2 Basic definitions and constructions

In this section, we recall the basic definitions from difference algebra and some results about difference algebraic groups. The reader with a solid grounding in difference algebra may prefer to skip this section.

We start by recalling some definitions from difference algebra. Standard references for difference algebra are [Reference CohnCoh65, Reference LevinLev08]. Our main references for difference algebraic groups are [Reference WibmerWib22, Reference WibmerWib21]. All rings are assumed to be commutative and unital. A difference ring (or $\sigma $ -ring for short) R is a ring together with a ring endomorphism $\sigma \colon R\to R$ . A morphism of $\sigma $ -rings is a morphism of rings that commutes with $\sigma $ .

Let k be a $\sigma $ -ring. A k- $ \sigma $ -algebra is a $\sigma $ -ring R together with a k-algebra structure such that $k\to R$ is a morphism of $\sigma $ -rings. A morphism of k- $ \sigma $ -algebras $R\to S$ is a morphism of $\sigma $ -rings that also is a morphism of k-algebras. We will write $\text {Hom}_{k,\sigma }(R,S)$ for the set of morphisms of k- $ \sigma $ -algebras.

If R and S are k- $ \sigma $ -algebras, then $R\otimes _k S$ is a k- $ \sigma $ -algebra via $\sigma (r\otimes s)=\sigma (r)\otimes \sigma (s)$ . A k- $ \sigma $ -algebra R is finitely $\sigma $ -generated if there exists a finite subset $B\subseteq R$ such that $R=k[B,\sigma (B),\ldots ]$ . We write $R=k\{B\}$ for a k- $ \sigma $ -algebra R if $B\subseteq R$ is such that $R=k[B,\sigma (B),\ldots ]$ .

An ideal $\mathfrak {a}$ of a $\sigma $ -ring R is a $\sigma $ -ideal if $\sigma (\mathfrak {a})\subseteq \mathfrak {a}$ . If $\sigma ^{-1}(\mathfrak {a})=\mathfrak {a}$ , then $\mathfrak {a}$ is a reflexive $\sigma $ -ideal. If $f\sigma (f)\in \mathfrak {a}$ implies that $f\in \mathfrak {a}$ , then $\mathfrak {a}$ is a perfect $\sigma $ -ideal. For a subset B of R, the $\sigma $ -ideal $[B]=(B,\sigma (B),\sigma ^2(B),\ldots )\subseteq R$ is the $\sigma $ -ideal $\sigma $ -generated by B.

Let k be a $\sigma $ -field. A $\sigma $ -polynomial over k in the $\sigma $ -variables $y_1,\ldots ,y_n$ is a polynomial in the variables $y_1,\ldots ,y_n,\sigma (y_1),\ldots ,\sigma (y_n),\sigma ^2(y_1),\ldots $ over k. The k- $ \sigma $ -algebra of all $\sigma $ -polynomials is denoted by $k\{y\}=k\{y_1,\ldots ,y_n\}$ .

Informally speaking, a $\sigma $ -variety X over k (or a k- $ \sigma $ -variety) is the set of solutions of a set $F\subseteq k\{y\}$ of $\sigma $ -polynomials. To make this precise, we can define X to be the functor from the category of k- $ \sigma $ -algebras to the category of sets that assigns to every k- $ \sigma $ -algebra R the R-valued solutions of F. A morphism of k- $ \sigma $ -varieties is then simply a natural transformation of functors. Alternatively, we can fix a sufficiently largeFootnote ¹ k- $ \sigma $ -algebra $\mathcal {U}$ and define a k- $ \sigma $ -variety to be the set of $\mathcal {U}$ -valued solutions of F. Then a morphism of k- $ \sigma $ -varieties is simply a map that is given by $\sigma $ -polynomials in the coordinates. A $\sigma $ -algebraic group Footnote ² (over k) is a group object in the category of $\sigma $ -varieties (over k). A morphism of $\sigma $ -algebraic groups (over k) is a morphism of $\sigma $ -varieties (over k) that respects the group structure.

In practice, it does not matter if we work with functors or inside $\mathcal {U}$ . In fact, we will suppress the R or the $\mathcal {U}$ from the notation most of the time. For example, we may write $\phi \colon \mathbb {G}_m\to \mathbb {G}_m,\ g\mapsto \sigma (g)g$ to specify a morphism $\phi $ of $\sigma $ -algebraic group from the multiplicative group to the multiplicative group. This can either be read as the transformation of functors such that $\phi _R\colon \mathbb {G}_m(R)\to \mathbb {G}_m(R),\ g\mapsto \sigma (g)g$ for every k- $ \sigma $ -algebra R or as the map $\mathbb {G}_m(\mathcal {U})\to \mathbb {G}_m(\mathcal {U}),\ g\mapsto \sigma (g)g$ .

For a k- $\sigma $ -variety $X,$

$$ \begin{align*}\mathbb{I}(X)=\{f\in k\{y\}|\ f(x)=0 \ \forall \ x\in X\}\end{align*} $$

is a $\sigma $ -ideal of $k\{y\}$ . The k- $ \sigma $ -algebra $k\{X\}=k\{y\}/\mathbb {I}(X)$ is called the $\sigma $ -coordinate ring of X. The functor $X\rightsquigarrow k\{X\}$ induces an equivalence of categories between the category of k- $ \sigma $ -varieties and the category of finitely $\sigma $ -generated k- $ \sigma $ -algebras. Moreover, $X\simeq \text {Hom}_{k,\sigma }(k\{X\},-)$ as functors (see [Reference WibmerWib22, Section 1.2]). Allowing ourselves the freedom of calling any functor isomorphic to a k- $ \sigma $ -variety (as defined above) a k- $ \sigma $ -variety, it follows that a functor from the category of k- $ \sigma $ -algebras to the category of sets is a k- $ \sigma $ -variety if and only if it is representable by a finitely $\sigma $ -generated k- $ \sigma $ -algebra. The category of $\sigma $ -algebraic groups over k is equivalent to the category of finitely $\sigma $ -generated k- $ \sigma $ -Hopf algebras, that is, Hopf algebras over k, that are finitely $\sigma $ -generated k- $ \sigma $ -algebras such that the Hopf algebra structure maps are morphisms of k- $ \sigma $ -algebras.

A $\sigma $ -ring R is called $\sigma $ -reduced if $\sigma \colon R\to R$ is injective. The quotient $R_{\sigma \text {-}\text {red}}=R/(0)^*$ of R by the reflexive closure $(0)^*=\{f\in R\ |\ \exists \ n\geq 1:\ \sigma ^n(f)=0\}$ of the zero ideal is $\sigma $ -reduced. A k- $ \sigma $ -variety X is $\sigma $ -reduced if $k\{X\}$ is $\sigma $ -reduced. If the zero ideal of $k\{X\}$ is perfect, then X is called perfectly $\sigma $ -reduced. If $k\{X\}$ is $\sigma $ -reduced and an integral domain, then X is called $\sigma $ -integral.

A subfunctor Y of a k- $ \sigma $ -variety X is a $\sigma $ -closed $\sigma $ -subvariety if it is defined by a $\sigma $ -ideal $\mathbb {I}(Y)\subseteq k\{X\}$ , that is,

commutes for every k- $ \sigma $ -algebra R. A morphism $\phi \colon X\to Y$ of k- $\sigma $ -varieties is a $\sigma $ -closed embedding if it induces an isomorphism between X and a $\sigma $ -closed $\sigma $ -subvariety of Y; equivalently, the dual map $\phi ^*\colon k\{Y\}\to k\{X\}$ is surjective [Reference WibmerWib22, Lemma 1.6].

If $\phi \colon X\to Y$ is a morphism of k- $ \sigma $ -varieties, we denote with $\phi (X)$ the smallest $\sigma $ -closed $\sigma $ -subvariety of Y such that $\phi $ factors through the inclusion $\phi (X)\hookrightarrow Y$ (cf. [Reference WibmerWib22, Lemma 1.5]).

If K is a $\sigma $ -field extension of k and X a $\sigma $ -variety over k, we denote the base change of X from k to K by $X_K$ . So $X_K(R)=X(R)$ for every K- $ \sigma $ -algebra R. Then $X_K$ is a $\sigma $ -variety over K with $\sigma $ -coordinate ring $K\{X_K\}=k\{X\}\otimes _k K$ .

If G is a $\sigma $ -algebraic group and H a $\sigma $ -closed $\sigma $ -subvariety such that $H(R)$ is a subgroup of $G(R)$ for every k- $ \sigma $ -algebra R, then H is a called a $\sigma $ -closed subgroup of G and we will write $H\leq G$ . If moreover $H(R)$ is a normal subgroup of $G(R)$ for every k- $ \sigma $ -algebra R, then H is normal in G and we indicate this by $H\unlhd G$ .

The forgetful functor from k- $ \sigma $ -algebras to k-algebras has a left adjoint $S\rightsquigarrow [\sigma ]_k S$ which we need to describe explicitly. For a k-algebra S and $i\geq 1,$ we set ${{\,}{\,}^{\sigma ^i}\! S}=S\otimes _k k,$ where the tensor product is formed by using $\sigma ^i\colon k\to k$ . More generally, if X is an object over k, we denote by ${{\,}{\,}^{\sigma ^i}\! X}$ the base change of X via $\sigma ^i\colon k\to k$ . We set $S[i]=S\otimes _k{{{\,}^\sigma \!} S}\otimes _{k}\dots \otimes _k {{\,}{\,}^{\sigma ^i}\! S}$ and let $[\sigma ]_k S$ denote the direct limit (i.e., the union) of the $S[i]$ ’s. We define the structure of a k- $ \sigma $ -algebra on $[\sigma ]_k S$ by setting

$$ \begin{align*}\sigma(f_0\otimes(f_1\otimes\lambda_1)\otimes\dots\otimes(f_i\otimes\lambda_i))&=1\otimes (f_0\otimes 1)\otimes(f_1\otimes\sigma(\lambda_1))\otimes\dots\\&\quad\otimes (f_i\otimes\sigma(\lambda_i))\in S[i+1]\end{align*} $$

for $f_0,\ldots ,f_i\in S$ and $\lambda _1,\ldots ,\lambda _i\in k$ .

