi },G)$ , we claim $C_b \circ \Xi \in Z^1_{\operatorname {def}}(\mathcal G_{\Psi },G)$ . Let $\sigma ,\tau \in \mathcal G$ ,

$$ \begin{align*} C_b \circ \Xi(\sigma\tau) &= C_b(\Xi(\sigma)\cdot \sigma *_{\Phi}\Xi(\tau))\\ &= C_b(\Xi(\sigma))\cdot C_b(\sigma *_{\Phi}\Xi(\tau))\\ &= C_b(\Xi(\sigma))\cdot \sigma *_{\Psi}C_b(\Xi(\tau)). \end{align*} $$

$C_b\circ \Xi $ is clearly a definable cocycle. It is straightforward to see that there are natural isomorphisms of pointed sets

$$ \begin{align*}C_b \circ -\ :Z^1_{\operatorname{def}}(\mathcal G_\Phi,G)\to Z^1_{\operatorname{def}}(\mathcal G_\Psi,G) \quad \text{ and }\quad H^i_{\operatorname{def}}(\mathcal G_\Phi,G)\to H^i_{\operatorname{def}}(\mathcal G_\Psi,G)\end{align*} $$

for $i = 0,1$ . Thus, for any $\mu \in H^1_{\operatorname {def}}(\mathcal G,G)$ , we may and will denote by $Z^1_{\operatorname {def}}(\mathcal G_\mu ,G)$ and $H^i_{\operatorname {def}}(\mathcal G_\mu ,G)$ the set of definable cocycles $Z^1_{\operatorname {def}}(\mathcal G_\Phi ,G)$ and cohomology sets $H^i_{\operatorname {def}}(\mathcal G_\Phi ,G)$ for $i=0,1$ , respectively, with $\Phi $ any definable cocycle having cohomology class $\mu $ . Similar observations apply to $Z^1_{\operatorname {def}}(\mathcal G_\mu ,N)$ and $H^i_{\operatorname {def}}(\mathcal G_\mu ,N)$ , and to $Z^1_{\operatorname {def}}(\mathcal G_\mu ,G/N)$ and $H^i_{\operatorname {def}}(\mathcal G_\mu ,G/N)$ , where N is an A-definable normal subgroup of G.

4 Further remarks on fibres and a finiteness result

We fix, throughout this section, an A-definable normal subgroup N of G. Let $\mu \in H^{1}_{\operatorname {def}}(\mathcal G,G)$ . In this section we deploy the results from the previous two sections to establish an isomorphism of pointed sets between the twisted cohomology $H^1_{\operatorname {def}}(\mathcal G_\mu ,N)$ (see Remark 3.5) and the non-twisted $H^1_{\operatorname {def}}(\mathcal G,N_\mu ),$ where $N_\mu $ is the natural A-form of N corresponding to $\mu $ (see details below). We then conclude with a finiteness result in definable cohomology.

But first we note an easy consequence of Corollary 3.4. Consider, once again, the short exact sequence of A-definable groups

$$ \begin{align*}1\to N \xrightarrow{\iota} G \xrightarrow{\pi} G/N\to 1.\end{align*} $$

For $\mu \in H^{1}_{\operatorname {def}}(\mathcal G,G)$ , we let

$$ \begin{align*}\mathfrak P(\mu):=(\pi^1)^{-1}(\pi^1(\mu))\end{align*} $$

denote the fibre of $\pi ^1:H^1_{\operatorname {def}}(\mathcal G,G)\to H^1_{\operatorname {def}}(\mathcal G,G/N)$ above $\pi ^1(\mu )$ . Then, $\mathfrak P(\mu )$ has the structure of a pointed set by specifying $\mu $ as its distinguished element. In the next statement we use the notation discussed in Remark 3.5.

