# On the exponent of geometric unipotent radicals of pseudo-reductive groups

Falk Bannuscher  Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780
Bochum,Germany<EMAIL_ADDRESS>,  Maike Gruchot  Lehrstuhl für Algebra
und Zahlentheorie, RWTH Aachen University, D-52062 Aachen, Germany
<EMAIL_ADDRESS>and  David I. Stewart  School of Mathematics,
Statistics and Physics, Newcastle University, UK
<EMAIL_ADDRESS>
###### Abstract.

Let $k^{\prime}/k$ be a finite purely inseparable field extension and let
$G^{\prime}$ be a reductive $k^{\prime}$-group. We denote by
$G=\mathrm{R}_{k^{\prime}/k}(G^{\prime})$ the Weil restriction of $G^{\prime}$
across $k^{\prime}/k$, a pseudo-reductive group. This article gives bounds for
the exponent of the geometric unipotent radical $\mathscr{R}_{u}(G_{\bar{k}})$
in terms of invariants of the extension $k^{\prime}/k$, starting with the case
$G^{\prime}=\operatorname{GL}_{n}$ and applying these results to the case
where $G^{\prime}$ is a simple group.

###### 2010 Mathematics Subject Classification:

20G15

##  1\. Introduction

Let $k$ be a field, which we assume is imperfect of characteristic $p$. In
this case there exist pseudo-reductive $k$-groups which are not reductive.
Recall that a smooth, connected, affine algebraic $k$-group $G$ is _pseudo-
reductive_ if the largest smooth connected unipotent normal $k$-subgroup
$\mathscr{R}_{u,k}(G)$ is trivial. ††* Funded by Leverhulme Trust Research
Project Grant number RPG-2021-080

We are interested in the structure of the geometric unipotent radical
$R:=\mathscr{R}_{u}(G_{\bar{k}})=\mathscr{R}_{u,\bar{k}}(G_{\bar{k}})$ of $G$.
Since $R$ is a $p$-group, it makes sense to study its exponent: the minimal
integer $s$ such that the $p^{s}$-power map on the geometric unipotent radical
factors through the trivial group. We denote the _exponent_ of $R$ by $e(R)$.

M. Bate, B. Martin, G. Röhrle and the third author previously gave some bounds
for $e(R)$ in [1]. For example, [1, Lem. 4.1] implies that if $k^{\prime}/k$
is a simple purely inseparable field extension with
$(k^{\prime})^{p^{e}}\subseteq k$, then $e(R)\leq e$.

The monographs [4] and [5] contain a classification of pseudo-reductive
groups. They are all related in some way or other to the Weil restrictions of
reductive groups, which are themselves pseudo-reductive. We focus on groups
which are Weil restrictions, since general pseudo-reductive groups contain
central pseudo-reductive subgroups whose classification is thought to be
intractable. Thus we assume unless stated otherwise that
$G=\mathrm{R}_{k^{\prime}/k}(G^{\prime})$ for some reductive
$k^{\prime}$-group $G^{\prime}$ where $k^{\prime}$ is a finite non-zero
reduced $k$-algebra. Since we are interested in $e(R)$, by the remarks in [1,
§4] we may as well consider the case that $k$ is separably closed and
$k^{\prime}$ is a purely inseparable field extension of $k$. In this case
$G^{\prime}$ is a split reductive group and as such is classified by its root
datum.

Recall that the exponent of the extension $k^{\prime}/k$ is the smallest
integer $e$ such that $(k^{\prime})^{p^{e}}\subseteq k$. To describe our
results, we need more sensitive data about the structure of $k^{\prime}/k$.
Since $k^{\prime}/k$ is purely inseparable, we can appeal to the results in
[10] and [11], which guarantee a _normal generating sequence_
$\alpha_{1},\ldots,\alpha_{l}$ of $k^{\prime}$ over $k$ with certain
properties—see Definition 2.6 below for more details. In particular, these
elements come with certain exponents which are invariants of $k^{\prime}/k$.
These are the integers $e_{1}\geq\ldots\geq e_{l}$ such that $e_{i}$ is
minimal subject to

$\alpha_{i}^{p^{e_{i}}}\in
k(\alpha_{1}^{p^{e_{i}}},\ldots,\alpha_{i-1}^{p^{e_{i}}}).$

It is immediate that $e_{1}=e$ and it follows from the Tower Law that
$[k^{\prime}:k]=\prod p^{e_{i}}$. Let $r\in\mathbb{N}$. We define integers

$\displaystyle\operatorname{m}(k^{\prime}/k)$
$\displaystyle:=\sum_{i=1}^{l}(p^{e_{i}}-1)+1=\sum_{i=1}^{l}p^{e_{i}}-l+1,$
$\displaystyle\operatorname{em}(k^{\prime}/k)$
$\displaystyle:=\lceil\log_{p}(\operatorname{m}(k^{\prime}/k))\rceil,$
$\displaystyle\operatorname{mr}(k^{\prime}/k,r)$
$\displaystyle:=\sum_{i=1}^{r}p^{e_{i}}\text{,\quad where we take $e_{i}=0$
for $i>l$},$ $\displaystyle\operatorname{emr}(k^{\prime}/k,r)$
$\displaystyle:=\lceil\log_{p}(\operatorname{mr}(k^{\prime}/k,r))\rceil,$
$\displaystyle\text{and finally }\operatorname{E}(k^{\prime}/k,r)$
$\displaystyle:=\min\\{\operatorname{em}(k^{\prime}/k),\operatorname{emr}(k^{\prime}/k,r)\\}.$

In other words $\operatorname{em}(k^{\prime}/k)$ is the minimal $s$ such that
$p^{s}\geq\operatorname{m}(k^{\prime}/k)$, and similarly for
$\operatorname{emr}$.

In answering a question of [1] on the exponent of the intersection of $R$ with
a pseudo-Borel subgroup, we discovered that these numbers give lower bounds on
$e(R)$ and for large enough rank determine the exponent of $R$ exactly. We
first state our results in the case that $G^{\prime}$ is a general linear
group.

######  Theorem 1.1.

Let $k^{\prime}/k$ be a finite purely inseparable field extension of a
separably closed field $k$ of characteristic $p$. Suppose $k^{\prime}/k$ has
exponent $e$ and a normal generating sequence $\alpha_{1},\ldots,\alpha_{l}$
with sequence of exponents $e_{1},\ldots,e_{l}$. For $r\in\mathbb{N}$, let
$G^{\prime}:=\operatorname{GL}_{r}$,
$G:=\mathrm{R}_{k^{\prime}/k}(G^{\prime})$ and
$R:=\mathscr{R}_{u}(G_{\bar{k}})$. Furthermore let $B^{\prime}$ denote a Borel
subgroup of $G^{\prime}$ and
$B:=\mathrm{R}_{k^{\prime}/k}(B^{\prime})\subseteq G$ the corresponding
pseudo-Borel subgroup of $G$. The following hold:

  1. (i)

if $s$ satisfies $p^{s}\geq p^{e}r$ then $e(R)$ is at most $s$; i.e.
$e(R)\leq\lceil e+\log_{p}(r)\rceil$;

  2. (ii)

we have $e(R\cap B_{\bar{k}})=\operatorname{E}(k^{\prime}/k,r)$;

  3. (iii)

in particular, $e(R)\geq\operatorname{E}(k^{\prime}/k,r)$;

  4. (iv)

if $\sum_{i=r+1}^{l}(p^{e_{i}}-1)<r-1$ then
$e(R)=\operatorname{E}(k^{\prime}/k,r)=\operatorname{em}(k^{\prime}/k)$.

######  Remarks 1.2.

(i) The bounds in Theorem 1.1 are both arithmetic and Lie-theoretic in
character: the integers $\operatorname{m}(k^{\prime}/k)$ and
$\operatorname{em}(k^{\prime}/k)$ depend only on the arithmetic structure of
$k^{\prime}/k$ but $\operatorname{mr}(k^{\prime}/k,r)$,
$\operatorname{emr}(k^{\prime}/k,r)$ and $\operatorname{E}(k^{\prime}/k,r)$
depend on both $k^{\prime}/k$ and the rank $r$ of $G^{\prime}$.

(ii). Part (i) of Theorem 1.1 improves [1, Lem. 4.4], which required
$p^{s}\geq r^{2}(p^{e}-1)$. As a corollary of (i) and (iv), if
$G=\mathrm{R}_{k^{\prime}/k}(\operatorname{GL}_{r})$ for $p\geq r\geq 2$ then
the exponent of $\mathscr{R}_{u}(G_{\bar{k}})$ is $e$ if $k^{\prime}/k$ is a
simple extension and is $e+1$ otherwise. If $r=1$, then $e(R)=e$ as
$\mathrm{R}_{k^{\prime}/k}(\operatorname{GL}_{1})$ is sent into the canonical
$\operatorname{GL}_{1}$ $k$-subgroup by the $p^{e}$-power map, since it also
sends $k^{\prime}$ into $k$.

(iii). In [1, Rem. 4.5(iv)], it was asked if the exponent $e(R)$ always
coincides with the exponent $e(R\cap B_{\bar{k}})$. Clearly Theorem
1.1(ii)-(iv) answers this question in the affirmative when
$G^{\prime}=\operatorname{GL}_{r}$ of large enough rank. We give an example to
show that in general the answer is ‘no’ (see Proposition 4.3), coming from
$\operatorname{SL}_{2}$ in characteristic $2$. In fact, in Section 4 we give a
complete description of $e(R)$ when $G^{\prime}$ is $\operatorname{SL}_{2}$,
$\operatorname{PGL}_{2}$ or $\operatorname{GL}_{2}$.

Together with the rank $1$ results in Section 4, the following (proved in
Section 5) confirms the large rank behaviour in all simple groups, with mild
conditions on the characteristic.