Example 2.1 Consider the univariate polynomial ring $S=k[y]$ . Then

$$ \begin{align*}S[i]=k[y, 1\otimes(y\otimes 1), \dots, 1\otimes \dots \otimes (y\otimes 1)]\end{align*} $$

and $\sigma \colon S[i]\to S[i+1]$ shifts each generator to the next one. Hence, $[\sigma ]_kS=k\{y\}$ , the difference polynomial ring in one variable.

For the sake of brevity and to be consistent with our nomenclature for difference algebraic geometry, we use the term variety over k to mean affine scheme of finite type over k. If $\mathcal {X}$ is a variety over k, then the functor $R\rightsquigarrow [\sigma ]_k\mathcal {X}(R)=\mathcal {X}(R)$ is a $\sigma $ -variety over k. Indeed, $k\{[\sigma ]_k\mathcal {X}\}=[\sigma ]_k k[\mathcal {X}],$ where $k[\mathcal {X}]$ is the coordinate ring of $\mathcal {X}$ (cf. [Reference WibmerWib22, Section 1.3]).

By an algebraic group over $k,$ we mean an affine group scheme of finite type over k. By a closed subgroup of an algebraic group, we mean a closed subgroup scheme. Let $\mathcal {G}$ be an algebraic group over k. Then $[\sigma ]_k\mathcal {G}$ is a $\sigma $ -algebraic group over k. If confusion is unlikely, we will simply write $\mathcal {G}$ for $[\sigma ]_k\mathcal {G}$ . In particular, by a $\sigma $ -closed subgroup of $\mathcal {G}$ , we mean a $\sigma $ -closed subgroup of $[\sigma ]_k \mathcal {G}$ .

Let G be a $\sigma $ -closed subgroup of an algebraic group $\mathcal {G}$ . Then G is defined by a $\sigma $ -ideal $\mathbb {I}(G)\subseteq [\sigma ]_k k[\mathcal {G}]$ . For $i\geq 0,$ the intersection $\mathbb {I}(G)\cap k[\mathcal {G}][i]\subseteq k[\mathcal {G}][i]=k[\mathcal {G}\times \cdots \times {{\,}{\,}^{\sigma ^i}\!\mathcal {G}}]$ defines a closed subgroup $G[i]$ of $\mathcal {G}\times \cdots \times {{\,}{\,}^{\sigma ^i}\!\mathcal {G}}$ . We call $G[i]$ the i-th-order Zariski closure of G in $\mathcal {G}$ . For $i=0,$ we simply speak of the Zariski closure. If $G[0]=\mathcal {G}$ , we say that G is Zariski-dense in $\mathcal {G}$ .

In [Reference WibmerWib22, Theorem 3.7], it is shown that there exist integers $d,e\geq 0$ such that $\dim (G[i])=d(i+1)+e$ for $i\gg 0$ . Moreover, every $\sigma $ -algebraic group G is isomorphic to a $\sigma $ -closed subgroup of an algebraic group $\mathcal {G}$ [Reference WibmerWib22, Proposition 2.16], for example, $\mathcal {G}$ can be chosen as $\text {GL}_m$ for some m, and the integer d does not depend on the choice of $\mathcal {G}$ or the choice of such an embedding. We call d the $\sigma $ -dimension of G and denote it by $\sigma \text {-}\dim (G)$ . Furthermore, if $\sigma \text {-}\dim (G)=0$ , then the integer $\text {ord}(G)=e\geq 0$ only depends on G and is called the order of G. If $\sigma \text {-}\dim (G)>0$ , we set $\text {ord}(G)=\infty $ .

Let G be a $\sigma $ -algebraic group and let $N\unlhd G$ be a normal $\sigma $ -closed subgroup. The quotient $G/N$ can be defined by a universal property [Reference WibmerWib21, Definition 3.1]. It exists (ibid., Theorem 3.3) and is well-behaved with respect to the $\sigma $ -dimension and the order (ibid. Corollary 3.13): Namely, $\sigma \text {-}\dim (G)=\sigma \text {-}\dim (N)+\sigma \text {-}\dim (G/N)$ and $\text {ord}(G)=\text {ord}(N)+\text {ord}(G/N)$ . Furthermore, the isomorphism theorems hold for $\sigma $ -algebraic groups (ibid., Section 5).

A morphism $\phi \colon G\to H$ of $\sigma $ -algebraic groups is called a quotient map if $\phi (G)=H$ . (For other equivalent characterizations of quotient maps, see Proposition 4.10 in [Reference WibmerWib21].) In particular, $\phi $ is a quotient map if and only if $H\simeq G/N$ for a normal $\sigma $ -closed subgroup N of G. Any morphism of $\sigma $ -algebraic groups factors uniquely as a quotient map followed by a $\sigma $ -closed embedding [Reference WibmerWib21, Theorem 4.14].

The kernel $\ker (\phi )$ of a morphism $\phi \colon G\to H$ of $\sigma $ -algebraic groups is given by $\ker (\phi )(R)=\ker (\phi _R)$ for every k- $ \sigma $ -algebra R. It is a normal $\sigma $ -closed subgroup of G. Indeed, $\mathbb {I}(\ker (\phi ))\subseteq k\{G\}$ is the ideal of $k\{G\}$ generated by $\phi ^*(\mathfrak {m}_H)$ , where $\mathfrak {m}_H$ is the kernel of the counit $k\{H\}\to k$ . A sequence $1 \to N \xrightarrow {\alpha } G\xrightarrow {\beta } H\to 1$ of $\sigma $ -algebraic groups is called exact if $\beta $ is a quotient map and $\alpha $ is a $\sigma $ -closed embedding that identifies N with the kernel of $\beta $ .

A $\sigma $ -algebraic group G is called $\sigma $ -étale if every element of $k\{G\}$ satisfies a separable polynomial over k. Recall that a finite-dimensional k-algebra S is étale if every element of S satisfies a separable polynomial over k. For a k-algebra R, let $\pi _0(R)$ denote the union of all étale k-subalgebras of R.

Let G be a $\sigma $ -algebraic group. Among all morphisms from G to a $\sigma $ -étale $\sigma $ -algebraic group, there exists a universal one $G\to \pi _0(G)$ (see [Reference WibmerWib22, Proposition 6.13]). Indeed $k\{\pi _0(G)\}=\pi _0(k\{G\})$ . The kernel $G^o$ of $G\to \pi _0(G)$ is called the identity component of G. If $G=G^o$ , then G is called connected. See [Reference WibmerWib21, Lemma 6.7] for other characterizations of connectedness.

Let G be a $\sigma $ -algebraic group over k. A G-torsor is a k- $ \sigma $ -variety X together with a morphism $G\times X\to X$ of k- $ \sigma $ -varieties such that $G(R)\times X(R)\to X(R)$ is a simply transitive group action of the group $G(R)$ on the set $X(R)$ for every k- $ \sigma $ -algebra R. A morphism of G-torsors is a G-equivariant morphism of k- $ \sigma $ -varieties. A G-torsor isomorphic to G with left translation is called trivial. A G-torsor X is trivial if and only if $X(k)\neq \emptyset $ [Reference Bachmayr and WibmerBW22, Remark 1.3].

Throughout this article k denotes a $\sigma $ -field. All varieties, $\sigma $ -varieties, algebraic groups, and $\sigma $ -algebraic groups are assumed to be over k unless the contrary is indicated.

3 The induction principle

Let $1\to N\to G\to H\to 1$ be an exact sequence of $\sigma $ -algebraic groups over k. The principle alluded to in the title of this section is the statement that $\lim _{n\to \infty }H^1_{\sigma ^n}(k,G_n)=0$ if $\lim _{n\to \infty }H^1_{\sigma ^n}(k,N_n)=0$ and $\lim _{n\to \infty }H^1_{\sigma ^n}(k,H_n)=0$ . The induction principle is crucial for the proof of (P3) $\Rightarrow $ (P1) because, after decomposing a general $\sigma $ -algebraic group into more manageable pieces, it allows us to restrict our considerations to those more manageable pieces. In this section, we also show that (P1) and (P2) are equivalent and similarly, their uniform variants (P1 $ ^u$ ) and (P2 $ ^u$ ) are equivalent.

Throughout Section 3, we denote with k a $\sigma $ -field (of arbitrary characteristic).

3.1 From $\sigma $ to $\sigma ^n$

In this section, we introduce the $\sigma ^n$ -varieties $X_n$ for every $\sigma $ -variety X. Throughout this section, we fix an integer $n\geq 1$ . We use the prefix k- $ \sigma ^n$ or $\sigma ^n$ instead of k- $ \sigma $ or $\sigma $ to indicate that we are working over the difference field $(k, \sigma ^n)$ instead of $(k,\sigma )$ .