For $\mu \in H^{1}_{\operatorname {def}}(\mathcal G,G)$ , let $\Phi \in Z^1_{\operatorname {def}}(\mathcal G,G)$ be a cocycle with cohomology class $\mu $ . Using the notation of Section 2, consider the map $\Lambda _\Phi :\mathcal G\to \operatorname {Aut}_{\operatorname {def}}(N)$ given by

$$ \begin{align*}\Lambda_\Phi(\sigma)(n)=C_{\Phi(\sigma)}(n)\end{align*} $$

for all $n \in N$ . Recall that $C_{h}:N\to N$ denotes conjugation by $h\in G$ ; namely, $C_{h}(n)=h\cdot n\cdot h^{-1}$ for all $n\in N$ . Since N is a normal in G, the image of $\Lambda _\Phi $ is indeed in $\operatorname {Aut}_{\operatorname {def}}(N)$ . Furthermore, $\Lambda _\Phi $ is a definable cocycle from $\mathcal G$ to $\operatorname {Aut}_{\operatorname {def}}(N)$ . It is clearly definable and for $\sigma _1,\sigma _2\in \mathcal G$ and $n \in N$ we have

$$ \begin{align*} \Lambda_\Phi(\sigma_1\sigma_2)(n) & =C_{\Phi(\sigma_1\sigma_2)}(n) \\ & = \Phi(\sigma_1\sigma_2) \cdot n\cdot (\Phi(\sigma_1\sigma_2))^{-1} \\ & = \Phi(\sigma_1)\cdot\sigma_1(\Phi(\sigma_2)) \cdot n \cdot (\sigma_1(\Phi(\sigma_2)))^{-1}\cdot (\Phi(\sigma_1))^{-1} \\ & = \Phi(\sigma_1) \cdot \sigma_1(C_{\Phi(\sigma_2)})(n) \cdot (\Phi(\sigma_1))^{-1} \\ & = C_{\Phi(\sigma_1)} (\sigma_1(C_{\Phi(\sigma_2)})(n))\\ & = \Lambda_\Phi(\sigma_1)\cdot \sigma_1(\Lambda_\Phi(\sigma_2))(n). \end{align*} $$

Furthermore, if $\Psi \in Z^1_{\operatorname {def}}(\mathcal G,G)$ is cohomologous to $\Phi $ , we have that $\Lambda _\Psi $ is cohomologous to $\Lambda _{\Phi }$ in $Z^1_{\operatorname {def}}(\mathcal G,\operatorname {Aut}_{\operatorname {def}}(N))$ . Indeed, if $b\in G$ witnesses the fact that $\Psi $ and $\Phi $ are cohomologous, then conjugation $C_{b}\in \operatorname {Aut}_{\operatorname {def}}(N)$ witnesses that $\Lambda _{\Psi }$ is cohomologous to $\Lambda _{\Phi }$ .

We thus have a well-defined map from $H^1_{\operatorname {def}}(\mathcal G,G)$ to $H^{1}_{\operatorname {def}}(\mathcal G,\operatorname {Aut}_{\operatorname {def}}(N))$ mapping $\mu $ to the cohomology class of $\Lambda _\Phi $ . By Proposition 2.4, this yields a map from $H^1_{\operatorname {def}}(\mathcal G,G)$ to $\mathfrak {F}_{\operatorname {def}}(N)$ (the collection of A-forms of N up to A-definable isomorphism). For $\mu \in H^1_{\operatorname {def}}(\mathcal G,G)$ we denote the associated A-form of N by $N_\mu $ . We note that when G is abelian $\Lambda _\Phi $ is the trivial cocycle and so $N_\mu $ is definably isomorphic to N over A.

We now state and prove the main result of this section.

Let $f: N_\mu \to N$ be the definable isomorphism (over M) constructed in Section 2 (right before Proposition 2.4). We recall the construction of f to obtain an explicit formula. Let $\Phi \in Z^1_{\operatorname {def}}(\mathcal G,G)$ be a definable cocycle with cohomology class $\mu $ . Let a be a tuple from M and $h(x,y)$ an A-definable function witnessing that $\Phi $ is definable; that is, $\Phi (\sigma )=h(a,\sigma (a))$ for all $\sigma \in \mathcal G$ . Recall that above we defined $\Lambda _\Phi :\mathcal G\to \operatorname {Aut}_{\operatorname {def}}(N)$ by $\Lambda _\Phi (\sigma )(-)=C_{\Phi (\sigma )}(-)$ and showed it is a definable cocycle from $\mathcal G$ to $\operatorname {Aut}_{\operatorname {def}}(N)$ . It follows that

$$ \begin{align*}\Lambda_\Phi(\sigma)(-)=h(a,\sigma(a))\cdot (-)\cdot h(a,\sigma(a))^{-1}.\end{align*} $$