######  Corollary 1.3.

Let $k^{\prime}/k$ be a finite, purely inseparable field extension of a
separably closed field $k$ of characteristic $p$. Suppose $k^{\prime}/k$ has
exponent $e$ and a normal generating sequence $\alpha_{1},\ldots,\alpha_{l}$
with sequence of exponents $e_{1},\ldots,e_{l}$. Let $G^{\prime}$ be a (split)
simple algebraic $k$-group of rank $r\geq 2$,
$G:=R_{k^{\prime}/k}(G^{\prime})$ and let $R:=\mathscr{R}_{u}(G_{\bar{k}})$.
If $G^{\prime}$ is type $B,C,D,F$ then assume $p\neq 2$ or $G^{\prime}$ is
$\operatorname{SO}_{n}$ or $\operatorname{Sp}_{2n}$, and if $G$ is of type
$E_{6}$ then assume $p\neq 3$. Then the exponent
$e(R)\geq\operatorname{E}(k^{\prime}/k,r)$; if
$\sum_{i=r+1}^{l}(p^{e_{i}}-1)<r-1$ then in fact,
$e(R)=\operatorname{E}(k^{\prime}/k,r)=\operatorname{em}(k^{\prime}/k)$.

##  2\. Preliminaries and Notation

Let $k$ be a field of characteristic $p$ and $G$ be a $k$-group, by which we
mean an affine algebraic group scheme of finite type over $k$. The _$k$
-unipotent radical_ is the maximal smooth connected normal unipotent
$k$-subgroup of $G$ and we denote this by $\mathscr{R}_{u,k}(G)$. The
_geometric unipotent radical_ is the $\bar{k}$-unipotent radical of the base
change $G_{\bar{k}}$ of $G$ to $\bar{k}$. This is denoted by
$\mathscr{R}_{u}(G_{\bar{k}})$. We call a smooth connected $k$-group $G$
_reductive_ if $\mathscr{R}_{u}(G_{\bar{k}})=1$ and _pseudo-reductive_ if
$\mathscr{R}_{u,k}(G)=1$. From now on $G$ will always be pseudo-reductive. We
recall the definition and key properties of Weil restriction from [4, A.5].

######  Definition 2.1.

Let $B\to B^{\prime}$ be a finite flat map of noetherian rings, and
$X^{\prime}$ a quasi-projective $B^{\prime}$-scheme. The _Weil restriction_ is
a finite type $B$-scheme satisfying the universal property

$\mathrm{R}_{B^{\prime}/B}(X^{\prime})(A)=X^{\prime}(B^{\prime}\otimes_{B}A)$

for all $B$-algebras $A$.

The following proposition makes use of the natural map

(2.2) $\displaystyle
q_{G^{\prime}}:\mathrm{R}_{k^{\prime}/k}(G^{\prime})_{k^{\prime}}\to
G^{\prime},$

which is induced by $k^{\prime}\otimes_{k}A\to A;a\otimes b\mapsto ab$ for any
$k^{\prime}$-algebra $A$.

######  Proposition 2.3 ([4, Prop. A.5.11, Thm. 1.6.2], [1, §2], [1, Lem.
3.4]).

Let $k^{\prime}/k$ be a finite and purely inseparable field extension,
$G^{\prime}$ a non-trivial smooth connected $k^{\prime}$-group and
$G=\mathrm{R}_{k^{\prime}/k}(G^{\prime})$.

  1. (i)

The kernel of $q_{G^{\prime}}$ is a smooth connected unipotent
$k^{\prime}$-subgroup and thus it is contained in
$\mathscr{R}_{u,k^{\prime}}(G_{k^{\prime}})$.

  2. (ii)

If $G^{\prime}$ is reductive over $k^{\prime}$, then the kernel of
$q_{G^{\prime}}$ has field of definition over k equal to
$k^{\prime}\subseteq\bar{k}$. Thus the kernel $\ker(q_{G^{\prime}})$ coincides
with $\mathscr{R}_{u,k^{\prime}}(G_{k^{\prime}})$, which is a
$k^{\prime}$-descent of $\mathscr{R}_{u}(G_{\bar{k}})$ (i.e.
$\mathscr{R}_{u,k^{\prime}}(G_{k^{\prime}})_{\bar{k}}\cong\mathscr{R}_{u}(G_{\bar{k}})$).

  3. (iii)

If $H^{\prime}$ is a reductive subgroup of $G^{\prime}$ then the geometric
unipotent radical of $\mathrm{R}_{k^{\prime}/k}(H^{\prime})$ is a subgroup of
the geometric unipotent radical of $\mathrm{R}_{k^{\prime}/k}(G^{\prime})$.

  4. (iv)

If $f:G_{1}\to G_{2}$ is an étale isogeny of reductive $k^{\prime}$-groups
then $\mathrm{R}_{k^{\prime}/k}(f)_{\bar{k}}$ induces an isomorphism of
geometric unipotent radicals of the respective Weil restrictions.

We are interested in the following invariant of a unipotent $k$-group:

######  Definition 2.4.

Let $k$ be a field of characteristic $p$ and $U$ a unipotent $k$-group. The
exponent $e(U)$ of $U$ is the minimal $s$ such that the $p^{s}$-power map on
$U$ factors through the trivial group.

Clearly, this definition is insensitive to base change. As in [1, §4] we may
therefore assume $k$ is separably closed for the remainder of the article.

When calculating with matrices, we make use of the fact that if $U(k)\subset
U$ is dense (e.g. if $k$ is perfect [6, Thm. 34.4]), then a map factors
through the trivial group if and only if it maps to the identity on $U(k)$.
Since $R=\mathscr{R}_{u}(G_{\bar{k}})$ descends to
$R^{\prime}:=\mathscr{R}_{u,k^{\prime}}(G_{k^{\prime}})$ whose
$k^{\prime}$-points are dense (by the fact that $k^{\prime}$ is separably
closed) we have $e(R)$ is the smallest $s$ such that $p^{s}$ kills
$R^{\prime}(k^{\prime})$.

Following [9], let $\mathfrak{B}:=k^{\prime}\otimes_{k}k^{\prime}$. Then
$\mathfrak{B}$ is a local ring with maximal ideal

$\mathfrak{m}=\langle 1\otimes x-x\otimes 1\mid x\in
k^{\prime}\rangle=\ker(\phi),$

where $\phi:k^{\prime}\otimes_{k}k^{\prime}\to k^{\prime};a\otimes b\mapsto
ab$. Moreover, $\mathrm{R}_{k^{\prime}/k}(G^{\prime})_{k^{\prime}}$ can be
identified with
$\mathrm{R}_{\mathfrak{B}/k^{\prime}}(G^{\prime}_{\mathfrak{B}})$ and the
$k^{\prime}$-points of the $k^{\prime}$-unipotent radical of the former is the
kernel $G^{\prime}(\mathfrak{m})$ of the map $G^{\prime}(\mathfrak{B})\to
G^{\prime}(k)$ induced by $\phi$. Suppose now
$G^{\prime}=\operatorname{GL}_{r}$ and
$G=\mathrm{R}_{k^{\prime}/k}(G^{\prime})$. As the entries of any
$M\in\operatorname{Mat}(r,\mathfrak{m})$ are nilpotent elements of
$\mathfrak{B}$, it follows that $I_{r}+M$ has invertible determinant, hence is
an element of $G^{\prime}(\mathfrak{m})$. Therefore

$\mathscr{R}_{u,k}(\mathrm{R}_{k^{\prime}/k}(\operatorname{GL}_{r})_{k^{\prime}})(k^{\prime})=\\{I_{r}+M\mid
M\in\operatorname{Mat}(r,\mathfrak{m})\\}.$

As $I_{r}$ and $M$ commute, we have $A^{p^{s}}=I_{r}+M^{p^{s}}=I_{r}$ is
equivalent to $M^{p^{s}}=0$. Hence:

######  Lemma 2.5.

The exponent of the geometric unipotent radical of
$\mathrm{R}_{k^{\prime}/k}(\operatorname{GL}_{r})$ is the smallest $s$ such
that $M^{p^{s}}=0$ for all $M\in\operatorname{Mat}(r,\mathfrak{m})$.

It was observed in [1, Lem. 4.1] that as $\mathfrak{m}^{n}=0$ for some
$n\in\mathbb{N}$, one knows that $s$ in the lemma above is bounded above by
the minimal $t$ such that $p^{t}\geq n$: the entries of $M^{p^{t}}$ are
homogeneous polynomials of degree $p^{t}$ in elements of $\mathfrak{m}$ and so
must vanish.

######  Definition 2.6.

Let $k^{\prime}/k^{\prime\prime}/k$ be a tower of finite purely inseparable
field extensions.

  1. (i)

If $x\in k^{\prime}$ then the minimal $s\in\mathbb{N}$ such $x^{p^{s}}\in
k^{\prime\prime}$ is called _the exponent of $x$ with respect to
$k^{\prime\prime}$_ and denoted by $e_{k^{\prime\prime}}(x)$.

  2. (ii)

We say that $x\in k^{\prime}$ is _normal in $k^{\prime}/k^{\prime\prime}$_ if
$e_{k^{\prime\prime}}(x)\geq e_{k^{\prime\prime}}(y)$ for all $y\in
k^{\prime}$.

  3. (iii)

A sequence $\alpha_{1},\ldots,\alpha_{l}$ in $k^{\prime}$ is called a _normal
sequence_ if for every $1\leq i\leq l$ and
$k_{i}=k[\alpha_{1},\ldots,\alpha_{i}]$ we have that $\alpha_{i}$ is normal in
$k^{\prime}/k_{i-1}$, and $\alpha_{i}\notin k_{i-1}$.

  4. (iv)

A normal sequence $\alpha_{1},\ldots,\alpha_{l}$ is called a _normal
generating sequence of $k^{\prime}/k$_ if
$k[\alpha_{1},\ldots,\alpha_{l}]=k^{\prime}$.

  5. (v)

For a normal generating sequence $\alpha_{1},\ldots,\alpha_{l}$ of
$k^{\prime}/k$ we set $e_{i}=e_{k_{i-1}}(\alpha_{i})$. We call
$e_{1},\ldots,e_{l}$ a _sequence of exponents for $k^{\prime}/k$_

Note that if $e_{1},\ldots,e_{l}$ is a sequence of exponents for
$k^{\prime}/k$ then we have $e_{1}\geq e_{2}\geq\ldots\geq e_{l}$ due to [11,
Prop. 5]. Crucially, by loc. cit. we also have

(2.7) $\alpha_{i}^{p^{e_{i}}}\in
k(\alpha_{1}^{p^{e_{i}}},\ldots,\alpha_{i-1}^{p^{e_{i}}})\text{ for all }1\leq
i\leq l.$

In particular, as the $e_{i}$ are decreasing,

(2.8) $(k^{\prime})^{p^{e_{i}}}\subseteq
k(\alpha_{1}^{p^{e_{i}}},\ldots,\alpha_{i-1}^{p^{e_{i}}}).$

Combining this with the main result of [10], we get that the sequence of
exponents is an invariant of the field extension $k^{\prime}/k$. More
precisely:

######  Lemma 2.9.

Let $k^{\prime}/k$ be a finite purely inseparable field extension. Suppose
that $\alpha_{1},\ldots,\alpha_{l}$ and $\beta_{1},\ldots,\beta_{l^{\prime}}$
are normal generating sequences of $k^{\prime}/k$ with sequences of exponents
$e_{1},\ldots,e_{l}$ and $e^{\prime}_{1},\ldots,e^{\prime}_{l^{\prime}}$. Then
$l=l^{\prime}$ and $e_{i}=e^{\prime}_{i}$ for $1\leq i<l$.