We need to be able to pass from k- $ \sigma $ -varieties to k- $ \sigma ^n$ -varieties and vice versa. The most conceptual way to do this is to consider the left adjoint and the right adjoint of the functor $(R,\sigma )\rightsquigarrow (R,\sigma ^n)$ from k- $ \sigma $ -algebras to k- $ \sigma ^n$ -algebras which we will now construct. Let $(S,\tau )$ be a k- $ \sigma ^n$ -algebra. We define

$$ \begin{align*}{_1S}=S\otimes_k{{{\,}^\sigma\!} S}\otimes_k\dots\otimes_k{{\,}{\,}^{\sigma^{n-1}\!}S}\end{align*} $$

and $\sigma \colon {_1S}\to {_1S}$ by

$$ \begin{align*} \sigma(a_0\otimes(a_1\otimes\lambda_1)\otimes\dots\otimes(a_{n-1}\otimes\lambda_{n-1}))=& \\ \tau(a_{n-1})\sigma(\lambda_{n-1})\otimes(a_0\otimes 1)\otimes & (a_1\otimes\sigma(\lambda_1))\otimes\dots\otimes (a_{n-2}\otimes\sigma(\lambda_{n-2})), \end{align*} $$

where $a_0,\ldots ,a_{n-1}\in S$ and $\lambda _1,\ldots ,\lambda _{n-1}\in k$ . Then ${_1S}$ is a k- $ \sigma $ -algebra and the inclusion into the first factor $S\to {_1S}$ is a morphism of k- $ \sigma ^n$ -algebras. If R is a k- $ \sigma $ -algebra and ${_1S}\to R$ is a morphism of k- $ \sigma $ -algebras, then $S\to {_1S} \to R$ is a morphism of k- $ \sigma ^n$ -algebras. Conversely, if $\psi \colon S\to R$ is a morphism of k- $ \sigma ^n$ -algebras, then

$$ \begin{align*}{_1S}&\to R,\ a_0\otimes(a_1\otimes\lambda_1)\otimes\dots\otimes(a_{n-1}\otimes\lambda_{n-1})\\ &\mapsto \psi(a_0)\sigma(\psi(a_1))\lambda_1\dots\sigma^{n-1}(\psi(a_{n-1}))\lambda_{n-1}\end{align*} $$

is a morphism of k- $ \sigma $ -algebras. It is straightforward to check that these two constructions are inverse to each other. So

$$ \begin{align*}\text{Hom}_{k,\sigma^n}(S,R)\simeq\text{Hom}_{k,\sigma}({_1S},R)\end{align*} $$

for every k- $ \sigma $ -algebra R. Thus, the functor $S\rightsquigarrow {_1 S}$ (with the obvious definition on morphisms) is the left adjoint of $(R,\sigma )\rightsquigarrow (R,\sigma ^n)$ . (This is why the $1$ is on the left side.)

Example 3.1 Consider the difference polynomial ring

$$ \begin{align*}S=k\{y\}_{\sigma^n}=k[y_1,\ldots,y_m,\sigma^n(y_1),\ldots,\sigma^n(y_m),\ldots]\end{align*} $$

over the difference field $(k,\sigma ^n)$ in the $\sigma ^n$ -variables $y_1,\ldots ,y_m$ . Then $_1(k\{y\}_{\sigma ^n})$ can be identified with $k\{y\}_\sigma $ , the difference polynomial ring over $(k,\sigma )$ in the $\sigma $ -variables $y_1,\ldots ,y_m$ . The identification is such that for $1\leq i\leq n-1$ and $1\leq j\leq m$ , the variable $\sigma ^i(y_j)\in k\{y\}_\sigma $ corresponds to the image of $y_j\otimes 1\in {{\,}{\,}^{\sigma ^i}\!}S$ under the inclusion ${{\,}{\,}^{\sigma ^i}\!}S\to {_1S}$ .

More generally, if $S=k\{y\}_{\sigma ^n}/[F]_{\sigma ^n}$ is a quotient of $k\{y\}_{\sigma ^n}$ by the $\sigma ^n$ -ideal $[F]_{\sigma ^n}$ of $k\{y\}_{\sigma ^n}$ that is $\sigma ^n$ -generated by some $F\subseteq k\{y\}_{\sigma ^n}$ , then $_1S$ can be identified with $k\{y\}_\sigma /[F]_\sigma $ , the quotient of $k\{y\}_\sigma $ by the $\sigma $ -ideal generated by $F\subseteq k\{y\}_{\sigma ^n}\subseteq k\{y\}_\sigma $ .

Next, we will construct the right adjoint of $(R,\sigma )\rightsquigarrow (R,\sigma ^n)$ . If S is any k-algebra, then we write ${_{\sigma ^{i}}S}$ for the k-algebra that equals S as a ring and where we multiply with elements in k via $\sigma ^i \colon k \to k$ . In other words, ${_{\sigma ^{i}}S}$ is obtained from S by restriction of scalars via $\sigma ^i \colon k \to k$ .

Let $(S,\tau )$ be a k- $ \sigma ^n$ -algebra. We define

$$ \begin{align*}S_1={_{\sigma^{n-1}}S}\times{_{\sigma^{n-2}}S}\times\cdots\times{_\sigma S}\times S\end{align*} $$

as a k-algebra and we consider $S_1$ as a k- $ \sigma $ -algebra via

$$ \begin{align*}\sigma(a_1,a_2,\ldots,a_n)=(\tau(a_n),a_1,\ldots,a_{n-1}).\end{align*} $$

The projection $S_1\to S$ onto the last factor is a morphism of k- $ \sigma ^n$ -algebras. So if ${R\to S_1}$ is a morphism of k- $ \sigma $ -algebras, then $R\to S_1\to S$ is a morphism of k- $ \sigma ^n$ -algebras. Conversely, if $\psi \colon R\to S$ is a morphism of k- $ \sigma ^n$ -algebras, then

$$ \begin{align*}R\to S_1,\ a\mapsto (\psi(\sigma^{n-1}(a)),\ldots,\psi(\sigma(a)),\psi(a))\end{align*} $$

is a morphism of k- $ \sigma $ -algebras. One can check that these two constructions are inverse to each other. So

$$ \begin{align*}\text{Hom}_{k,\sigma^n}(R,S)\simeq\text{Hom}_{k,\sigma}(R,S_1)\end{align*} $$

for every k- $ \sigma $ -algebra R. In other words, $S\rightsquigarrow S_1$ is the right adjoint of $(R,\sigma )\rightsquigarrow (R,\sigma ^n)$ . (This is why the one is on the right.) Note that with $k^{[n]}$ defined as on page 4, we have $k^{[n]}=(k,\sigma ^n)_1$ . More generally, for a k- $ \sigma $ -algebra $R,$ we define the k- $ \sigma $ -algebra $R^{[n]}$ by

$$ \begin{align*}R^{[n]}=(R,\sigma^n)_1={_{\sigma^{n-1}}R}\times \cdots\times R.\end{align*} $$

One can verify directly that, for $l\geq 1,$ the map

$$ \begin{align*}R^{[n]}\to R^{[nl]},\ (a_1,\ldots,a_n)\mapsto & \\ (\sigma^{n(l-1)}(a_1),\ldots,&\sigma^{n(l-1)}(a_n),\ldots ,\sigma^n(a_1),\ldots,\sigma^n(a_n),a_1,\ldots,a_n)\end{align*} $$

is a morphism of k- $ \sigma $ -algebras.

If X is a k- $ \sigma $ -variety, then we can define a functor $X_n$ from k- $ \sigma ^n$ -algebras to sets by

$$ \begin{align*}X_n(S)=X(S_1)\end{align*} $$

for any k- $ \sigma ^n$ -algebra S. Conversely, if Z is a k- $ \sigma ^n$ -vari $Z_1$ from k- $ \sigma $ -algebras to sets by

$$ \begin{align*}Z_1(R)=Z(R,\sigma^n)\end{align*} $$

for any k- $ \sigma $ -algebra R.

Lemma 3.2

1. (i) If X is a k- $ \sigma $ -variety, then $X_n$ is a k- $ \sigma ^n$ -variety. Indeed, the $\sigma ^n$ -coordinate ring $k\{X_n\}$ of $X_n$ is $(k\{X\},\sigma ^n)$ .

2. (ii) If Z is a k- $ \sigma ^n$ -variety, then $Z_1$ is a k- $ \sigma $ -variety. Indeed, $k\{Z_1\}={_1 (k\{Z\})}$ .

Proof (i) For a k- $ \sigma ^n$ -algebra $S,$ we have $X_n(S)=\text {Hom}_{k,\sigma }(k\{X\},S_1)=\text {Hom}_{k,\sigma ^n}(k\{X\},S)$ . As $(k\{X\},\sigma )$ is finitely $\sigma $ -generated, $(k\{X\},\sigma ^n)$ is finitely $\sigma ^n$ -generated. So $X_n$ is indeed a k- $ \sigma ^n$ -variety.

(ii) For a k- $ \sigma $ -algebra $R,$ we have $Z_1(R)=\text {Hom}_{k,\sigma ^n}(k\{Z\},R)=\text {Hom}_{k,\sigma }({_1 (k\{Z\})},R)$ . As $k\{Z\}$ is finitely $\sigma ^n$ -generated, $_1(k\{Z\})$ is finitely $\sigma $ -generated. So $Z_1$ is indeed a k- $ \sigma $ -variety.

So $X\rightsquigarrow X_n$ is a functor from the category of k- $ \sigma $ -varieties to the category of k- $ \sigma ^n$ -varieties and $Z\rightsquigarrow Z_1$ is a functor from the category of k- $ \sigma ^n$ -varieties to the category of k- $ \sigma $ -varieties. For a k- $ \sigma $ -variety X and a k- $ \sigma ^n$ -variety $Z,$ we have

$$ \begin{align*}\text{Hom}(X,Z_1)=\text{Hom}_{k,\sigma}({_1(k\{Z\})}, k\{X\})=\text{Hom}_{k,\sigma^n}(k\{Z\},k\{X\})=\text{Hom}(X_n,Z).\end{align*} $$

So the functors $X\rightsquigarrow X_n$ and $Z\rightsquigarrow Z_1$ are adjoint. Note that we can identify $(X_n)_m$ with $X_{nm}$ , a fact that we will use frequently.

Example 3.3 Note that the $\sigma $ -polynomial ring $k\{y\}_\sigma $ over k in the $m\, \sigma $ -variables $y_1,\ldots ,y_m$ is, as a k- $ \sigma ^n$ -algebra, the $\sigma ^n$ -polynomial ring over $(k,\sigma ^n)$ in the $nm\, \sigma ^n$ -variables

$$ \begin{align*}(z_1,\ldots,z_{nm})=(y_1,\ldots,y_m,\sigma(y_1),\ldots,\sigma(y_m),\ldots,\sigma^{n-1}(y_1),\ldots,\sigma^{n-1}(y_m)).\end{align*} $$

We may therefore write $k\{y\}_\sigma =k\{z\}_{\sigma ^n}$ . Furthermore, if $F\subseteq k\{y\}\, \sigma $ -generates the $\sigma $ -ideal I of $k\{y\}_\sigma $ , then $F,\sigma (F),\ldots ,\sigma ^{n-1}(F)\subseteq k\{y\}_\sigma =k\{z\}_{\sigma ^n}\, \sigma ^n$ -generates the $\sigma ^n$ -ideal I of $k\{z\}_{\sigma ^n}$ . Thus, if X is the k- $ \sigma $ -variety defined by $F\subseteq k\{y\}_\sigma $ , then $X_n$ is the k- $ \sigma ^n$ -variety defined by $F,\sigma (F),\ldots ,\sigma ^{n-1}(F)\subseteq k\{z\}_{\sigma ^n}$ .