In other words, if we set $\tilde h$ to be the A-definable (partial) function

$$ \begin{align*}\tilde h(x,y,z)=h(x,y)\cdot z\cdot h(x,y)^{-1}\end{align*} $$

we get

$$ \begin{align*}\Lambda_{\Phi}(\sigma)(-)=\tilde h(a,\sigma(a),-), \quad \text{ for all }\sigma\in \mathcal G,\end{align*} $$

that is, a and $\tilde h(x,y,z)$ witness that the cocycle $\Lambda _\Phi $ is definable. Now let Y be the realisations of $tp(a/A)$ in M, then

$$ \begin{align*}N_\mu=(Y\times N)/R,\end{align*} $$

where R is the equivalence relation

$$ \begin{align*}(b,d)\,R\, (c,e)\quad \iff \quad \tilde h(b,c,e)=d \quad \iff \quad h(b,c)\cdot e \cdot h(b,c)^{-1}=d.\end{align*} $$

The group structure on $N_\mu $ is the one induced by

$$ \begin{align*}(b,d) * (c,e)=(b,d\cdot \tilde h(b,c,e))=(b,d\cdot h(b,c)\cdot e \cdot h(b,c)^{-1})\end{align*} $$

for $(b,c), (c,e)\in Y\times N$ . Finally, the function $f:N_{\mu }\to N$ is given by

$$ \begin{align*}f(m)=\tilde h(a,\tau(a),n)=\Lambda_\Phi(\tau)(n)=C_{\Phi(\tau)}(n),\end{align*} $$

where m is the R-equivalence class of $(\tau (a),n)\in Y\times N$ . Recall that Y is the $\operatorname {Aut}(M/A)$ -orbit of a, so every element in Y is of the form $\tau (a)$ for some $\tau \in \operatorname {Aut}(M/A)$ .

The upshot is that now Theorem 4.2 is a consequence of the following lemma.

Lemma 4.3. Let $f:N_\mu \to N$ be as above. Set

$$ \begin{align*}t_f:Z^1_{\operatorname{def}}(\mathcal G,N_\mu)\to Z^{1}_{\operatorname{def}}(\mathcal G_\mu,N)\end{align*} $$

to be the map defined by $t_f(\Psi )=f\circ \Psi $ . Then, $t_f$ is a bijection. Furthermore, it induces an isomorphism of pointed sets $T_f:H^1_{\operatorname {def}}(\mathcal G,N_\mu )\to H^{1}_{\operatorname {def}}(\mathcal G_\mu ,N).$

Proof We first show that f commutes with the actions of $\mathcal G$ on $N_\mu $ and $\mathcal G_\mu $ on N; that is, we prove that for $\sigma \in \mathcal G$ and $m\in N_\mu $ we have

(4.1) $$ \begin{align} f(\sigma(m))=\sigma* f(m). \end{align} $$

Indeed, let m be the R-equivalence class of $(\tau (a), n)\in Y\times N$ , by construction of f we have (see discussion above)

$$ \begin{align*}f(m)=C_{\Phi(\tau)}(n)\quad \text{ and }\quad f(\sigma(m))=C_{\Phi(\sigma\tau)}(\sigma(n)).\end{align*} $$

The latter follows as $\sigma (m)$ is the R-equivalence class of $(\sigma \tau (a),\sigma (n))$ . Using this, we get

$$ \begin{align*} \sigma * (f(m))&= \sigma* (C_{\Phi(\tau)}(n)) \\ &= C_{\Phi(\sigma)}\left(\sigma\left(C_{\Phi(\tau)}(n)\right)\right) \\ &= C_{\Phi(\sigma)}C_{\sigma(\Phi(\tau))}\sigma(n) \\ &= C_{\Phi(\sigma\tau)}\sigma(n) \\ &= f(\sigma(m)), \end{align*} $$

as desired.

Let $\Psi \in Z^1_{\operatorname {def}}(\mathcal G,N_\mu )$ . It follows from (4.1), as we have done in previous sections, that $f\circ \Psi $ is a cocycle from $\mathcal G_\mu $ to N. As f is a definable function, $t_f(\Psi )=f\circ \Psi $ is in fact in $Z^1_{\operatorname {def}}(\mathcal G_\mu ,N)$ . One can analogously consider $f^{-1}:N\to N_\mu $ and easily argue that the map

$$ \begin{align*}t_{f^{-1}}:Z^1_{\operatorname{def}}(\mathcal G_\mu,N)\to Z^{1}_{\operatorname{def}}(\mathcal G,N_\mu)\end{align*} $$

given by $t_{f^{-1}}(\Lambda )=f^{-1}\circ \Lambda $ is the inverse of $t_f$ . Note here that one can use (4.1) to argue that $f^{-1}$ commutes with the relevant actions.