By the lemma, the numbers $\operatorname{m}(k^{\prime}/k)$,
$\operatorname{mr}(k^{\prime}/k,r)$ defined in the introduction are both
invariants of $k^{\prime}/k$ and $(k^{\prime}/k,r)$ respectively. To relate
these with $\mathfrak{m}$, we first prove:

######  Lemma 2.10.

For a normal generating sequence $\alpha_{1},\ldots,\alpha_{l}$ of
$k^{\prime}/k$, define $a_{i}:=1\otimes\alpha_{i}-\alpha_{i}\otimes
1\in\mathfrak{m}$. Then:

  1. (i)

the ideal $\mathfrak{m}$ is generated by the $a_{i}$ over $k^{\prime}$;

  2. (ii)

the $p^{e_{i}}$-power map takes $\mathfrak{m}$ into the $k^{\prime}$-subspace
$k^{\prime}[a_{1}^{p^{e_{i}}},\ldots,a_{i-1}^{p^{e_{i}}}]\cap\mathfrak{m}$.

###### Proof.

(i) Since $\mathfrak{m}$ is a codimension $1$ (left) $k^{\prime}$-subspace of
$\mathfrak{B}$, it is a codimension $[k^{\prime}:k]$ $k$-subspace of
$\mathfrak{B}$. Define $J=\\{(b_{1},b_{2},\ldots,b_{l})\in\mathbb{Z}^{l}\mid
0\leq b_{i}\leq p^{e_{i}}-1\\}$. For $b=(b_{1},...,b_{l})\in J$ we set
$x^{b}=\prod_{i=1}^{l}\alpha_{i}^{b_{i}}$. Then $\\{x^{b}\\}_{b\in J}$ is a
$k$-basis of $k^{\prime}$ and $\\{1\otimes x^{b}\\}$ is a $k^{\prime}$-basis
of $\mathfrak{B}$. For $b\neq(0,\ldots,0)$, the elements $a_{b}:=1\otimes
x^{b}-x^{b}\otimes 1=1\otimes x^{b}-x^{b}(1\otimes 1)$ are therefore linearly
independent, generating a codimension $1$ subspace of $\mathfrak{B}$, which is
contained in the codimension $1$ kernel $\mathfrak{m}$ of the multiplication
map $x\otimes y\mapsto xy$; hence they form a $k^{\prime}$-basis of
$\mathfrak{m}$. We wish to show
$a_{b}\in\mathfrak{m}^{\prime}:=\mathfrak{m}\cap
k^{\prime}[a_{1},\ldots,a_{l}]$ by induction on $\sum b_{i}$, the case $\sum
b_{i}=1$ being trivial. So let $\sum b_{i}>1$. Without loss of generality,
suppose $b_{1}>0$ and let $b^{\prime}=b-(1,0,\ldots,0)$. Then inductively
$a_{b^{\prime}}=1\otimes x^{b^{\prime}}-x^{b^{\prime}}\otimes
1\in\mathfrak{m}^{\prime}$. Now one checks directly that
$a_{b}=x^{b^{\prime}}a_{1}+\alpha_{1}x^{b^{\prime}}-a_{b^{\prime}}\cdot
a_{1}\in\mathfrak{m}^{\prime}$ as required. (And it follows that
$\mathfrak{m}=\mathfrak{m}^{\prime}$.)

(ii) Define
$\mathfrak{m}^{\prime}:=k^{\prime}[a_{1}^{p^{e_{i}}},\ldots,a_{i-1}^{p^{e_{i}}}]\cap\mathfrak{m}$.
From (i), we can express any element of $\mathfrak{m}$ as a polynomial in
$a_{1},\ldots,a_{l}$ over $k^{\prime}$. Since $k^{\prime}$ has characteristic
$p$, the $p^{e_{i}}$-power map distributes over addition and so it suffices to
check that $(a_{1}^{r_{1}}\cdots
a_{l}^{r_{l}})^{p^{e_{i}}}\in\mathfrak{m}^{\prime}$ for
$r_{1},\ldots,r_{l}\in\mathbb{Z}_{\geq 0}$. For this it suffices to check that
$a_{j}^{p^{e_{i}}}\in\mathfrak{m}^{\prime}$ for $j\geq i$. Now
$a_{j}^{p^{e_{i}}}=(1\otimes\alpha_{j}-\alpha_{j}\otimes
1)^{p^{e_{i}}}=1\otimes\alpha_{j}^{p^{e_{i}}}-\alpha_{j}^{p^{e_{i}}}\otimes 1$
and by (2.8) we can write $\alpha_{j}^{p^{e_{i}}}$ as a $k$-polynomial $P$ in
$k[\alpha_{1}^{p^{e_{i}}},\ldots,\alpha_{i-1}^{p^{e_{i}}}]$ with no constant
term. But applying part (i) with $l=i-1$, $a_{j}^{p^{e_{i}}}$ in place of
$a_{j}$ for $1\leq j\leq i-1$ and $k[a_{1}^{p^{e_{i}}},\dots
a_{i-1}^{p^{e_{i}}}]$ in place of $k^{\prime}$, we deduce that $1\otimes
P-P\otimes 1$ is indeed a polynomial in the $a_{j}^{p^{e_{i}}}$ with $1\leq
j\leq i-1$ as required.∎

We may now identify the minimal integer $n$ such that $\mathfrak{m}^{n}=0$
with the integer $\operatorname{m}(k^{\prime}/k)$. In fact we also prove
something stronger:

######  Lemma 2.11.

With notation as above, let $d\leq l$ and suppose
$m_{1},\ldots,m_{d}\in\mathfrak{m}$ and $f_{1},\ldots,f_{d}\in\mathbb{Z}_{\geq
0}$.

If $\sum f_{i}\geq\operatorname{mr}(k^{\prime}/k,d)-d+1=\sum p^{e^{i}}-d+1$
then $m_{1}^{f_{1}}\cdots m_{d}^{f_{d}}=0$.

In particular, the minimum $n$ such that $\mathfrak{m}^{n}=0$ is
$\operatorname{m}(k^{\prime}/k)$.

###### Proof.

Assume for a contradiction that $m:=m_{1}^{f_{1}}\cdots m_{d}^{f_{d}}\neq 0$.
We proceed by induction on $d$, observing that the product of any factors of
$m$ is also non-zero. It does no harm to assume that $f_{1}\geq\ldots\geq
f_{d}$. If $d=1$, then $\operatorname{mr}(k^{\prime}/k,1)=p^{e_{1}}=p^{e}$ and
indeed $m=0$, a contradiction. Otherwise, let $c$ be maximal such that
$f_{c}\geq p^{e_{c}}$. This implies
$\sum_{i=1}^{c}f_{i}\geq\operatorname{mr}(k^{\prime}/k,c)-c+1$ so if $c<d$,
then $m_{1}^{f_{1}}\cdots m_{c}^{f_{c}}$ is zero by induction, a
contradiction. Therefore $c=d$. Let $(q_{i},r_{i})$ be the quotient and
remainder when $p^{e_{d}}$ is divided into $f_{i}$. As $f_{i}\geq f_{d}\geq
p^{e_{d}}$ we have each $q_{i}\geq 1$. Now

$f_{1}+\ldots+f_{d}=p^{e_{d}}(q_{1}+\ldots+q_{d})+(r_{1}+\ldots+r_{d})\geq
p^{e_{1}}+\ldots+p^{e_{d}}-d+1$

and so

$\displaystyle p^{e_{d}}(q_{1}+\ldots+q_{d})$ $\displaystyle\geq
p^{e_{d}}(q_{1}+\ldots+q_{d-1}+1)$ $\displaystyle\geq
p^{e_{1}}+\ldots+p^{e_{d}}-d+1-d(p^{e_{d}}-1)$
$\displaystyle=p^{e_{d}}(p^{e_{1}-e_{d}}+\ldots+p^{e_{d-1}-e_{d}}-d+1)+1.$

Since the second expression above is a multiple of $p^{e_{d}}$, we have

$q_{1}+\ldots+q_{d}\geq p^{e_{1}-e_{d}}+\ldots+p^{e_{d-1}-e_{d}}-(d-2).$

Now as $q_{i}p^{e_{d}}\leq f_{i}$ for each $i$, we have
$m^{\prime}=m_{1}^{q_{1}p^{e_{d}}}\cdots m_{d}^{q_{d}p^{e_{d}}}\neq 0$. By
Lemma 2.10(ii), we may write each $m_{i}^{p^{e_{d}}}$ as a polynomial in
$k^{\prime}[a_{1}^{p^{e_{d}}},\ldots,a_{d-1}^{p^{e_{d}}}]$. After expanding
$m^{\prime}$, any constituent monomial will be of the form
$a_{1}^{\nu_{1}}\cdots a_{d-1}^{\nu_{d-1}}$ where
$\sum_{i=1}^{d-1}\nu_{i}=p^{e_{d}}\sum_{i=1}^{d}q_{i}\geq
p^{e_{1}}+\ldots+p^{e_{d-1}}-p^{e_{d}}(d-2)$ and each $\nu_{i}$ is divisible
by $p^{e_{d}}$. Furthermore, as $a_{d-1}^{p^{e_{d-1}}}$ can be re-expressed in
terms of the $a_{i}$ with $i<d-1$, we may assume $\nu_{d-1}\leq
p^{e_{d-1}}-p^{e_{d}}$. It follows that

$\sum_{i=1}^{d-2}\nu_{i}\geq p^{e_{1}}+\ldots+p^{e_{d-2}}-p^{e_{d}}(d-3).$

Continuing inductively, $\nu_{1}\geq p^{e_{1}}-p^{e_{d}}+1$. Thus $\nu_{1}\geq
p^{e_{1}}$ and hence all monomials in $m^{\prime}$ are zero, a contradiction.