For a concrete example, let us consider, as on page 4, the k- $ \sigma $ -variety $X=\{x\in \text {GL}_m|\ \sigma (x)=(x^T)^{-1}A\}$ , where $A\in \text {GL}_m(k)$ . As X is defined by $\sigma (x)-(x^T)^{-1}A=0$ , $X_2$ is defined by $\sigma (x)-(x^T)^{-1}A=0$ and $\sigma ^2(x)-(\sigma (x)^T)^{-1}\sigma (A)=0$ , where x and $y=\sigma (x)$ are to be considered as $\sigma ^2$ -variables. Thus,

$$ \begin{align*} X_2&=\{(x,y)\in\text{GL}_m\times\text{GL}_m|\ y=(x^T)^{-1}A,\ \sigma^2(x)=(y^T)^{-1}\sigma(A) \} \\ &\simeq\{x\in\text{GL}_m|\ \sigma^2(x)=x(A^T)^{-1}\sigma(A)\} \end{align*} $$

as claimed on page 4.

Example 3.4 If X comes from an affine variety $\mathcal {X}$ over k, then the $\sigma ^n$ -variety $X_n$ corresponds to $\mathcal {X}\times {{{\,}^\sigma \!} \mathcal {X}}\times \cdots \times {{\,}{\,}^{\sigma ^{n-1}}\!\mathcal {X}}$ . In other words,

$$ \begin{align*}([\sigma]_k\mathcal{X})_n\simeq[\sigma^n]_k(\mathcal{X}\times\cdots\times{{\,}{\,}^{\sigma^{n-1}}\!\mathcal{X}}).\end{align*} $$

Proof This follows from Example 3.3 but can also be seen in a more functorial fashion as follows: For every k- $ \sigma ^n$ -algebra $S,$ we have

$$ \begin{align*} ([\sigma]_k\mathcal{X})_n(S) & =\mathcal{X}({_{\sigma^{n-1}}}S\times\cdots\times S)\simeq\text{Hom}_k(k[\mathcal{X}],{_{\sigma^{n-1}}}S\times\cdots\times S) \\ & \simeq\text{Hom}_k(k[\mathcal{X}],{_{\sigma^{n-1}}}S)\times\cdots\times \text{Hom}_k(k[\mathcal{X}],S) \\ &\simeq\text{Hom}_k({{\,}{\,}^{\sigma^{n-1}}}(k[\mathcal{X}]),S)\times\cdots\times \text{Hom}_k(k[\mathcal{X}],S) \\ & = \text{Hom}_k({{\,}{\,}^{\sigma^{n-1}}}(k[\mathcal{X}])\otimes_k\dots\otimes_k k[\mathcal{X}],S)\simeq (\mathcal{X}\times\cdots\times{{\,}{\,}^{\sigma^{n-1}}\!\mathcal{X}})(S).\quad\\[-34pt] \end{align*} $$

Example 3.5 If Z is the k- $ \sigma ^n$ -variety defined by $F\subseteq k\{y\}_{\sigma ^n}$ . Then $Z_1$ is the k- $ \sigma $ -variety defined by $F\subseteq k\{y\}_{\sigma ^n}\subseteq k\{y\}_\sigma $ (cf. Example 3.1). Thus, concretely, to compute $Z_1$ from $Z,$ we simply have to interpret the defining $\sigma ^n$ -equations of Z as $\sigma $ -equations.

Remark 3.6 Let X be a k- $ \sigma $ -variety. We have $X(k^{[n]})\neq \emptyset $ if and only if $X_n(k)\neq \emptyset $ .

Proof In fact, $X_n(k)=X(k^{[n]})$ .

If G is a $\sigma $ -algebraic group and X a G-torsor, then $G_n$ is a $\sigma ^n$ -algebraic group and $X_n$ is a $G_n$ -torsor. Remark 3.6 then shows that $X(k^{[n]})\neq \emptyset $ if and only if $X_n$ is trivial.

The following three lemmas relate properties of the $\sigma $ -algebraic group G to properties of the $\sigma ^n$ -algebraic group $G_n$ .

Lemma 3.7 Let G be a $\sigma $ -algebraic group over k. If G is connected, then $G_n$ is connected.

Proof Recall (e.g., from [Reference WibmerWib21, Lemma 6.7]) that G is connected if and only if $\pi _0(k\{G\})=k$ . As $\pi _0(k\{G\})$ does not depend on $\sigma $ the claim follows.

Lemma 3.8 Let G be a $\sigma $ -algebraic group over k. Then $\sigma ^n\text {-}\dim (G_n)=\sigma \text {-}\dim (G)\cdot n$ and $\text {ord}(G_n)=\text {ord}(G)$ .

Proof We consider G as a Zariski-dense, $\sigma $ -closed subgroup of some algebraic group $\mathcal {G}$ via a $\sigma $ -closed embedding $G \hookrightarrow \mathcal {G}$ . Algebraically, this means that we have an inclusion of Hopf algebras $k[\mathcal {G}]\subseteq k\{G\}$ such that $k[\mathcal {G}]$ generates $k\{G\}$ as a k- $ \sigma $ -algebra. In particular, if B is a finite subset of $k[\mathcal {G}]$ such that $k[\mathcal {G}]=k[B]$ , then $k\{G\}=k\{B\}$ .

The $\sigma $ -closed embedding $G\hookrightarrow [\sigma ]_k\mathcal {G}$ yields a $\sigma ^n$ -closed embedding $G_n\hookrightarrow ([\sigma ]_k\mathcal {G})_n=[\sigma ^n]_k(\mathcal {G}\times \cdots \times {{\,}{\,}^{\sigma ^{n-1}}\!\mathcal {G}}$ (Example 3.4). So we can consider $G_n$ as a $\sigma ^n$ -closed subgroup of the algebraic group $\mathcal {G}\times {{{\,}^\sigma \!}\mathcal {G}}\times \cdots \times {{\,}{\,}^{\sigma ^{n-1}}\!\mathcal {G}}$ . Algebraically, this corresponds to the inclusion of Hopf algebras $k[B,\ldots ,\sigma ^{n-1}(B)]\subseteq k\{G\}$ . For $i\geq 0,$ let $d_i=\dim (k[B,\ldots ,\sigma ^{i}(B)])$ . Then, for $i\gg 0$ , we have $d_i=\sigma \text {-}\dim (G)(i+1)+e$ for some integer $e\geq 0$ .

Similarly, for $i\gg 0$ , we have $d_{(i+1)n-1}=\sigma ^n\text {-}\dim (G_n)(i+1)+e'$ for some integer $e'\geq 0$ . Therefore,

$$ \begin{align*}\sigma\text{-}\dim(G)((i+1)n)+e=\sigma^n\text{-}\dim(G_n)(i+1)+e'\end{align*} $$

and thus $\sigma ^n$ - $\dim (G_n)=\sigma \text {-}\dim (G)\cdot n$ . Moreover, $\sigma \text {-}\dim (G)=0$ if and only if $\sigma ^n$ - $\dim (G_n)=0$ and in this case $e=e'$ , that is, $\text {ord}(G)=\text {ord}(G_n)$ .

Lemma 3.9 If $1\to N\to G\to H\to 1$ is an exact sequence of $\sigma $ -algebaic groups, then $1\to N_n\to G_n\to H_n\to 1$ is an exact sequence of $\sigma ^n$ -algebraic groups for every $n\geq 1$ .

Proof The exactness of $1\to N\to G\to H\to 1$ can be reformulated in terms of the corresponding morphisms $k\{H\}\to k\{G\}\to k\{N\}$ of k- $ \sigma $ -Hopf algebras. Exactness at N means that $k\{G\}\to k\{N\}$ is surjective, exactness at H means that $k\{H\}\to k\{G\}$ is injective, and exactness at G means that the kernel of $k\{G\}\to k\{N\}$ agrees with the ideal of $k\{G\}$ generated by the image of $\mathfrak {m}_H$ in $k\{G\}$ , where $\mathfrak {m}_H$ is the kernel of the counit $k\{H\}\to k$ . These properties do not depend on the power of $\sigma $ considered. Thus, $1\to N_n\to G_n\to H_n\to 1$ is exact.

3.2 Exact sequences and $H^1$

In this section, we establish the induction principle. We also observe that (P1) $\Leftrightarrow $ (P2) and (P1 $ ^u$ ) $\Leftrightarrow $ (P2 $ ^u$ ).

Let G be a $\sigma $ -algebraic group over k. A cohomology set $H^1(k,G)$ that classifies G-torsors was introduced in [Reference Bachmayr and WibmerBW22]. Far more general and elaborate approaches to cohomology in difference algebraic geometry can be found in [Reference Chalupnik and KowalskiCK23, Reference TomašićTom]. For our purposes, the elementary approach from [Reference Bachmayr and WibmerBW22] is sufficient.