It also follows from (4.1), as we have explicitly done in previous sections, that $t_f$ induces a (well defined) map $T_f:H^1_{\operatorname {def}}(\mathcal G,N_\mu )\to H^{1}_{\operatorname {def}}(\mathcal G_\mu ,N)$ . Clearly, since $f:N_\mu \to N$ is a group homomorphism, the map $t_f$ maps the trivial cocycle of $Z^{1}_{\operatorname {def}}(\mathcal G,N_\mu )$ to that of $Z^{1}_{\operatorname {def}}(\mathcal G_\mu ,N)$ . Thus, $T_{f}$ maps distinguished element to distinguished element. Furthermore, one readily checks that the induced map $T_{f^{-1}}:H^1_{\operatorname {def}}(\mathcal G_\mu ,N)\to H^{1}_{\operatorname {def}}(\mathcal G,N_\mu )$ is the inverse of $T_f$ , and hence $T_f$ is indeed an isomorphism of pointed sets.

Here is the promised finiteness result.

Proof Since $H^1_{\operatorname {def}}(\mathcal G, G/N)$ is finite, there are $\mu _1,\dots ,\mu _r\in H^1_{\operatorname {def}}(\mathcal G, G)$ such that

$$ \begin{align*}H^1_{\operatorname{def}}(\mathcal G, G/N)= \{\pi^1(\mu_1),\dots, \pi^1(\mu_r)\}.\end{align*} $$

Note that the $\pi ^1$ -fibre in $H^1_{\operatorname {def}}(\mathcal G, G)$ over $\pi ^1(\mu _i)$ coincides with the (pointed) set $\mathfrak P(\mu _i)$ as defined above. By Lemma 4.1 and Theorem 4.2, this set is a homomorphic image of the cohomology $H^1_{\operatorname {def}}(\mathcal G, N_{\mu _i}),$ where $N_{\mu _i}$ is the A-form of N associated with $\mu _i$ . By our assumption, the latter cohomology set is finite, and so the $\pi ^1$ -fibre in $H^1_{\operatorname {def}}(\mathcal G, G)$ over $\pi ^1(\mu _i)$ is finite for all $i=1,\dots ,r$ . It follows that $H^1_{\operatorname {def}}(\mathcal G, G)$ is finite.

5 Differential Galois cohomology

In this section we complete the results on differential constrained cohomology of differential algebraic groups over bounded, differentially large, fields from [Reference León Sánchez and Pillay3]. We also clarify a proof from [Reference Kamensky and Pillay1] about the algebraic Galois cohomology of algebraic groups over bounded fields.

For model-theoretic notions, differential algebraic notions and their inter-relations, see the introductions to [Reference León Sánchez and Pillay3] and [Reference Pillay7], and the various references there (such as Poizat’s [Reference Poizat9]). But we recall some key definitions: The field K (which we are assuming to be of characteristic $0$ ) is bounded or equivalently has Serre’s property (F), if K has only finitely many extensions of degree n, for each n. A field K is said to be large if whenever W is a K-irreducible variety with a smooth K-point, then V has a Zariski-dense set of K-points. A differential field $(K,\partial )$ is said to be differentially large if K is large as a field, and for any differential field extension $(L,\partial )$ , if K is existentially closed in L (as fields), then $(K,\partial )$ is existentially closed in $(L,\partial )$ . We will also be using notation from Example 1.3.

We will make use of a slight generalization or version of Corollary 4.4. So here we are in the same general context as the previous section: M is a structure, A a subset of the universe of M as in Assumption 1.1, G is a group definable in M over A, N a normal subgroup of G definable in M over A, $\pi $ denotes the surjective homomorphism $G\to G/N$ , and $\mathcal G = Aut(M/A)$ .

The following was proved in [Reference Kamensky and Pillay1, Theorem 5.2], but some of the subtleties around exact sequences of cohomology sets were overlooked. We take the opportunity to give the proof again.