We now tackle the second statement. To see that
$n\geq\operatorname{m}(k^{\prime}/k)$, observe that
$a:=\prod_{i=1}^{l}a_{i}^{p^{e_{i}}-1}$ is a product of
$\operatorname{m}(k^{\prime}/k)-1$ elements from $\\{a_{1},\ldots,a_{l}\\}$.
Expanding this product in terms of the $k$-basis $\\{x^{b}\otimes x^{c}\mid
b,c\in J\\}$ of $\mathfrak{B}$, the coefficient of $1\otimes x^{b}$ where
$b=(p^{e_{1}}-1,\ldots,p^{e_{l}}-1)$ is easily seen to be $1$. It follows that
$a\neq 0$.

For the upper bound, take $d=l$ in the first part of the lemma, noting that
$\operatorname{m}(k^{\prime}/k)=\operatorname{mr}(k^{\prime}/k,l)-l+1$. ∎

##  3\. Bounds in the case of the general linear group

In this section, $G^{\prime}=\operatorname{GL}_{r}$ and
$G=\mathrm{R}_{k^{\prime}/k}(G^{\prime})$, where $k^{\prime}/k$ is purely
inseparable of exponent $e$. In [1, Lem. 4.4] it was shown that the exponent
$e(R)$ of $R:=\mathscr{R}_{u}(G_{\bar{k}})$ is at most $s$, where $s$ is
chosen such that $p^{s}\geq r^{2}(p^{e}-1)$. Using the Cayley–Hamilton theorem
in its full generality (see [2, III.8.11]) yields an improvement.

###### Proof of Theorem 1.1(i).

Let $M\in\operatorname{Mat}(r,\mathfrak{m})$, as in Lemma 2.5. By the
Cayley–Hamilton theorem, we have $\chi_{M}(M)=0$, where
$\chi_{M}(\lambda)=\lambda^{r}+\sum_{t=0}^{r-1}f_{t}\lambda^{t}$ is the
characteristic polynomial of $M$ with degree $r$. Observe that the coefficient
$f_{t}$ is either zero or a homogeneous polynomial in the entries of
$\mathfrak{m}$ with degree $r-t$. Since the matrices $M^{r-1},\ldots,M,I_{r}$
all commute and $k^{\prime}\otimes_{k}k^{\prime}$ has characteristic $p$, we
get

$M^{rp^{e}}=-(f_{r-1}^{p^{e}}M^{(r-1)p^{e}}+\ldots+f_{1}^{p^{e}}M^{p^{e}}+f_{0}^{p^{e}}I_{r})=0,$

since $f_{t}^{p^{e}}=0$ for all $f_{t}\in\mathfrak{m}$. Choosing $s$ such that
$p^{s}\geq p^{e}r$ we get $M^{p^{s}}=M^{p^{e}r}M^{p^{s}-p^{e}r}=0$. ∎

Let $B^{\prime}\subseteq G^{\prime}$ be the upper Borel subgroup, whose points
are upper triangular matrices, and
$B=\mathrm{R}_{k^{\prime}/k}(B^{\prime})\subseteq G$ its Weil restriction—a
pseudo-Borel subgroup in the terminology of [5, C.2]. We now specify certain
elements $M\in\operatorname{Mat}(r,\mathfrak{m})$ with $M\in
B(k^{\prime})=B^{\prime}(\mathfrak{B})$ to construct lower bounds on $e(R)$
which depend on $r$ and the exponents $e_{i}$ of a normal generating sequence.
Moreover, we calculate exactly the exponent of the intersection $R\cap
B_{\bar{k}}$.

Before we give the proof, we make some elementary remarks about powers of
matrices. If $A$ is a commutative ring and $E_{i,j}\in\operatorname{Mat}(r,A)$
is the $r\times r$ elementry matrix with $e_{i,j}=1$ and zeros everywhere
else, then we have $E_{i,j}E_{l,s}=E_{i,s}$ if and only if $j=l$ and $0$ else.
Write $M=\sum a_{ij}E_{i,j}$ for a general element of
$\operatorname{Mat}(r,A)$ . Then after expansion, any non-zero term in the
power $M^{n}$ is of the form $a_{w}\cdot E_{i_{1},i_{n+1}}$ where
$a_{w}=a_{i_{1},i_{2}}\cdot a_{i_{2},i_{3}}\cdots a_{i_{l},i_{n+1}}$, and
$w\in(i_{1},\ldots,i_{n+1})\in\\{1,\ldots,r\\}^{n+1}$. Furthermore, the
$(i,j)$-th entry of $M^{n}$ is the sum over all $a_{w}$ such that $i_{1}=i$
and $i_{n+1}=j$. For $w\in\\{1,\ldots,r\\}^{n+1}$, say $n$ is the _length_ of
$w$. Now, if $M$ is an upper triangular matrix, then $a_{w}$ will be zero
unless $i_{j}\leq i_{j+1}$ for each $1\leq j\leq n+1$. Hence the only $w$
yielding non-zero $a_{w}$ are non-decreasing sequences of integers from
$\\{1,\ldots,r\\}$, whose length is bounded above by
$\operatorname{m}(k^{\prime}/k)$ (Lemma 2.11).

###### Proof of Theorem 1.1(ii)-(iv).

We work inside $B^{\prime}(\mathfrak{m})$. There are two cases. Suppose we
have $\sum_{i=r+1}^{l}(p^{e_{i}}-1)<r-1$. Then there is an integer $q<r$ with
$q$ maximal such that $(p^{e_{q+1}}-1)+\ldots+(p^{e_{l}}-1)\geq q-1$. Define
the matrix $M\in B^{\prime}(\mathfrak{m})$ as follows.

(i) Start with $M=0$;

(ii) From the top left corner, fill the first $q$ spaces on the diagonal with
$a_{1},\ldots,a_{q}$ respectively;

(iii) Fill up the first $q-1$ elements of the superdiagonal with $p^{e_{l}}-1$
entries of $a_{l}$, then $p^{e_{l-1}}-1$ entries of $a_{l-1}$ and so on;

(iv) If one has now written down $p^{e_{q+1}}-1$ entries $a_{q}$, then stop.
Otherwise put $a_{q}$ in the $q$th position of the superdiagonal. If, after
this, there are only $p^{e_{q+1}}-1-\tau$ entries $a_{q}$ with $\tau\geq 1$ on
the superdiagonal, we set also the $(q+1)$st diagonal element equal to
$a_{q}$.

We claim $M^{\operatorname{m}(k^{\prime}/k)-1}\neq 0$. Take the sequence

$w=(\underbrace{1,\ldots,1}_{p^{e_{1}}\text{
times}},\underbrace{2,\ldots,2}_{p^{e_{2}}\text{
times}},\ldots,\underbrace{q,\ldots,q}_{p^{e_{q}}\text{
times}},\underbrace{q+1,\ldots,q+1}_{\tau+1\text{ times}})$

and observe that $a_{w}\neq 0$. A little thought shows that this is the unique
sequence $w$ of this length with $a_{w}\neq 0$ starting at $1$ and ending at
$q+1$. This proves the claim. Hence we conclude $e(R\cap
B_{\bar{k}})\geq\operatorname{em}(k^{\prime}/k)=\operatorname{E}(k^{\prime}/k,r)$.
Since $M^{\operatorname{m}(k^{\prime}/k)}=0$ by Lemmas 2.11 and 2.5, we have
$e(R\cap B_{\bar{k}})\leq\operatorname{em}(k^{\prime}/k)$ and so this case is
done.

Now suppose $\sum_{i=r+1}^{l}(p^{e_{i}}-1)\geq r-1$. One constructs a similar
$M$. Start as before with $M=0$. One fills the diagonal with
$a_{1},\ldots,a_{r}$ and the superdiagonal with any $r-1$ elements from the
list $(\underbrace{a_{r+1},\ldots,a_{r+1}}_{p^{e_{r+1}}-1\text{
times}},\ldots,\underbrace{a_{l},\ldots,a_{l}}_{p^{e_{l}}-1\text{ times}})$.
This gives a matrix $M$ such that $M^{m}\neq 0$ where
$m=\sum_{i=1}^{r}(p^{e_{i}}-1)+r-1=\operatorname{mr}(k^{\prime}/k,r)-1$. Hence
$e(R\cap B_{\bar{k}})\geq\operatorname{emr}(k^{\prime}/k,r)$.

We need to demonstrate the upper bound for this case. Take a matrix $M\in
B^{\prime}(\mathfrak{m})$ and any sequence $w$ of length
$\operatorname{mr}(k^{\prime}/k)$ such that $a_{w}\neq 0$ in a product of
$\operatorname{mr}(k^{\prime}/k,r)$ copies of $M$. Suppose the entries on the
diagonal of $M$ are $m_{1},\ldots,m_{r}\in\mathfrak{m}$. Then
$a_{w}=m_{1}^{f_{1}}\cdots m_{r}^{f_{r}}\cdot\mu$ for non-negative integers
$f_{1},\ldots,f_{r}$ and $\mu$ a product of $s$ entries from above the
diagonal; we have $s\leq r-1$. Since $\operatorname{mr}(k^{\prime}/k)=\sum
f_{i}+s$, we have $\sum f_{i}\geq\operatorname{mr}(k^{\prime}/k,r)-r+1$. We
are now done by Lemma 2.11.

For part (iii) of the theorem, we note that any lower bound for $e(R\cap
B_{\bar{k}})$ is automatically one for $e(R)$. For part (iv) we just observe
that $\operatorname{E}(k^{\prime}/k,r)=\operatorname{em}(k^{\prime}/k)$ is
also an upper bound for $e(R)$ by Lemmas 2.11 and 2.5.∎

##  4\. Extended example of $\operatorname{SL}_{2}$ and
$\operatorname{PGL}_{2}$

In this section we give a complete description of $e(R)$ in case
$G^{\prime}=\operatorname{SL}_{2},\operatorname{PGL}_{2}$ or
$\operatorname{GL}_{2}$.