We begin by recalling the definition of the cohomology set $H^1(A/k,G)$ , which, for a k- $ \sigma $ -algebra A, classifies G-torsors X with $X(A)\neq \emptyset $ . Consider the canonical morphisms of k- $ \sigma $ -algebras

where the upper arrow on the left denotes the morphism $a\mapsto 1\otimes a$ , the lower arrow on the left denotes the morphism $a\mapsto a\otimes 1$ , and the upper arrow on the right denotes the morphism $a\otimes b\mapsto 1\otimes a\otimes b$ and so on. These morphisms induce morphisms of groups

An element $\chi \in G(A\otimes _k A)$ is called a cocycle if $\partial _2(\chi )=\partial _1(\chi )\partial _3(\chi )$ . Two cocycles $\chi _1$ and $\chi _2$ are called equivalent if there exists a $g\in G(A)$ such that $\chi _1=\delta _1(g)\chi _2\delta _2(g)^{-1}$ . The set of all equivalence classes of cocycles is denoted by $H^1_\sigma (A/k,G)$ . It is a pointed set with the usual functorial properties. The distinguished element is the equivalence class of $1\in G(A\otimes _k A)$ .

We set $H^1_\sigma (k,G)=\lim \limits _{\longrightarrow }H^1_\sigma (A/k,G)$ , where the limit is taken over the set of isomorphism classes of finitely $\sigma $ -generated k- $ \sigma $ -algebras. (See [Reference Bachmayr and WibmerBW22, Section 5] for more details). Let $\Sigma _\sigma (G)$ denote the pointed set of all isomorphism classes of G-torsors. The distinguished element in $\Sigma _\sigma (G)$ is the class of the trivial torsor. According to [Reference Bachmayr and WibmerBW22, Theorem 4.2], we have an isomorphism of pointed sets

$$ \begin{align*}\Sigma_\sigma(G)\simeq H^1_\sigma(k,G).\end{align*} $$

Fix $n\geq 1$ . If X is a G-torsor, then $X_n$ is a $G_n$ -torsor. As $X(k)\subseteq X(k^{[n]})=X_n(k)$ , we see that $X_n$ is trivial if X is trivial. Thus, we have a map of pointed sets

$$ \begin{align*}\Sigma_\sigma(G)\to \Sigma_{\sigma^n}(G_n).\end{align*} $$

Example 3.10 Let $G=\{g\in \text {GL}_m|\ \sigma (g)=(g^T)^{-1}\}$ . For every $A\in \text {GL}_m(k),$ the k- $ \sigma $ -variety $X=\{x\in \text {GL}_m|\ \sigma (x)=(x^T)^{-1}A\}$ is a G-torsor under left multiplication. Its image $X_2$ under $\Sigma _\sigma (G)\to \Sigma _{\sigma ^2}(G_2)$ is the $G_2$ -torsor $X_2=\{x\in \text {GL}_m|\ \sigma ^2(x)=x(A^T)^{-1}\sigma (A)\}$ , where $G_2=\{g\in \text {GL}_m|\ \sigma ^2(g)=g\}$ (see Example 3.3).

We also have a canonical map

$$ \begin{align*}H^1_\sigma(k,G)\to H^1_{\sigma^n}(k,G_n),\end{align*} $$

which we will describe next. Let X be a k- $ \sigma $ -variety and A a k- $ \sigma $ -algebra. The morphism $A\to A^{[n]},\ a \mapsto (\sigma ^{n-1}(a),\ldots ,a)$ of k- $ \sigma $ -algebras induces a map $X(A)\to X(A^{[n]})=X((A,\sigma ^n)_1)=X_n(A)$ . Under the identifications $X(A)=\text {Hom}_{k,\sigma }(k\{X\},A)$ and $X_n(A)=\text {Hom}_{k,\sigma ^n}(k\{X\},A)$ , this map is simply interpreting a morphism $k\{X\}\to A$ of k- $ \sigma $ -algebras as a morphism of k- $ \sigma ^n$ -algebras.

If $A\to B$ is a morphism of k- $ \sigma $ -algebras, then

commutes. Similarly, if $X\to Y$ is a morphism of k- $ \sigma $ -varieties, then

(3.1)

commutes. Moreover, if G is a $\sigma $ -algebraic group over k, then $G(A)\to G_n(A)$ is a morphism of groups. The commutative diagram

shows that $G(A\otimes _k A)\to G_n(A\otimes _k A)$ maps cocycles to cocycles and equivalent cocycles to equivalent cocycles. We therefore have a well-defined map $H^1_\sigma (A/k,G)\to H^1_{\sigma ^n}(A/k,G_n)$ , which we may compose with the inclusion $H^1_{\sigma ^n}(A/k,G_n)\to H^1_{\sigma ^n}(k,G_n)$ . If $A\to B$ is a morphism of k- $ \sigma $ -algebras, then

commutes. Thus, we have an induced map of pointed sets $H^1_\sigma (k,G)\to H^1_{\sigma ^n}(k,G_n)$ . The construction of this map also applies with $(k,\sigma )$ replaced by $(k,\sigma ^d)$ (for some $d\geq 1$ ) and so we have maps $H^1_{\sigma ^d}(k,G_d)\to H^1_{(\sigma ^d)^n}(k,(G_{d})_n)=H^1_{\sigma ^{dn}}(k,G_{dn})$ . If X is a k- $ \sigma ^n$ -variety, $(A,\tau )$ a k- $ \sigma ^n$ -algebra, and $d,d'\geq 1$ , then the composition $X(A,\tau )\to X_{d}(A,\tau ^{d})\to (X_d)_{d'}(A,(\tau ^d)^{d'})=X_{dd'}(A,\tau ^{dd'})$ agrees with $X(A,\tau )\to X_{dd'}(A,\tau ^{dd'})$ because all maps are just reinterpreting the same morphism from $X(A,\tau )=\text {Hom}_{k,\sigma ^n}(k\{X\},A)$ . From this, it follows that for integers $n,n',n"\geq 1$ with $n|n'$ and $n'|n"$ , the composition $H^1_{\sigma ^n}(k,G_n)\to H^1_{\sigma ^{n'}}(k,G_{n'})\to H^1_{\sigma ^{n"}}(k,G_{n"})$ agrees with $H^1_{\sigma ^n}(k,G_n)\to H^1_{\sigma ^{n"}}(k,G_{n"})$ . The maps $H^1_{\sigma ^n}(k,G_n)\to H^1_{\sigma ^{n'}}(k,G_{n'})$ for $n|n'$ thus form a directed system over the directed set of all integers greater or equal to one, partially ordered by $n\leq n'$ if and only if $n|n'$ . We can therefore consider the direct limit $\lim \limits _{n\to \infty } H^1_{\sigma ^n}(k,G_n)$ .

Example 3.11 Let $\mathcal {G}$ be an algebraic group over k and $G=[\sigma ]_k\mathcal {G}$ . By [Reference Bachmayr and WibmerBW22, Proposition 3.5], we have $H^1_\sigma (k,G)=H^1(k,\mathcal {G})$ . Furthermore, $G_n=[\sigma ^n]_k(\mathcal {G}\times \cdots \times {{\,}{\,}^{\sigma ^{n-1}}\!\mathcal {G}})$ (Example 3.4) and so $H^1_{\sigma ^n}(k,G_n)=H^1(k,\mathcal {G}\times \cdots \times {{\,}{\,}^{\sigma ^{n-1}}\!\mathcal {G}})=H^1(k,\mathcal {G})\times \cdots \times H^1(k,{{\,}{\,}^{\sigma ^{n-1}}\!\mathcal {G}})$ . To understand the maps $H^1_{\sigma ^n}(k,G_n)\to H^1_{\sigma ^{n'}}(k,G_{n'})$ for $n|n'$ , we first need an explicit description of the map $X_n(A,\tau )\to (X_n)_d(A,\tau ^d)$ for $d\geq 1$ , X a k- $ \sigma ^n$ -variety, and $(A,\tau )$ a k- $ \sigma ^n$ -algebra. Now $X_n(A,\tau )\to (X_n)_d(A,\tau ^d)=X_n((A,\tau )^{[d]})$ is obtained by applying the functor $X_n$ to the morphism $(A,\tau )\to (A,\tau )^{[d]}$ of k- $ \sigma ^n$ -algebras. As $X_n(A,\tau )=X((A,\tau )_1)$ , this means that we need to apply the functor X to the morphism $(A,\tau )_1\to ((A,\tau )^{[d]})_1$ of k- $ \sigma $ -algebras. As

$$ \begin{align*}(A,\tau)\to (A,\tau)^{[d]}={_{(\sigma^n)^{d-1}}}A\times\cdots\times{_{\sigma^n}}A\times A,\quad a\mapsto (\tau^{d-1}(a),\ldots,\tau(a),a),\end{align*} $$

we see that

$$ \begin{align*}(A,\tau)_1&={_{\sigma^{n-1}}}A\times\cdots\times A\longrightarrow ((A,\tau)^{[d]})_1\\ &={_{\sigma^n}}({_{(\sigma^{n(d-1)}}}A\times\cdots\times A)\times\cdots\times {_{\sigma^{n(d-1)}}}A\times\cdots\times A\end{align*} $$

is given by

$$ \begin{align*}(a_1,\ldots,a_n)\longmapsto(\tau^{d-1}(a_1),\ldots,a_1,\ldots,\tau^{d-1}(a_n)\ldots,a_n).\end{align*} $$

Thus, after reordering the factors as $_{\sigma ^{nd-1}}A\times \cdots \times {_{\sigma }A}\times A$ , the map is given by

$$ \begin{align*}(a_1,\ldots,a_n)\mapsto (\tau^{d-1}(a_1),\ldots,\tau^{d-1}(a_n),\ldots,a_1,\ldots,a_n).\end{align*} $$

Applying $X=[\sigma ]_k\mathcal {G}$ to this (as in Example 3.4), we find the map

$$ \begin{align*}\mathcal{G}(A)\times\cdots\times{{\,}{\,}^{\sigma^{n-1}}\!\mathcal{G}}(A)\longrightarrow \mathcal{G}(A)\times\cdots\times{{\,}{\,}^{\sigma^{nd-1}}\!\mathcal{G}}(A),\end{align*} $$

(3.2) $$ \begin{align}(g_1,\ldots,g_n)\longmapsto (g_1,\ldots,g_n,\ldots,\tau^{d-1}(g_1),\ldots,\tau^{d-1}(g_n)), \end{align} $$

where $\tau ^j\colon {{\,}{\,}^{\sigma ^i}\!}\mathcal {G}(A)\to {{\,}^{\sigma ^{i+n}}\!\mathcal {G}}(A)$ (for $0\leq i\leq n-1$ and $1\leq j\leq d-1$ ) is given by applying $\tau $ to the coordinates. Without coordinates, if $g\colon {{\,}^{\sigma ^i}\!}(k[\mathcal {G}])\to A$ , then $\tau ^j(g)\colon {\,}^{\sigma ^{n}}\!({{\,}^{\sigma ^{i}}\!(k[G])})\to A,\ f\otimes \lambda \mapsto \tau ^j(g(f))\lambda $ .