Proof The first point is to reduce to the case when G is connected. Let $G^0$ be the connected component of G. The exact sequence $1 \to G^{0} \to G \to G/G^{0} \to 1$ of algebraic groups over K induces an exact sequence

$$ \begin{align*}H^{1}(K, G_{0}) \to H^{1}(K,G) \to H^{1}(K, G/G^{0}).\end{align*} $$

The last term, $H^{1}(K, G/G^{0})$ , is finite by Proposition 8, Chapter III of [Reference Serre11]. Hence by Corollary 5.1, $H^{1}(K,G)$ is countable if and only if $H^{1}(K,\text {}_{\mu }G^{0})$ is countable for all $\mu \in H^{1}(K,G)$ . Since every K-form of $G^{0}$ is connected, the reduction is complete.

The Chevalley–Barsotti theorem (Theorem 16 of [Reference Rosenlicht10]) yields an exact sequence $1\to L\to G\to A \to 1$ of connected algebraic groups over K where L is linear and A is an abelian variety. As shown in the proof in Theorem 5.2 of [Reference Kamensky and Pillay1], $H^{1}(K, A)$ is countable (as it coincides with $H^{1}(K_{0},A),$ where $K_{0}$ is a countable elementary substructure of K). It is proved in [Reference Serre11] that $H^{1}(K, L)$ is finite for a connected linear algebraic group L. Since every K-form of L is linear, each set $H^{1}(K, \text {}_{\mu }L)$ is finite. By Corollary 5.1, $H^{1}(K,G)$ is countable.

We now pass to differential algebraic groups over differential fields. As in [Reference León Sánchez and Pillay3] we work with one derivation (so the ambient theory is DCF $_{0}$ ), although everything generalizes suitably to the case of a finite set $\Delta $ of commuting derivations (and the corresponding theory DCF $_{0,m}$ ). In [Reference León Sánchez and Pillay3, Theorem 4.1] we proved finiteness of the differential constrained cohomology $H^{1}_{\partial }(K,G)$ when G is a linear differential algebraic group over a bounded field K with $(K,\partial )$ differentially large.

Here we aim towards proving:

We will go through a few lemmas. For us, G will denote a differential algebraic group over a differential field $(K,\partial )$ ; equivalently, a group definable over K in a differentially closed field $(\mathcal U,\partial )$ containing K. There is no harm in taking $\mathcal U$ to be a differential closure $K^{diff}$ of K.

Before completing the proof of Theorem 5.3 let us recall the “Manin maps” and their properties (with precise references):

We finish with the proof of Theorem 5.3.

Proof of Theorem 5.3

We reduce to the connected case as in the beginning of Lemma 5.5. Let G be our connected definable (over K) group. By Corollary 4.2 of [Reference Pillay6], we may assume that G is a subgroup of $G_{1}({\mathcal U})$ for some connected algebraic group $G_{1}$ over K. The Chevalley–Barsotti theorem mentioned earlier gives an exact sequence $1\to L_1 \to G_{1} \to A_1 \to 1$ of connected algebraic groups over K where $L_{1}$ is linear and $A_{1}$ is an abelian variety. Let $L = G\cap L_{1}$ . So we obtain an exact sequence $1\to L\to G \to A \to 1$ of differential algebraic groups over K, where $L = G\cap L_{1}(\mathcal U)$ and $A\subseteq A_{1}({\mathcal U})$ . Theorem 4.1 of [Reference León Sánchez and Pillay3] says that $H^{1}_{\partial }(K,L')$ is finite for any K-form $L'$ of L (as $L'$ will also be linear). So, by Corollary 5.1, it suffices to prove that $H^{1}_{\partial }(K, A)$ is countable. Let $\mu : A_{1}({\mathcal U}) \to ({\mathcal U}^{d},+)$ be the Manin map given by Fact 5.6. Then, restricting $\mu $ to A gives an exact sequence of definable over K groups

$$ \begin{align*}1 \to ker(\mu)\cap A \to A \to N \to 1,\end{align*} $$

where $ker(\mu )\cap A$ has finite Morley rank and N is linear.

By Lemma 5.5 (and 5.4), $H^{1}_{\partial }(K, ker(\mu )\cap A))$ is countable, and the same holds also for any K-form of $ker(\mu )\cap A$ (as it also has finite Morley rank). By Theorem 4.1 of [Reference León Sánchez and Pillay3], $H^{1}_{\partial }(K,N)$ is finite. Hence, by Corollary 5.1, $H^{1}_{\partial }(K,A)$ is countable.