######  Proposition 4.1.

Let $k^{\prime}/k$ be purely inseparable of characteristic $p$ and exponent
$e$. Let $G$ be $\mathrm{R}_{k^{\prime}/k}(\operatorname{GL}_{2})$,
$\mathrm{R}_{k^{\prime}/k}(\operatorname{PGL}_{2})$ or
$\mathrm{R}_{k^{\prime}/k}(\operatorname{SL}_{2})$, the last of these only if
$p\neq 2$. Then $e(R)$ is $e$ if $k^{\prime}/k$ is simple and $e+1$ otherwise.

###### Proof.

If $G^{\prime}$ is $\operatorname{GL}_{2}$, then Theorem 1.1(iv) gives the
exponent $e(R)=e$ for $k^{\prime}/k$ simple of exponent $e$, and at least
$e+1$ when $k^{\prime}/k$ is non-simple. However, by Theorem 1.1(i) we have
$e(R)$ at most $e+1$. As the map
$\operatorname{GL}_{2}\to\operatorname{PGL}_{2}$ is smooth, and
$\operatorname{SL}_{2}\subset\operatorname{GL}_{2}$, we immediately obtain $e$
or $e+1$ respectively as upper bounds for both the cases
$G^{\prime}=\operatorname{SL}_{2}$ and $G^{\prime}=\operatorname{PGL}_{2}$. In
fact, the matrices constructed in the proof of Theorem 1.1(ii)-(iv) and their
relevant powers are clearly non-zero in their image in
$\operatorname{PGL}_{2}(\mathfrak{B})$ and so the exponents of the radicals in
case $G^{\prime}=\operatorname{GL}_{2}$ and
$G^{\prime}=\operatorname{PGL}_{2}$ coincide. Now by Proposition 2.3(iv) these
also coincide for $\operatorname{SL}_{2}$ outside of characteristic $2$, since
the isogeny $f:\operatorname{SL}_{2}\to\operatorname{PGL}_{2}$ is smooth. ∎

We see some different behaviour for $\operatorname{SL}_{2}$ in characteristic
$2$; in particular, the exponents of $R_{u}(G_{\bar{k}})\cap
R_{u}(B_{\bar{k}})$ and $R_{u}(G_{\bar{k}})$ do not coincide.

Recall that a unital ring $R$ has characteristic $2$ if the equation $1+1=0$
holds in $R$. We prove a general formula for $2^{s}$-th powers of matrices
over rings of characteristic $2$.

######  Lemma 4.2.

Let $R$ be a commutative ring of characteristic $2$ and
$M=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ with $a,b,c,d\in R$. Then for all
$s\in\mathbb{N}_{0}$ we have

$\displaystyle
M^{2^{s}}=\begin{pmatrix}a^{2^{s}}+\sum_{i=0}^{s-1}b^{2^{i}}c^{2^{i}}t^{2^{s}-2^{i+1}}&bt^{2^{s}-1}\\\
ct^{2^{s}-1}&d^{2^{s}}+\sum_{i=0}^{s-1}b^{2^{i}}c^{2^{i}}t^{2^{s}-2^{i+1}}\end{pmatrix},$

where $t=a+d$.

###### Proof.

We argue by induction on $s$. For $s=0$, we just get $M$ on both sides. So
assume the result for $s$ and set
$l_{s}=\sum_{i=0}^{s-1}b^{2^{i}}c^{2^{i}}t^{2^{s}-2^{i+1}}$. We note that

$\displaystyle
l_{s}^{2}+bct^{2^{s+1}-2}=\sum_{i=1}^{s}b^{2^{i}}c^{2^{i}}t^{2^{s+1}-2^{i+1}}+bct^{2^{s+1}-2}=l_{s+1}$

holds. And so we get

$\displaystyle M^{2^{s+1}}=(M^{2^{s}})^{2}$
$\displaystyle=\begin{pmatrix}a^{2^{s+1}}+l_{s}^{2}+bct^{2^{s+1}-2}&bt^{2^{s}-1}(a^{2^{s}}+d^{2^{s}})\\\
ct^{2^{s}-1}(a^{2^{s}}+d^{2^{s}})&d^{2^{s+1}}+l_{s}^{2}+bct^{2^{s+1}-2}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}a^{2^{s+1}}+l_{s+1}&bt^{2^{s+1}-1}\\\
ct^{2^{s+1}-1}&d^{2^{s+1}}+l_{s+1}\end{pmatrix}.\qed$

######  Proposition 4.3.

Let $G^{\prime}=\operatorname{SL}_{2}$, $p=2$ and suppose $k^{\prime}/k$ has
normal generating sequence $\alpha_{1},\ldots,\alpha_{l}$ with exponents
$e_{1}\geq e_{2}\geq\ldots\geq e_{l}$. Let $B^{\prime}$ be an upper Borel
subgroup of $G^{\prime}$ and $B=\mathrm{R}_{k^{\prime}/k}(B^{\prime})$ a
pseudo-Borel.

(i) We have $e(R\cap B_{\bar{k}})=e$;

(ii) We have $e(R)=\begin{cases}e+1\text{ if }e_{1}=e_{2}\\\ e\text{
otherwise.}\end{cases}$

###### Proof.

We first calculate the exponent of $B^{\prime}(\mathfrak{m})=\ker
B^{\prime}(\mathfrak{B})\to B^{\prime}(k^{\prime})$. A general element is
$I+M$ with $M=\begin{pmatrix}a&b\\\ 0&d\\\ \end{pmatrix}$, where
$a,b,d\in\mathfrak{m}=\langle a_{1},\ldots,a_{l}\rangle$, satisfying
$a+d+ad=0$ from the condition $\det(I+M)=1$. As $a\in\mathfrak{m}$, $1+a$ is
invertible in $\mathfrak{B}$, and so it follows that $d=a(1+a)^{-1}$ and
$a+d=ad=a^{2}(1+a)^{-1}$. By Lemma 4.2, the $2^{s}$-th power of this matrix is

$\begin{pmatrix}a^{2^{s}}&b(a^{2}(1+a)^{-1})^{2^{s}-1}\\\ 0&d^{2^{s}}\\\
\end{pmatrix}.$

Choosing for example $a=a_{1}=1\otimes\alpha_{1}-\alpha_{1}\otimes 1$, one
sees the minimal $s$ for which $M^{2^{s}}=0$ is at least $e_{1}=e$ and also
that if $s\geq e$ then the diagonal entries vanish. For $s=e$, the off-
diagonal entry is a multiple of $(a^{2})^{2^{e}-1}$, which has at least
$2^{e}$ factors of $a$, hence is also. So $e(R\cap B_{\bar{k}})=e$, proving
(i).

For (ii), a general element of $G^{\prime}(\mathfrak{m})$ is a matrix $I+M$
where $M=\begin{pmatrix}a&b\\\ c&d\\\ \end{pmatrix}$, satisfying
$a+d+ad+bc=0$, since $\det(I+M)=1$. If $l=1$, (i.e. $k^{\prime}/k$ is simple
of exponent $e=e_{1}$), then it is clear that $e(R)=e$. Suppose $e_{2}\leq
e_{1}-1=e-1$. We may write $d=(1+a)^{-1}(a+bc)$, hence
$t=a+d=(1+a)^{-1}(a^{2}+bc)$. Therefore
$t^{2^{e-1}}=(1+a)^{-2^{e-1}}(a^{2^{e}}+b^{2^{e-1}}c^{2^{e-1}})=(1+a)^{-2^{e-1}}(b^{2^{e-1}}c^{2^{e-1}})$.
Now by Lemma 2.10 we have $m^{2^{e-1}}\in\langle
a_{1}^{2^{e-1}}\rangle\subseteq\mathfrak{m}$ for any $m\in\mathfrak{m}$. It
follows that $b^{2^{e-1}}c^{2^{e-1}}$ has a factor of $a_{1}^{2^{e-1}}\cdot
a_{1}^{2^{e-1}}=a_{1}^{2^{e}}=0$. By the same token, $t^{2^{e-1}}=0$ and
hence, using the lemma, $M^{2^{e}}=0$. Therefore $e(R)=e$ as required.

Now suppose $e_{1}=e_{2}$. Since
$G\subseteq\mathrm{R}_{k^{\prime}/k}(\operatorname{GL}_{2})$ we have $e(R)\leq
e+1$. Set $b=a_{1}$, $c=a_{2}$, $d=0$ and $a=a_{1}a_{2}$. Then

$M^{2^{e}}=\begin{pmatrix}(a_{1}a_{2})^{2^{e}}+\sum_{i=1}^{e-1}(a_{1}a_{2})^{2^{i}+2^{e}-2^{i+1}}&a_{1}(a_{1}a_{2})^{2^{e}-1}\\\
a_{2}(a_{1}a_{2})^{2^{e}-1}&\sum_{i=1}^{e-1}(a_{1}a_{2})^{2^{i}+2^{e}-2^{i+1}}\\\
\end{pmatrix}$

The off-diagonal entries are zero, but
$\sum_{i=1}^{e-1}(a_{1}a_{2})^{2^{i}+2^{e}-2^{i+1}}=\sum_{i=1}^{e}(a_{1}a_{2})^{2^{e}-2^{i}}$
and the $i=e-1$ entry is $(a_{1}a_{2})^{2^{e-1}}$. Provided $e_{2}=e_{1}$,
this element is non-zero by the same argument as in the proof of Lemma 2.11—in
a $k$-basis of $\mathfrak{B}$, the coefficient of
$1\otimes\alpha_{1}^{2^{e-1}}\alpha_{2}^{2^{e-1}}$ in $(a_{1}a_{2})^{2^{e-1}}$
is evidently $1$. It follows that $M^{2^{e}}\neq 0$ and $e(R)=e+1$. ∎