If $k\to k'$ is a morphism of fields, we have a map $H^1(k,\mathcal {G})\to H^1(k',\mathcal {G}_{k'})$ , which on isomorphism classes of torsors corresponds to base change. Applying this with $\sigma ^n\colon k\to k$ , we obtain maps $\sigma ^n\colon H^1(k,{{\,}^{\sigma ^i}\!\mathcal {G}})\to H^1(k,{{\,}^{\sigma ^{i+n}}\!\mathcal {G}})$ .

Base change via $\sigma ^n\colon k\to k$ takes $g\colon {{\,}^{\sigma ^i}\!(k[\mathcal {G}])}\to A$ and returns ${\,}^{\sigma ^n}\! g\colon {{\,}^{\sigma ^{i+n}}\!{(k[\mathcal {G}])}}\to {{\,}^{\sigma ^n}\! A}$ . The map $\tau ^j(g)\colon {{\,}^{\sigma ^{i+n}}\!{(k[\mathcal {G}])}}\to A$ is then obtained from ${\,}^{\sigma ^n}\! g$ by composing with the morphism ${{\,}^{\sigma ^n}\! A}\to A,\ a\otimes \lambda \mapsto \tau (a)\lambda $ of k-algebras. Since a morphism of k-algebras does not matter when we pass to cohomology sets, we see that $\tau ^j\colon {{\,}^{\sigma ^i}\!}\mathcal {G}(A)\to {{\,}^{\sigma ^{i+n}}\!\mathcal {G}}(A)$ induces $\sigma ^n\colon H^1(k,{{\,}^{\sigma ^i}\!\mathcal {G}})\to H^1(k,{{\,}^{\sigma ^{i+n}}\!\mathcal {G}})$ .

Based on (3.2), we thus find that for $n|n'$ , the map

$$ \begin{align*}H^1_{\sigma^n}(k,G_n)&=H^1(k,\mathcal{G})\times\cdots\times H^1(k,{{\,}^{\sigma^{n-1}}\!\mathcal{G}})\longrightarrow H^1_{\sigma^{n'}}(k,G_{n'})\\ &=H^1(k,\mathcal{G})\times\cdots\times H^1(k,{{\,}^{\sigma^{n'-1}}\!\mathcal{G}})\end{align*} $$

is given by

$$ \begin{align*}(a_1,\ldots,a_n)\longmapsto (a_1,\ldots,a_n,\sigma^n(a_1),\ldots,\sigma^n(a_n),\ldots,\sigma^{n'-n}(a_1),\ldots,\sigma^{n'-n}(a_n)).\end{align*} $$

Therefore, the limit $\lim \limits _{n\to \infty } H^1_{\sigma ^n}(k,G_n)$ can be identified with the set of all “periodic” elements of $\prod _{n\geq 1}H^1(k,{{\,}^{\sigma ^n}\!\mathcal {G}})$ , that is, all elements of the form $(a_1,\ldots ,a_n,\sigma ^n(a_1),\ldots ,\sigma ^n(a_n),\sigma ^{2n}(a_1),\ldots ,\sigma ^{2n}(a_n),\ldots )$ for some $n\geq 1$ .

We will need two more commutative diagrams.

Lemma 3.12 Let G be a $\sigma $ -algebraic group over k. Then

commutes.

Proof Let X be a G-torsor and consider its isomorphism class $[X]$ in $\Sigma _\sigma (G)$ . We first describe explicitly to which element h in $H^1_\sigma (k,G)$ our element $[X]$ maps via the vertical arrow: Fix a finitely $\sigma $ -generated k- $ \sigma $ -algebra A such that $X(A)\neq \emptyset $ . Choose $x\in X(A)$ and consider $f_1,f_2\colon X(A)\to X(A\otimes _k A)$ . Then $f_1(x)$ and $f_2(x)$ are both contained in $X(A\otimes A)$ , so there exists a unique $\chi \in G(A\otimes _k A)$ with $f_1(x)=\chi .f_2(x)$ . In fact, $\chi $ is a cocycle and its equivalence class inside $H^1_\sigma (A/k,G)\to H^1_\sigma (k,G)$ defines h (see the proof of [Reference Bachmayr and WibmerBW22, Theorem 4.1]).

The upper horizontal arrow maps $[X]$ to the isomorphism class $[X_n]$ of the $G_n$ -torsor $X_n$ . Let $\tilde x\in X_n(A)$ be the image of x under $X(A)\to X_n(A)$ and let $\widetilde {\chi }\in G_n(A\otimes _k A)$ be the unique element that satisfies $f_1(\widetilde {x})=\widetilde {\chi }.f_2(\widetilde {x})$ . Then the image $\tilde h \in H^1_{\sigma ^n}(k, G_n)$ of $[X_n]$ under the vertical arrow on the right is defined by the equivalence class of $\tilde \chi $ inside $H^1_{\sigma ^n}(A/k,G_n)\to H^1_{\sigma ^n}(k,G_n)$ .

The lower horizontal map is induced by $G(A\otimes A)\to G_n(A\otimes A)$ . Let $\chi _n \in G_n (A\otimes A)$ be the image of $\chi $ under $G(A\otimes A)\to G_n(A\otimes A)$ . Then $\chi _n$ satisfies $f_1(\widetilde {x})={\chi _n}.f_2(\widetilde {x})$ and we conclude $\chi _n=\widetilde {\chi }$ , so the diagram commutes.

Example 3.13 The map $H^1_\sigma (k,G)\to H^1_{\sigma ^n}(k,G_n)$ is in general neither injective nor surjective, that is, there may exist a $G_n$ -torsor that does not come from a G-torsor and there may be two non-isomorphic G-torsors that become isomorphic as $G_n$ -torsors. For a non-surjective example, let $\mathcal {G}$ be an algebraic group over k and ${G=[\sigma ]_k\mathcal {G}}$ . Then $H^1_\sigma (k,G)=H^1(k,\mathcal {G})$ and $H^1_{\sigma ^n}(k,G_n)=H^1(k,\mathcal {G})\times \cdots \times H^1(k,{{\,}^{\sigma ^{n-1}}\!\mathcal {G}})$ by Example 3.11, and $H^1_\sigma (k,G)\to H^1_{\sigma ^n}(k,G_n)$ is the map $a\mapsto (a,\sigma (a),\dots ,\sigma ^{n-1}(a))$ which is not surjective for suitable choices of $(k,\sigma )$ and $\mathcal {G}$ (e.g., take $\mathcal {G}=\text {SO}_2$ , $k=\mathbb {R,}$ and $\sigma =\text {id})$ .

For a non-injective example, consider the $\sigma $ -algebraic group $G=\{g\in \mathbb {G}_m|\ g^2=1,\ \sigma (g)=g\}$ and the G-torsor $X=\{x\in \mathbb {G}_m|\ x^2=1,\ \sigma (x)=-x\}$ (under left multiplication). Then X is nontrivial but $X_2$ is trivial because $X_2(k)=X(k^{[2]})\neq \emptyset $ .

Lemma 3.14 Let $G\to H$ be a morphism of $\sigma $ -algebraic groups over k. Then

commutes.

Proof This follows from the commutativity of (3.1).

Definition 3.15 Let G be a $\sigma $ -algebraic group over k. To simplify our statements, we call k

• • G-trivial if for every G-torsor X, there exists an integer $n\geq 1$ such that $X(k^{[n]})\neq \emptyset $ . If in addition the integer n can be chosen uniformly for all X, then we call k uniformly G-trivial.

• • (uniformly) strongly G-trivial if $(k,\sigma ^n)$ is (uniformly) $G_n$ -trivial for all $n\geq 1$ .

Thus, k is uniformly strongly G-trivial if for every $n\geq 1$ and every $\sigma $ -algebraic group G over k, there exists a $d\geq 1$ such that, for every $G_n$ -torsor, we have $X((k,\sigma ^n)^{[d]})=X((k,(\sigma ^n)^d)_1)=X_d(k,(\sigma ^n)^d)\neq \emptyset $ . The notion of being (uniformly) strongly G-trivial is important for making the induction principle work.

Property (P1) means that k is G-trivial for every $\sigma $ -algebraic group G over k, while (P1 $ ^u$ ) means that k is uniformly G-trivial for every $\sigma $ -algebraic group G over k. The following lemma is needed for (P1) $\Rightarrow $ (P2).

Lemma 3.16 If k is G-trivial for every $\sigma $ -algebraic group G over k, then k is strongly G-trivial for every $\sigma $ -algebraic group G over k.

Proof Let G be a $\sigma $ -algebraic group over k and let X be a $G_n$ -torsor for some $n\geq 1$ . Then $X_1$ is a $(G_n)_1$ -torsor, and by assumption, there exists a $d\geq 1$ such that $X_1(k^{[d]})\neq \emptyset $ . As $X_1(k^{[d]})=X(k^{[d]},\sigma ^n),$ there exists a morphism $k\{X\}\to k^{[d]}$ of k- $ \sigma ^n$ -algebras. The projection $k^{[d]}\to k$ to the last factor is a morphism of k- $ \sigma ^d$ -algebras. Thus, the composition $k\{X\}\to k^{[d]}\to k$ is a morphism of k- $ \sigma ^{nd}$ -algebras. Therefore, $X_d(k,(\sigma ^n)^d)=X((k,(\sigma ^n)^d)_1)=X((k,\sigma ^n)^{[d]})\neq \emptyset $ as desired.

