Standard block embeddings and natural-module semilinear obstructions

Problem

Must a standard homomorphism between elementary groups admit the printed description by a semilinear isomorphism of their original natural modules, or by a dual-module semilinear isomorphism in the correlation case?

\Lambda:E_n(R)\longrightarrow E_m(S),\qquad R^n\stackrel{?}{\cong}_{\sigma}S^m \quad\text{or}\quad R^n\stackrel{?}{\cong}_{\sigma}(S^m)^*.

The distinction between an isomorphism and an embedding with a supplementary summand is essential.

The standard block embedding

For a commutative ring $R$ , let $E_n(R)$ denote the subgroup of $\operatorname{GL}_n(R)$ generated by the elementary transvections

t_{ij}(a)=1+a e_{ij}\qquad (i\ne j,\ a\in R).

The standardness definition preceding Problem 10.44 of the Kourovka Notebook (1) allows a decomposition of the target module as $P^n\oplus Q$ . Problem 10.44 asks for a description by semilinear isomorphisms between the original natural modules, with a dual module in the correlation branch. A nonzero supplementary summand already obstructs this conclusion in the example below.

Put

k=\mathbb F_3,\qquad V=k^3,\qquad W=k^3\oplus k, \qquad W^*=\operatorname{Hom}_k(W,k).

For a unital ring homomorphism $\sigma:k\to k$ , a $\sigma$ -semilinear map $g$ is an additive map satisfying $g(av)=\sigma(a)g(v)$ .

Theorem 1. The map

\Lambda:E_3(k)\longrightarrow E_4(k),\qquad x\longmapsto \begin{pmatrix}x&0\\0&1\end{pmatrix},

is a standard injective homomorphism with nontrivial proper image. For every unital ring homomorphism $\sigma:k\to k$ , there is no bijective $\sigma$ -semilinear map $V\to W$ and no bijective $\sigma$ -semilinear map $V\to W^*$ . Thus the natural-module semilinear isomorphisms required by the literal formulation of Problem 10.44 do not exist for $\Lambda$ .

Proof. Block multiplication shows that $x\mapsto\operatorname{diag}(x,1)$ is a homomorphism on general linear groups. It is injective because its upper-left block is $x$ . For $1\leq i\ne j\leq3$ ,

\Lambda(t_{ij}(a))=t_{ij}(a)\in E_4(k),

so it restricts to the displayed elementary-group homomorphism. The element $\Lambda(t_{12}(1))$ is not the identity. Every matrix in the image has zero $(1,4)$ -entry, whereas $t_{14}(1)$ has $(1,4)$ -entry equal to $1$ . Hence the image is proper.

We specify the standardness data. Take

P=k,\qquad Q=k,\qquad B=k,\qquad \delta=1_k,\qquad f=1.

Here $B$ is the coefficient-image ring, and its embedding into $\operatorname{End}_k(P)$ is

\iota:B\xrightarrow{\ \cong\ }\operatorname{End}_k(P), \qquad a\longmapsto (z\mapsto az).

The inverse sends an endomorphism $u$ to $u(1)$ . Commutativity gives the antiisomorphism $\nu:B\to B^{\mathrm{op}}$ , $a\mapsto a^{\mathrm{op}}$ . The element $f$ is central and idempotent. Let $g:W\to P^3\oplus Q$ be the coordinate identification, and let

\tau:\operatorname{GL}_3(\operatorname{End}_k(P)) \longrightarrow\operatorname{GL}_k(P^3\oplus Q)

act by the given matrix on $P^3$ and by the identity on $Q$ . This map is injective. After identifying $B$ with its endomorphism image, the matrix in the standard formula has entries

\iota\!\left(\delta(x_{ij})f+ \nu\bigl(\delta((x^{-1})_{ji})\bigr)^{\mathrm{op}}(1-f)\right) =\iota(x_{ij}).

In this display the superscript $\mathrm{op}$ on an element of $B^{\mathrm{op}}$ denotes its underlying element of $B$ . Applying $\tau$ gives $\operatorname{diag}(x,1)$ under $g$ . These data therefore realize the standard formula for $\Lambda$ .

Finally,

|V|=3^3=27,\qquad |W|=3^4=81,\qquad |W^*|=81.

The last equality follows from the dual basis of the coordinate basis of $W$ . A bijective semilinear map would induce a bijection of the underlying finite sets, contradicting $27\ne81$ . ◻

The example has source dimension $3$ , target dimension $4$ , and $2\in k^\times$ , as in the hypotheses attached to the standardness definition. It retains the original natural modules in the conclusion. Replacing them by modules transported through an equivalence of module categories changes the formulation under consideration; no assertion about that reformulation is made here.

References

Preprint · Lean (GitHub)

E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved problems in group theory, 21st ed., September 2026 update, Problem 10.44 and the preceding definition, arXiv:1401.0300v46.