##  5\. Simple groups of rank at least $2$

For a positive integer $r$, recall that the algebraic $k$-group $\mu_{r}$ is
the functor returning the $r$-th roots of unity in any $k$-algebra $A$; it is
the kernel of the $r$-power map
$\phi_{r}:\operatorname{\mathbb{G}_{m}}\to\operatorname{\mathbb{G}_{m}};x\mapsto
x^{r}$. (We have $\mu_{r}$ étale if and only if $p\nmid r$.) Say a
$k^{\prime}$-group $L$ is a _skewed $\operatorname{GL}_{n}$_ if
$L\cong\operatorname{GL}_{n}/K$, where $K\cong\mu_{r}$, with $r\nmid n$; the
quotient by $\mu_{r}$ induces the $r$-power map on the central torus of
$\operatorname{GL}_{n}$. Since $K\cap\operatorname{SL}_{n}=1$, the derived
subgroup $\mathscr{D}(L)$ of $L$ is still isomorphic to
$\operatorname{SL}_{n}$ and notably there is still a central torus $C\subset
L$ which contains $Z(\operatorname{SL}_{n})$. In [12, Ex. 5.17] it is
explained that if $\mu_{n}\subset T$ is a finite subgroup of a split $k$-torus
of rank at least $2$, then any automorphism of $\mu_{n}$ (which is some
$r$-power map on $\mu_{n}$ with $(r,n)=1$) lifts to an automorphism of $T$. In
our case, the quotient map by $K\times 1$ on
$\operatorname{GL}_{n}\times\operatorname{\mathbb{G}_{m}}$, lifts to an
automorphism of $\operatorname{GL}_{n}\times\operatorname{\mathbb{G}_{m}}$
defined on the central torus
$\operatorname{\mathbb{G}_{m}}\times\operatorname{\mathbb{G}_{m}}$ by
$(x,y)\mapsto(x^{r}y^{-n},x^{b}y^{a})$ where $a$ and $b$ are integers such
that $ar+bn=1$. It follows that if $L$ is a skewed
$\operatorname{GL}_{n}$-group, then
$L\times\operatorname{\mathbb{G}_{m}}\cong\operatorname{GL}_{n}\times\operatorname{\mathbb{G}_{m}}$.

The group $\operatorname{SL}_{n+1}$ contains a (Levi) subgroup
$L:=\operatorname{GL}_{n}$ embedded block diagonally via
$g\mapsto\operatorname{diag}(g,\det{g}^{-1})$. If $G^{\prime}$ is a split
simple $k$-group of type $A_{n}$ then
$G^{\prime}\cong\operatorname{SL}_{n+1}/\mu_{r}$ with $\mu_{r}\subseteq
Z(\operatorname{SL}_{n+1})\cong\mu_{n+1}$, so $r\mid n+1$. But then $r\nmid
n$, hence $L$ maps under the quotient by $\mu_{r}$ to a skewed
$\operatorname{GL}_{n}$-subgroup of $G^{\prime}$. This observation establishes
the type $A$ case of the following lemma.

######  Lemma 5.1.

Let $G^{\prime}$ be a split simple $k^{\prime}$-group of rank $r$. Then one of
the following holds:

  1. (i)

$G^{\prime}$ contains a skewed $\operatorname{GL}_{r}$ subgroup $L$;

  2. (ii)

There are étale isogenies $\pi:L\to L/Z$, and $\phi:\operatorname{GL}_{r}\to
L/Z$ or $\phi:L\to\operatorname{GL}_{r}$;

  3. (iii)

$p=2$ and either $G^{\prime}$ is of type $F_{4}$ or $G^{\prime}$ is of type
$B$, $C$ or $D$, but not isomorphic to $\operatorname{SO}_{2r}$,
$\operatorname{SO}_{2r+1}$ or $\operatorname{Sp}_{2r}$;

  4. (iv)

$p=3$ and $G$ is of type $E_{6}$.

In cases (i) and (ii) if we let
$R=\mathscr{R}_{u}(\mathrm{R}_{k^{\prime}/k}(L)_{\bar{k}})$, then we have
$e(R)=e(\mathscr{R}_{u}(\mathrm{R}_{k^{\prime}/k}(\operatorname{GL}_{r})_{\bar{k}}))$.

###### Proof.

If $G^{\prime}=\operatorname{SO}_{2r}$, $\operatorname{SO}_{2r+1}$ or
$\operatorname{Sp}_{2r}$ then [7, 4.5] promises a Levi subgroup
$L\cong\operatorname{GL}_{r}$ as required so (i) holds in all these cases. The
central isogenies $\operatorname{Sp}_{2r}\to\operatorname{PSp}_{2r}$,
$\operatorname{Spin}_{n}\to\operatorname{SO}_{n}$,
$\operatorname{SO}_{n}\to\operatorname{PSO}_{n}$,
$\operatorname{Spin}_{2r}\to\operatorname{HSpin}_{2r}$ or
$\operatorname{HSpin}_{2r}\to\operatorname{PSO}_{2r}$ (where
$\operatorname{HSpin}_{2r}$ occurs only if $n$ even) are all quotients by
central subgroups isomorphic to $\mu_{2}$, hence are étale maps when $p>2$ and
hence (ii) holds for all classical groups when $p>2$. Otherwise (iii) holds.

Let now $G^{\prime}$ be exceptional. Note that groups of type $E_{7}$ and
$E_{8}$ contain simple $A_{7}$ and $A_{8}$ maximal rank subgroups, evident
from the Borel–de-Siebenthal algorithm. Hence each contains a skewed
$\operatorname{GL}_{7}$ and $\operatorname{GL}_{8}$ subgroup $L$ respectively,
satisfying (i) as required. By [8, Thm. A], any Levi subgroup $L$ of $G_{2}$
is isomorphic to $\operatorname{GL}_{2}$ (rather than
$\operatorname{SL}_{2}\times\operatorname{\mathbb{G}_{m}}$, for example),
hence satisfies (i). If $G^{\prime}$ is of type $F_{4}$, then we take $L$ to
be an $A_{3}$ Levi subgroup of the $\operatorname{Spin}_{9}$ subgroup of
$G^{\prime}$. Now $L$ admits an étale isogeny with the $\operatorname{GL}_{4}$
Levi subgroup of
$\operatorname{SO}_{9}\cong\operatorname{Spin}_{9}/Z(\operatorname{Spin}_{9})$
if $p\neq 2$; so $L$ satisfies (ii) in that case, and if $p=2$ then $L$ is
listed in (iii). It remains to deal with the case $G^{\prime}=E_{6}$. Using
Bourbaki notation [3], we take $L$ to be the $A_{5}$-Levi subgroup
corresponding to the nodes $\\{1,3,4,5,6\\}$ of the Dynkin diagram explicitly.
Assume first $G^{\prime}$ is simply connected. Then $\mathscr{D}(L)$ is simply
connected. Since $L$ centralises the $\operatorname{SL}_{2}$ subgroup
generated by the $\pm\tilde{\alpha}$ root groups, where $\tilde{\alpha}$ is
the longest root, the central torus $C$ of $L$ is the image of the cocharacter
$h_{\tilde{\alpha}}:\operatorname{\mathbb{G}_{m}}\to G^{\prime};t\mapsto
h_{\tilde{\alpha}}(t)$. One calculates
$h_{\tilde{\alpha}}(t)=h_{\alpha_{1}}(t)h_{\alpha_{2}}(t^{2})h_{\alpha_{3}}(t^{2})h_{\alpha_{4}}(t^{3})h_{\alpha_{5}}(t^{2})h_{\alpha_{6}}(t)$.
This is inside $\mathscr{D}(L)\cong\operatorname{SL}_{6}$ iff $t^{2}=1$;
therefore $\mathscr{D}(L)\cap C\cong\mu_{2}$—so $L$ is not a skewed
$\operatorname{GL}_{r}$. However, if $p\neq 3$ then there is an étale isogeny
from $L$ to $L/Z\cong\operatorname{GL}_{6}$ where $Z$ is a diagonal copy of
$\mu_{3}$ in both factors $\mathscr{D}(L)$ and $C$ of $L$, so that (ii) holds.
Besides which, since $Z(G^{\prime})\cong\mu_{3}$, then if $p\neq 3$, the
quotient by $Z(G^{\prime})$ is étale and $L/Z(G^{\prime})$ satisfies (ii) for
the adjoint group $G/Z(G^{\prime})$. Otherwise $p=3$ and (iv) holds.

We prove the last statement. Suppose (ii) holds. Then we note that if
$\pi:A\to B$ is an étale central isogeny of reductive groups, then
$\mathrm{R}_{k^{\prime}/k}(A)\to\mathrm{R}_{k^{\prime}/k}(B)$ is an isogeny
whose kernel has trivial intersection with the unipotent radical. As in
Proposition 2.3(iv), it follows that the induced map
$\mathscr{R}_{u}(\mathrm{R}_{k^{\prime}/k}(A)_{\bar{k}})\to\mathscr{R}_{u}(\mathrm{R}_{k^{\prime}/k}(B)_{\bar{k}})$
is an isomorphism as required. If (i) holds, then $L$ is a skewed
$\operatorname{GL}_{r}$-subgroup and by the remarks preceding the lemma,
$L\times\operatorname{\mathbb{G}_{m}}\cong\operatorname{GL}_{r}\times\operatorname{\mathbb{G}_{m}}$.
Observe
$\mathscr{R}_{u}(\mathrm{R}_{k^{\prime}/k}(L\times\operatorname{\mathbb{G}_{m}})_{\bar{k}})\cong\mathscr{R}_{u}(\mathrm{R}_{k^{\prime}/k}(L)_{\bar{k}})\times\mathscr{R}_{u}(\mathrm{R}_{k^{\prime}/k}(\operatorname{\mathbb{G}_{m}})_{\bar{k}})$;
since $L$ already contains $C\cong\operatorname{\mathbb{G}_{m}}$, the exponent
of the product is equal to that of the first factor, $e(R)$. The same argument
with $\operatorname{GL}_{r}\times\operatorname{\mathbb{G}_{m}}$ in place of
$L\times\operatorname{\mathbb{G}_{m}}$ implies the claim. ∎

###### Proof of Corollary 1.3.

With the hypotheses on $p$, by Lemma 5.1 we get a reductive subgroup $L$ of
$G^{\prime}$, with

$e(\mathscr{R}_{u}(\mathrm{R}_{k^{\prime}/k}(L))_{\bar{k}})=e(\mathscr{R}_{u}(\mathrm{R}_{k^{\prime}/k}(\operatorname{GL}_{r}))_{\bar{k}})\geq\operatorname{E}(k^{\prime}/k,r).$

By Proposition 2.3(iii) it follows that
$e(R)\geq\operatorname{E}(k^{\prime}/k,r)$. If
$\sum_{i=r+1}^{l}(p^{e_{i}}-1)<r-1$, then Weil restricting a faithful
representation $G^{\prime}\to\operatorname{GL}_{n}$ from $k^{\prime}$ to $k$
gives an embedding of $G$ in $\operatorname{GL}_{m}$ with
$m=[k^{\prime}:k]\cdot n\geq r$. Now by Proposition 2.3(iii) again, $e(R)\leq
e(\mathscr{R}_{u}(\mathrm{R}_{k^{\prime}/k}(\operatorname{GL}_{m}))_{\bar{k}})$
which is equal to
$\operatorname{E}(k^{\prime}/k,m)=\operatorname{em}(k^{\prime}/k)$ by Theorem
1.1.∎

Acknowledgments: We thank Michael Bate and Gerhard Röhrle for helpful comments
on early versions of the paper. Many thanks to Jay Taylor for discussions
about root data. Finally, thanks to the referee for suggesting improvements to
the exposition.