The following lemma will allow us to pass to higher powers of $\sigma $ in the proof of (P3) $\Rightarrow $ (P1).

Lemma 3.17 Let G be a $\sigma $ -algebraic group over k. If $(k,\sigma ^n)$ is (strongly) $G_n$ -trivial, then k is (strongly) G-trivial.

Proof Assume that $(k,\sigma ^n)$ is $G_n$ -trivial and let X be a G-torsor. Then $X_n$ is a $G_n$ -torsor, so there exists an integer $d\geq 1$ with $(X_n)_d(k)\neq \emptyset $ (Remark 3.6). Define $m=nd$ . As $(X_n)_d=X_m$ , we have $X_m(k)\neq \emptyset $ and thus $X(k^{[m]})\neq \emptyset $ (Remark 3.6). We conclude that k is G-trivial.

If $(k,\sigma ^n)$ is strongly $G_n$ -trivial, then $(k,(\sigma ^n)^d)$ is $(G_n)_d$ -trivial for every integer $d\geq 1$ . In other words, $(k,(\sigma ^d)^n)$ is $(G_d)_n$ -trivial for every integer $d\geq 1$ . By the statement proven in the first paragraph, $(k,\sigma ^d)$ is $G_d$ -trivial for every $d\geq 1$ . Hence, k is strongly G-trivial.

Lemma 3.18 Let G be a $\sigma $ -algebraic group over k, X a $G_n$ -torsor, and $d\geq 1$ . Then $X((k,\sigma ^n)^{[d]})\neq \emptyset $ if and only if the image of X in $H^1_{\sigma ^n}(k,G_n)$ maps to the distinguished element under $H^1_{\sigma ^{n}}(k,G)\to H^1_{\sigma ^{nd}}(k,G_{nd})$ .

Proof We have $X((k,\sigma ^n)^{[d]})\neq \emptyset $ if and only if $X_d(k,(\sigma ^n)^d)\neq \emptyset $ by Remark 3.6. So the claim follows from Lemma 3.12.

At this point, the equivalence of (P1) and (P2) is self-evident.

Proposition 3.19 Properties (P1) and (P2) are equivalent. Similarly, properties (P1 $ ^u$ ) and (P2 $ ^u$ ) are equivalent.

Proof The equivalence (P1 $ ^u$ ) $\Leftrightarrow $ (P2 $ ^u$ ) follows from Lemma 3.18. According to that lemma, property (P2) means that k is strongly G-trivial for every $\sigma $ -algebraic group G over k. Thus, (P1) $\Leftrightarrow $ (P2) follows from Lemma 3.16.

We are now prepared to establish the induction principle. Essentially, this is an application of the exact cohomology sequence. Recall that a sequence $M'\to M\to M"$ of pointed sets is exact at M if the kernel of $M\to M"$ (i.e., the inverse image of the distinguished element of $M"$ ) agrees with the image of $M'\to M$ .

Proposition 3.20 (Induction principle)

Let $1\to N\to G\to H\to 1$ be an exact sequence of $\sigma $ -algebraic groups over k.

1. (i) If k is (uniformly) H-trivial and (uniformly) strongly N-trivial, then k is (uniformly) G-trivial.

2. (ii) If k is (uniformly) strongly N-trivial and (uniformly) strongly H-trivial, then k is (uniformly) strongly G-trivial.

Proof According to Lemma 3.14, we have a commutative diagram for all integers $n,m,l \geq 1$ :

We prove (i) and (ii) simultaneously and set $n=1$ in case (i) and $n\geq 1$ arbitrary in case (ii). Because $1\to N_{nm}\to G_{nm}\to H_{nm}\to 1$ is exact by Lemma 3.9, the exact cohomology sequence [Reference Bachmayr and WibmerBW22, Proposition 5.6] shows that the middle row in the above diagram is exact.

Start with an element in $\gamma \in H^1_{\sigma ^n}(k,G_n)$ . As k is (strongly) H-trivial, there exists an $m\geq 1$ such that its image in $H^1_{\sigma ^n}(k,H_n)$ lies in the kernel of the vertical map $H^1_{\sigma ^{n}}(k,H_{n})\to H^1_{\sigma ^{nm}}(k,H_{nm})$ . Moreover, m can be chosen uniformly for all elements in $H^1_{\sigma ^n}(k,G_n)$ if k is uniformly (strongly) H-trivial. By the exactness of the middle row, $\alpha (\gamma )$ is contained in the image of $H^1_{\sigma ^{nm}}(k,N_{nm})\to H^1_{\sigma ^{nm}}(k,G_{nm})$ . Since k is strongly N-trivial, there exists an $l\geq 1$ such that its preimage in $H^1_{\sigma ^{nm}}(k,N_{nm})$ is mapped to the trivial element in $H^1_{\sigma ^{nml}}(k,N_{nml})$ and we conclude that $\beta (\alpha (\gamma ))$ is trivial. Moreover, if N is uniformly strongly N-trivial, then l does not depend on the choice of $\gamma $ . All claims now follow with Lemma 3.18.

By induction, we obtain from Proposition 3.20(ii).

Corollary 3.21 If a $\sigma $ -algebraic group G has a subnormal series

$$ \begin{align*}G=G_0\supseteq G_1\supseteq\cdots\supseteq G_n=1\end{align*} $$

such that k is (uniformly) strongly $G_i/G_{i+1}$ -trivial for $i=0,\ldots ,n-1$ , then k is (uniformly) strongly G-trivial.

4 First observations regarding property (P3)

In this section, we discuss some equivalent reformulations of property (P3)(b) (and (P3 $ ^u$ )(b)). We also show that (P1) $\Rightarrow $ (P3) and similarly (P1 $ ^u$ ) $\Rightarrow $ (P3 $ ^u$ ).

Throughout Section 4, we denote with k a $\sigma $ -field (of arbitrary characteristic).

Proof Let us first show that (i) implies (ii). Finding a solution in $k^{[n]}$ for some set of difference equations is equivalent to finding a morphism of k- $ \sigma $ -algebras $R\to k^{[n]}$ for some appropriate k- $ \sigma $ -algebra R. If $n|\tilde n$ , say $\tilde n=nl$ , then we have a morphism of k- $ \sigma $ -algebras $k^{[n]}\to k^{[\tilde n]}$ :

$$ \begin{align*}(a_1,\ldots,a_n)\mapsto (\sigma^{n(l-1)}(a_1),\ldots,\sigma^{n(l-1)}(a_n),\ldots,\sigma^n(a_1),\ldots,\sigma^n(a_n),a_1,\ldots,a_n).\end{align*} $$

Thus, if (i) holds with some integer $n\geq 1$ , then it also holds for every integer $\tilde n$ that is a multiple of n and in particular for $\tilde n=nd$ . We can therefore assume that $d|n$ , say $n=dl$ . By assumption, there exists a $Y\in \mathcal {G}(k^{[n]})$ with $\sigma ^d(Y)=AY$ , and thus $\sigma ^{dl}(Y)=\sigma ^{(l-1)d}(A)\dots \sigma ^d(A)AY$ . The projection $k^{[n]}\to k$ onto the last factor is a morphism of k- $ \sigma ^n$ -algebras. Therefore, the equation $\sigma ^{dl}(Y)=\sigma ^{(l-1)d}(A)\dots \sigma ^d(A)AY$ has a solution in $\mathcal {G}(k)$ .

Next, we will show that (ii) implies (i). Let $Z\in \mathcal {G}(k)$ be a solution to $\sigma ^{dl}(Y)=\sigma ^{(l-1)d}(A)\dots \sigma ^d(A)AY$ . Define $n=dl$ and $A_i=\sigma ^{i-1}(A)\dots \sigma (A)A$ for all integers $i\geq 1$ . Then

$$ \begin{align*}Z'&=( \sigma^{d-1}(A_{l-1}Z), \ldots, A_{l-1}Z,\ldots,\sigma^{d-1}(A_{2}Z), \ldots A_{2}Z,\sigma^{d-1}\\&\quad(A_{1}Z), \ldots A_{1}Z,\sigma^{d-1}(Z),\ldots,Z)\in\mathcal{G}(k^{[n]})\end{align*} $$

is a solution to $\sigma ^d(Y)=AY$ .

Note that if n does not depend on A, then l does not depend on A and vice versa, so the second statement also holds.

Proof The equivalence of (b) and (b’) follows (in both cases (P3) and (P3 $ ^u$ )) from Lemma 4.1 with $\mathcal {G}=\text {GL}_m$ and $d=1$ . Similarly, the equivalence of (b $ _d$ ) and (b $ _d$ ’) follows from Lemma 4.1 with $\mathcal {G}=\text {GL}_m$ in both cases. So it remains to show that (b’) implies (b $ _d$ ’) in both cases. We start with the (P3 $ ^u$ ) case and assume that (P3 $ ^u$ )(b’) holds and fix integers $m,d\geq 1$ . Then, by (P3 $ ^u$ )(b’), there exists an integer $l_d$ such that for every ${B\in \text {GL}_{md}(k)}$ , there exists a $Z\in \text {GL}_{md}(k)$ with $\sigma ^{l_d}(Z)=\sigma ^{(l_d-1)}(B)\dots \sigma (B) B Z$ . By possibly replacing $l_d$ with $dl_d$ , we may assume that d divides $l_d$ and define $l=\frac {l_d}{d}$ . We claim that (P3 $ ^u$ )(b $ _d$ ’) holds for this l. Fix a matrix $A \in \text {GL}_m(k)$ . It remains to show that there exists a matrix $Y\in \text {GL}_m(k)$ with $\sigma ^{dl}(Y)=\hat AY,$ where $\hat A=\sigma ^{(l-1)d}(A)\dots \sigma ^d(A)A$ . By assumption, there exists a $Z\in \text {GL}_{md}(k)$ with $\sigma ^{l_d}(Z)=\hat B Z$ , where we define B as block matrix

$$ \begin{align*}B=\begin{pmatrix} 0 & I_m & 0 & & \\ \vdots & & \ddots & & 0 \\ 0 & & & & I_m \\ A & 0& \dots & & 0 \\ \end{pmatrix} \in \text{GL}_{md}(k)\end{align*} $$

and set $\hat B=\sigma ^{(l_d-1)}(B)\dots \sigma (B) B$ . Recall that d divides $l_d$ , so the product $\hat B$ with $l_d$ factors is the block diagonal matrix:

$$ \begin{align*}\hat B= \begin{pmatrix} \hat A & & & \\ & \sigma(\hat A) & & \\ & & \ddots & \\ & & & \sigma^{d-1}(\hat A) \end{pmatrix}.\end{align*} $$