## References

  * [1] Michael Bate, Benjamin Martin, Gerhard Röhrle, and David I. Stewart, _On unipotent radicals of pseudo-reductive groups_ , Michigan Math. J. 68 (2019), no. 2, 277–299. MR 3961217
  * [2] Nicolas Bourbaki, _Algebra I. Chapters 1–3_ , Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998, Translated from the French, Reprint of the 1989 English translation [ MR0979982 (90d:00002)]. MR 1727844
  * [3] by same author, _Lie groups and Lie algebras. Chapters 4–6_ , Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002, Translated from the 1968 French original by Andrew Pressley. MR 1890629
  * [4] Brian Conrad, Ofer Gabber, and Gopal Prasad, _Pseudo-reductive groups_ , second ed., New Mathematical Monographs, vol. 26, Cambridge University Press, Cambridge, 2015. MR 3362817
  * [5] Brian Conrad and Gopal Prasad, _Classification of pseudo-reductive groups_ , Annals of Mathematics Studies, vol. 191, Princeton University Press, Princeton, NJ, 2016. MR 3379926
  * [6] James E. Humphreys, _Linear algebraic groups_ , Springer-Verlag, New York-Heidelberg, 1975, Graduate Texts in Mathematics, No. 21. MR 0396773
  * [7] J. C. Jantzen, _Nilpotent orbits in representation theory_ , Lie theory, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1–211. MR 2042689 (2005c:14055)
  * [8] Peter B. Kleidman, _The maximal subgroups of the Chevalley groups $G_{2}(q)$ with $q$ odd, the Ree groups ${}^{2}G_{2}(q)$, and their automorphism groups_, J. Algebra 117 (1988), no. 1, 30–71. MR 955589
  * [9] Joseph Oesterlé, _Nombres de Tamagawa et groupes unipotents en caractéristique $p$_, Invent. Math. 78 (1984), no. 1, 13–88. MR 762353
  * [10] Günter Pickert, _Eine Normalform für endliche reininseparable Körpererweiterungen_ , Math. Z. 53 (1950), 133–135. MR 37837
  * [11] Richard Rasala, _Inseparable splitting theory_ , Trans. Amer. Math. Soc. 162 (1971), 411–448. MR 284421
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# Computer Science for Future
Sustainability and Climate Protection in the Computer Science Courses of the
HAW Hamburg

Elina Eickstädt  Martin Becke  Martin Kohler  Julia Padberg111email:
<EMAIL_ADDRESS>
Hamburg University of Applied Sciences

###### Abstract

Computer Science for Future (CS4F) is an initiative in the Department of
Computer Science at HAW Hamburg. The aim of the initiative is a paradigm shift
in the discipline of computer science, thus establishing sustainability goals
as a primary leitmotif for teaching and research. The focus is on teaching
since the most promising multipliers are the students of a university. The
change in teaching influences our research, the transfer to business and civil
society as well as the change in our own institution. In this article, we
present the initiative CS4F and reflect primarily on the role of students as
amplifiers in the transformation process of computer science.

###### Contents

  1. 1 Computer Science is Cross-Sectional
  2. 2 CS4F Experiments
  3. 3 Teaching as a Multiplier
  4. 4 Ongoing Activities

##  1 Computer Science is Cross-Sectional

The scientific discipline of computer science has played a steadily growing
role in shaping society over the past decade. Computer science defines
processes in companies, increasingly models modes of interaction in society
and determines how services and goods are handled. In an increasingly granular
networked and automated world, the infrastructure of our coexistence is
determined by decisions made in computer science.

In short, computer science has affected, if not already permeated, almost
every area of our existence. The success of the digital technologies in recent
decades has been strongly influenced by ideas and concepts of the Silicon
Valley, which, with mantras like “move fast and break things”, have shaped an
understanding of computer science as a disruptive technology supplier. Slogans
such as “make the world better” were often misunderstood in simply doing it
differently.

This attitude is increasingly being challenged, and there is a growing
recognition that digitisation and informatics are systematically integrated
into the Global sustainability should be at the service of global
sustainability. The relationship between informatics and sustainability is
also the focus of various science policy organisations (e. g. the
”‘Gesellschaft für Informatik”’222https://informatik2021.gi.de/informatik2021
(visited 2. 1. 2023) (Society for Informatics), ”‘Bits und
Bäume”’333https://fahrplan22.bits-und-baeume.org/bitsundbaeume/ (visited 2. 1.
2023) (Bits and Trees), FIfF444Forum Computer Scientists for Peace and Social
Responsibility (visited 2. 1. 2023) ,ISOC.DE555Internet Society, German
Chapter (visited 2. 1. 2023) ,netzpolitik.org666netzpolitik. org (visited 2.
1. 2023)  or within the UN777Promotion and protection of all human rights,
civil, political, economic, social and cultural rights, including the right to
development  (A/HRC/32/L. 20)  (visited 2. 1. 2023)).

This change can also be seen in research, for example, the number of research
publications on net zero emissions targets increased to 1. 6 million between
2001 and 2020, representing 4.9% of global publications compared to 1.2% in
2001 [3].

Although there are already various approaches to establish sustainability
aspects in higher computer science education, e.g. [14, 13, 4, 1, 12] these
questions are still far from being established in teaching. It must be
questioned whether today’s computer scientists have sufficient knowledge to
consider other targets such as efficiency, optimization or innovation. In the
context of current global challenges such as climate change, decisions in
information systems can have far-reaching consequences. For example, the
selection of the transaction technology can have a significant impact on the
CO2 fingerprint [8].

These costs are rarely part of the requirements process because there is
little or no knowledge about the dependencies that can be expected. Much
depends on the use to which the technology is optimized and the extent to
which users and producers are aware of its influence. The avoidance of
greenhouse gases in the context of climate change is only a very small part of
what can be achieved with new objectives, new projects and new tools. Teaching
is the our main focus, but its content hast to be kept up to date basically
through the connection to research.

##  2 CS4F Experiments

In the Department of Computer Science at the Hamburg University of Applied
Sciences (HAW), we would like to react to the demands of our time and have
launched initiatives at the beginning of 2022 that have found a conceptual
superstructure under the title Computer Science for Future (CS4F). Through
CS4F, the Department of Computer Science would like to initiate a paradigm
shift with its teachers and researchers, with the staff and students, and not
least with the existing laboratories.

With CS4F, we are aiming for a paradigm shift in the discipline of computer
science by establishing sustainability goals as the primary guiding principle
for teaching and research. In order to address these sustainability goals with
diverse methodologies, in heterogeneous groupings and with different
objectives, we want to conceive of concrete strategies, measures and
activities as experiments that are subject to permanent feedback and gain our
insights in small, tangible experiments. We accept our responsibility and
focus on the future viability of computer science. That is why CS4F launches a
transformation process to address the imperative of climate protection and
sustainability from the perspective of informatics:

enhancement of the curriculum:

In teaching, new courses are to be developed as well as existing ones adapted
to meet the requirements of the CS4F.

permanent integration into research and organization:

It is important to create reward systems for the respective institution, as
well as to establish structures that consolidate the orientation of CS4F in
the long term.

communication with others:

The aim is to offer points of exchange and linkages in order to set in motion
a multilateral process of problem awareness and action awareness.

The process of change will affect different aspects on a large and small scale
and thus take place at different levels of granularity. In particular, CS4F
derives its focus from the Sustainable Development Goals set by the United
Nations, also known as the Sustainable Development Goals (SDGs).

However, CS4F does not only see itself as responsible for minimising damage,
but also wants to make clear to all designers of and participants in
information technologies the high level of responsibility that goes hand in
hand with their use and offer solutions to meet this responsibility.
Strategically, CS4F thus pursues various goals, which, as illustrated in
Figure 1, unfold the desired effect in the interaction of different sub-areas:
teaching, research, transfer and civil society initiatives. Thus, the CS4F
initiative is also revitalised by the students who have pursued a different
educational focus. The figure illustrates that teaching is the focus and
triggers further work in the other sub-areas of research, transfer and civil
society initiatives.

Figure 1: Effect of CS4F

For research in particular, it is to be expected that research tasks will
result from questions in teaching. Teaching must result from established
scientific foundations that can be addressed in the context of research on
sustainability topics in the department. Currently, the teaching material for
wide areas of sustainability and computer science can be considered
insufficient.

In the area of transfer, the teaching has a direct effect firstly through our
graduates, who take the knowledge of connections between computer science and
sustainability with them to their future jobs. In addition, we strive for
cooperations within the framework of the interdisciplinary laboratory
”Creative Space For Technical Innovations” (CSTI)888https://csti.haw-
hamburg.de/(last visited 6.6.2022), which result from approaches to solutions
from research and teaching.

But teaching is at the heart of the changes. One of the main goals of CS4F is
to strengthen the motivation and problem awareness of students, the future
developers. We see the students at the universities as multipliers in order to
achieve sustainable computer science in the long term.

The transformation process in the Department of Computer Science driven by
CS4F is supported by all those who participate, i.e. students, staff and
professors, who are involved in the following areas ideas of sustainability
(from the perspective of computer science), especially climate protection.
CS4F thus serves the representation, information and coordination in order to
develop to develop visions and goals, to implement and support measures and to
promote continuity and funding. The activation of students is a central
component of the initiative in order to generate activity from the intrinsic
motivation. CS4F builds on the demands and initiatives of Fridays for Future
or Students for Future and provides already active students with a framework
in which ideas for implementation in computer science can be scientifically
expressed and discussed. Furthermore, students who have not been involved so
far should be informed about the effects of their decisions and involved in
the discussion. For CS4F, strengthening self-responsibility is an essential
key to establishing new goals, i. e. in the creation, operation and
safeguarding of software and hardware.

##  3 Teaching as a Multiplier

With Computer Science for Future (CS4F), the Department of Computer Science
aims to become sustainable in the sense of the UN’s Sustainable Development
Goals. This target is demanding: very comprehensive, very complex and also
very controversial. Therefore, its implementation requires an adaptable
transformation process that affects teaching, research and our infrastructure.
Teaching has a special role to play here, because our students represent both
the greatest innovative potential and the greatest multiplier. Both forces can
be used in our teaching and integrated into the transformation. The teachers
see themselves as learning coaches in the sense of [5] and focus their
teaching more on education for sustainable development as well as on the
concrete social challenges and on interdisciplinarity in teaching.

### Research Learning

In [6] it is stated that digitisation as well as education for sustainable
development in the field of higher education needs such teaching/learning
formats to provide students with those competences that are needed to solve
problems in the future that are currently not yet known. The concept of
inquiry learning seems suitable to combine both directions:

  * •

collaborative learning processes include learning and knowledge achieved in
non-hierarchical groups, as well as.

  * •

dealing with complexity and uncertainties and reflective questioning also aim
at action competences of sustainable development.

The experience of the last two years of the pandemic has clearly shown that
collaborative learning under poor conditions can lead to more stress among
students. CS4F will critically reflect on how to design the curriculum to give
students space to think and time to engage with sustainability issues. This
also applies to the implementation of collaborative learning in the true
sense. This requires a constant exchange with students as well as a flexible
adaptation of teaching. Existing structures must be strengthened and expanded.
The members of the interdisciplinary laboratory ”Creative Space For Technical
Innovations” (CSTI) are contact persons for research, teaching and transfer
and, due to the experience in research-based teaching already available in the
laboratory, an important initial point of reference for the desired
transformation.

In [7] a suitable classification of the possibilities of implementation is
given (see Figure 2). It is particularly profitable that different target
groups, also with different objectives and backgrounds, can be addressed.

Figure 2: Research Learning

### Transformative Science

The serious turn towards integrating the UN’s Sustainable Development Goals
(SDGs) and accepting the societal role that informatics plays in shaping the
world and its relationships also means transforming what we understand as
science principles. The German Advisory Council on Global Change (WBGU) made
this clear programmatically as early as 2011 with the catchphrase
”transformative research”[2]. In the wake of global crises and the need for
rapid change, knowledge gain and theory building must themselves become part
of the transformation. The aim of transformation research is to describe,
explain, evaluate and support transformations towards a sustainable society. A
transformative science supports and is itself part of such transformation
processes and can thus contribute to change in a very concrete way[10]. The
advantage is also that science can and must change itself under the impression
of the examined processes, as it also offers a deeper understanding and
possibility to examine inner regularities and processes as part of the
transformation movement. Transformative questions can influence the processes
(what changes within transformations), the dynamics of change (how do
transformation processes take place) or the driving forces (through what / by
whom are transformation processes supported). Sustainability transformations
are open, non-linear processes characterised by high uncertainty of prediction
[11]. By embedding such processes, systemic thinking, concrete questions and
transdisciplinary working methods are necessary. The real-world reference and
the inclusion of different, also non-academic experts and voices favours
research formats such as workshops, reallabs or field experiments. For
computer science, this means a further strengthening of agile and incremental
approaches. Beyond this, however, is the involvement of civil society and
individual non-academic actors. The clarification of relevance (what is
important for a successful transformation instead of what is important for the
scientific discipline) benefits from cooperations in the sense of a co-design
both for the relevant research questions and for the research goals.
Especially for the clarification of relevance (what is important for a
successful transformation instead of what is important for the scientific
discipline), such cooperations in the sense of co-research and co-design of
the research goals, but especially of the relevant research questions, are
essential.

With the CS4F initiative, we want to give the emerging activities and
initiatives an essential role in helping to clarify and develop what actually
needs to be researched and transformed. Here, research can then emerge from
teaching, with student initiatives becoming the questioners for research
projects or with interest groups conducting non-academic research.

### Persistence in Student Projects

At the heart of the activities around CS4F is the future viability of computer
science and the postulate that computer science students are the ones who
drive innovative and implementable solutions. Individual projects become
prototypical experiments that serve as concrete learning and practical
experiences that can lead to long-term change [9]. However, this is contrasted
by the short length of time students stay in such activities due to workload
and degree attainment. CS4F is therefore also an initially rudimentary
platform to bundle and continue student activities.

##  4 Ongoing Activities

Since the beginning of the year, we have started these initiatives and are
thus still in the first phase, the implementation of our trials. The focus of
the first stage in particular is on two goals. One is to increase visibility
internally and externally, and the other is to rebuild the curriculum within
the framework of the existing examination regulations . This generally means
creating communication platforms and new teaching offers to drive projects and
inform about your progress:

### Collaboration in CS4F

The initiatives and actions for the internal projects are developed and
communicated in this MS team CS4F Computer Science for Future. In this way, we
have established a communication structure which, among other things, serves
to

  * •

to publicise the current and planned measures,

  * •

to develop new measures from suggestions and ideas,

  * •

cooperation within the HAW and

  * •

to organise the cooperation with others.

In order to be able to respond to current requirements, the process is carried
out in an agile manner, i.e. measures are regularly discussed throughout the
department across all status groups, addressed and considered in
retrospectives. This includes initiatives in teaching, in research and in the
organisation of the Department of Computer Science.

### Podcast CS4F

Accompanying the implementation of the project is a podcast created by
students professors and employees of the university. In each episode, the team
talks to activists, experts and scientists about climate change mitigation,
sustainability in the context of computer science. The podcast also features
some basic discussions about ethical issues surrounding the impact of
technology on society. The aim of the podcast is on the one hand to report on
the work on the project, but also to give those interested in computer science
an insight into the different facets and possibilities of sustainable computer
science. In the best case, an intrinsic motivation can be triggered to become
active oneself. Of course, the podcast will also offer an interface to
directly address young students and to get in touch with schools and
companies. The goal is not to make the podcast primarily for insiders, but to
introduce listeners to the topics. It also focuses less on persuasion and more
on providing broad information about how computer science works and what
impact computer science can have on society.

### Climate Orientation Unit

CS4F has accompanied student activities from the context Students for Future
and supported them with its own offers in the Climate Orientation Unit (Klima
OE), a student event series. The Climate OE offer was cross-university and
various events were held in one week to draw attention to the topic of
sustainability, to show possible solutions and to discuss room for manoeuvre.
CS4F has formulated the position from within the department and fed it into
the process. In this discussion, the enormous need for an interdisciplinary
exchange once again became more than clear.

### Sustainability goals in courses

Currently, elective courses are held on various sustainability topics. These
cover the following topics, among others:

  * •

Relationships between human rights and protocols,

  * •

Challenges and solutions to the climate crisis

  * •

climate and information technology

  * •

ethical responsibility of computer scientists

  * •

machine ethics

  * •

environmental informatics

The relationship between human rights and protocols is relevant to our
teaching. protocols are relevant to our teaching, because protocols have an
effective influence on rights such as freedom of expression or the right of
assembly. right of assembly. The lecture series Our House Is On Fire by the
AstA of the University of HH and Fridays for Future is already taking place
for the third semester and focuses on the challenges and solutions of the
climate crisis. It is offered as an elective course in the bachelor’s degree
programmes in computer science. Another course focuses on climate and computer
science and examines the challenges in IT in the area of tension between the
environment, technology and the economy. One course focuses on the
responsibility of computer scientists and requires fundamental reflections on
computer science and its use. Another deals with machine ethics, i.e. the
question of what it means when machines are increasingly thought of and
constructed more autonomously and intelligently, so that their activities
become increasingly ethically relevant. There is also a seminar on
environmental informatics, which deals with the generation and processing of
environmental data as well as environmental processes.

In addition, we have started to introduce various CS4F topics in three
mandatory courses:

  * •

In the first semester, the course ”Fundamentals of Computer Engineering” was
expanded to include interdisciplinary concepts. In particular, concepts from
sociology were used to illustrate the contrast between solutionism and
technology criticism. In addition, a basic overview of ethics was given and
the role of ethics in computer science was discussed. This included an
introduction to various ethical theories, such as contractualism, autonomy
orientation, and John Rawls’ modern contract theory, and how these concepts
relate to computer science.

  * •

The course ”‘Fundamentals of Computer Science”’ takes place in the first
semester. The lecture has been extended to include the topic of sustainable
software development. Sustainable software development refers to the creation
of software that is sustainable and resource-efficient in the long term and
considers not only technical aspects, but also social, ecological and economic
factors. CO2 emissions associated with the use of ICT systems are a
significant contributor to global greenhouse gas emissions, and addressing
these emissions is an important part of efforts to combat climate change. By
implementing CO2-conscious computing practices, businesses, governments, and
individuals can help reduce their greenhouse gas emissions and contribute to a
more sustainable future.

  * •

In the third semester in the ”Applied Computer Science” program, graph theory
and graph algorithms are taught. These algorithms are used to analyzeanalyse
and optimize complex systems in areas such as transportation, energy and
communication. The lab will provide practice in implementing such algorithms,
but students will also learn to make their design decisions consciously and
communicate them appropriately. This semester, this approach has been extended
to make students aware of the potential socially relevant implications of
their own implementation. They are expected to learn about the potential
impact of their own work on society and to use ethical guidelines from the
”Gesellschaft für Informatik” (German Informatics Society) to address ethical
decisions and be aware of the potential consequences. Students should also
learn to communicate their work and findings in an understandable way and to
participate in societal discussions.

These cases are some of the learning formats developed or adapted under the
CS4F paradigm at the HAW. The CS4F framework is conceptualized as an open
platform for students and teachers alike to create and experiment with new
course modules, projects and learning formats that integrate sustainability
issues, transdisciplinary and cross-disciplinary thinking and concrete
projects tackling sustainability issues in or by the use of computer science.

The first results are promising and can develop into a solid path how to
integrate sustainability issues in the education of future computer scientists
and thus impacting the way these students act as professionals in their
respective fields.

## References

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