Now the $md$ columns of $Z\in \text {GL}_{md}(k)$ are k-linearly independent, so we can choose m columns $z_1,\dots ,z_m$ of Z such that their projections $y_1,\dots ,y_m \in k^m$ to the first m coordinates are still k-linearly independent. Define $Y\in \text {GL}_m(k)$ as the matrix with columns $y_1,\dots ,y_m$ . By construction, $\sigma ^{dl}(z_i)=\sigma ^{l_d}(z_i)=\hat B z_i$ , so $\sigma ^{dl}(y_i)=\hat A y_i$ for all i and thus $\sigma ^{dl}(Y)=\hat A Y$ as claimed. The proof in the (P3) case is analogous, but here we start with a fixed matrix $A\in \text {GL}_m(k)$ , define $B\in \text {GL}_{md}(k)$ as above and obtain an integer $l_d$ from (P3)(b’) (depending on B and thus on A) and a matrix $Z\in \text {GL}_{md}(k)$ with $\sigma ^{l_d}(Z)=\hat B Z$ (with $\hat B$ defined as above). Again, we may assume that d divides $l_d$ and set $l=l_d$ (which now depends on A) and the same computations as above yield a matrix $Y\in \text {GL}_m(k)$ with $\sigma ^{dl}(Y)=\hat A Y$ as claimed in (P3)(b $ _d$ ’).

The following lemma shows that property (P3) is preserved when passing to higher powers of $\sigma $ .

Proof No need to worry about (P3)(a) and (P3 $ ^u$ )(a). For property (b), we use Lemma 4.2 to conclude that it suffices to show that (b’) holds for the difference field $(k,\sigma ^d)$ in both cases. But that condition equals (b $ _d$ ’) for the difference field $(k,\sigma )$ which does hold by Lemma 4.2 in both cases (P3) and (P3 $ ^u$ ).

For (c), let $\alpha _0,\ldots ,\alpha _m\in \mathbb {Z}$ with $\alpha \neq 0$ and for $a\in k^\times $ consider the multiplicative $\sigma ^d$ -equation

(4.1) $$ \begin{align} y^{\alpha_0}\sigma^d(y)^{\alpha_1}\dots\sigma^{dm}(y)^{\alpha_m}=a. \end{align} $$

If (P3)(c) holds, there exists an $n\geq 1$ such that (4.1), when considered as a $\sigma $ -equation, has a solution in $k^{[n]}$ . So (4.1), when considered as a system of $\sigma ^d$ -equations, has a solution in the k- $ \sigma ^d$ -algebra $(k^{[n]},\sigma ^d)$ . As in the proof of Lemma 4.1, we can assume that $d|n$ , say $n=dl$ . Then we have a morphism

$$ \begin{align*}(k^{[n]},\sigma^d)\to (k,\sigma^d)^{[l]},\ (a_1,\ldots,a_n)\mapsto(a_d,a_{2d},\ldots,a_n)\end{align*} $$

of k- $ \sigma ^d$ -algebras. Thus, (4.1), when considered as a system of $\sigma ^d$ -equations, has a solution in $(k,\sigma ^d)^{[l]}$ . We conclude that $(k,\sigma ^d)$ satisfies (P3)(c). Moreover, if $(k,\sigma )$ satisfies the stronger condition (P3 $ ^u$ )(c), then n does not depend on the choice of a and thus l does not depend on it either and thus $(k,\sigma ^d)$ satisfies the stronger condition (P3 $ ^u$ )(c).

As indicated in the introduction, to show that (P1) implies (P3) it suffices to consider (P1) for a few particular $\sigma $ -algebraic groups, namely, algebraic groups, $\text {GL}_m^\sigma $ and one-relator subgroups of $\mathbb {G}_m$ .

Proof To prove (a) in both cases, we need to show that (P1) implies that k is algebraically closed. As every finite Galois extension of k defines a torsor for a finite (constant) algebraic group over k, it is clear that a field with the property that every torsor for every algebraic group over k is trivial, has to be separably closed. Moreover, if k has positive characteristic p, then $\mathcal {G}=\{g\in \mathbb {G}_m|\ g^p=1\}$ is a (non-reduced) algebraic group and for every $a\in k^\times $ , we have a $\mathcal {G}$ -torsor $\mathcal {X}=\{x\in \mathbb {G}_m|\ x^p=a\}$ . Thus, if all these torsors are trivial, k must be algebraically closed.

So let $\mathcal {G}$ be an algebraic group over k and let $\mathcal {X}$ be a $\mathcal {G}$ -torsor. We have to show that $\mathcal {X}(k)\neq \emptyset $ . Set $G=[\sigma ]_k\mathcal {G}$ and $X=[\sigma ]_k\mathcal {X}$ . Then X is a G-torsor. So, by assumption, there exists $n\geq 1$ such that $X(k^{[n]})=\mathcal {X}(k^{[n]})\neq \emptyset $ . In other words, there exists a morphism of k-algebras $k[\mathcal {X}]\to k^{[n]}$ . Composing with the last projection $k^{[n]}\to k$ yields a morphism of k-algebras $k[\mathcal {X}]\to k$ . So $\mathcal {X}(k)\neq \emptyset $ and k is algebraically closed.

To prove (b), consider the $\sigma $ -algebraic group $G=\text {GL}_m^\sigma =\{g\in \text {GL}_m|\ \sigma (g)=g\}$ . For $A\in \text {GL}_m(k)$ consider the $\sigma $ -variety X given by $X=\{x\in \text {GL}_m|\ \sigma (x)=Ax\}$ . It is straightforward to check that X is a G-torsor under the action $(g,x)\mapsto xg^{-1}$ . We conclude that (P1) implies (P3)(b) and (P1 $ ^u$ ) implies (P3 $ ^u$ )(b).

To prove (c), consider the $\sigma $ -algebraic group

$$ \begin{align*}G=\{g\in \mathbb{G}_m |\ g^{\alpha_0}\sigma(g)^{\alpha_1}\dots\sigma^m(g)^{\alpha_m}=1\}.\end{align*} $$

Then the $\sigma $ -variety $X=\{x\in \mathbb {G}_m |\ x^{\alpha _0}\sigma (x)^{\alpha _1}\dots \sigma ^m(x)^{\alpha _m}=a\}$ is a G-torsor. We conclude that (P1) implies (P3)(c) and (P1 $ ^u$ ) implies (P3 $ ^u$ )(c).

5 The proof of (P3) $\Rightarrow $ (P1)

By now we have seen that (P1) and (P2) are equivalent (Proposition 3.19) and that (P1) implies (P3) (Proposition 4.5) and similarly for their uniform variants. Thus, to complete the proof of our main result, it suffices to prove (P3) $\Rightarrow $ (P1) (and (P3 $ ^u$ ) $\Rightarrow $ (P1 $ ^u$ )). This however is the most difficult part.

From now on, we assume that k is a $\sigma $ -field of characteristic zero.

It is an interesting question to ask for a generalization of Theorem 1.3 to positive characteristic. It may seem plausible to expect that the theorem also holds for a difference field k of positive characteristic p if (P3 $ ^u$ )(b) is replaced by

(P3 $ ^u$ )( $b_p$ ) For every $m\geq 1$ and every power q of p, there exists an integer $n\geq 1$ such that for every $A\in \text {GL}_m(k),$ the equation $\sigma _q(Y)=AY$ has a solution in $\text {GL}_m(k^{[n]})$ .

Here, $\sigma _q$ is defined by $\sigma _q(y)=\sigma (y)^q$ . Note that the equation $\sigma _q(Y)=AY$ defines a torsor for the $\sigma $ -algebraic group $G=\text {GL}_m{\,}^{\sigma _q}=\{g\in \text {GL}_m|\ \sigma _q(g)=g\}$ . Moreover, any Frobenius difference field, that is, an algebraically closed field of positive characteristic equipped with a Frobenius endomorphism, satisfies (P3 $ ^u$ )( $b_p$ ) with $n=1$ (due to the Lang–Steinberg theorem).

Many of our intermediate results (and proofs) in this section also work in arbitrary characteristic. However, there are also several steps where we make use of the characteristic zero assumption. At those steps, in positive characteristic, one usually would have to contend with non-reduced algebraic or $\sigma $ -algebraic groups and the Frobenius endomorphism. We hope to establish a variant of Theorem 1.3 in positive characteristic in a future project dedicated to the study of torsors for $\sigma $ -algebraic groups over Frobenius difference fields.

One of the advantages of working in characteristic zero is that every Hopf algebra is reduced. Therefore, a $\sigma $ -algebraic group G is connected if and only if it is integral [Reference WibmerWib21, Lemma 6.7].

In this section, we show that if k satisfies (P3) (or (P3 $ ^u$ )), then k is (uniformly) strongly G-trivial for any $\sigma $ -algebraic group G over k. The proof heavily relies on the induction principle from Section 3 and the structure theory of $\sigma $ -algebraic groups. Any $\sigma $ -algebraic group can be decomposed into $\sigma $ -algebraic groups belonging to certain more manageable classes of $\sigma $ -algebraic groups and we first show that if k satisfies (P3), then k is uniformly strongly G-trivial for any G belonging to one of those more manageable classes. We start from classes of small (e.g., finite, order zero, and $\sigma $ -dimension zero) $\sigma $ -algebraic groups and then build up to the general case.