I am looking at the paper "p-adic Groups" by Bruhat (in the Boulder Proceedings, 1965). I have a question about one of the statements. Let $k$ be the quotient field of a complete discrete valuation ring $\mathcal{O}$. Let $V$ be a vector space over $k$. Let $L$ be a lattice in $V$. Choose a basis for $L$.


Bruhat states the following in pp. 63-64:


(1) The algebra $\mathcal{O}[GL]:=\mathcal{O}[g\_{ij},(det(g\_{ij}))^{-1}]$ is an $\mathcal{O}$-structure for $GL(V)$.


(2) More generally, if $G$ is a linear algebraic group over $k$, then given a faithful rational representation $\rho:G \rightarrow GL(V)$, the image of $\mathcal{O}[GL(V)]$ in $k[G]$ is an $\mathcal{O}$-structure for $G$. 


(3) Any $\mathcal{O}$-structure for $G$ may be obtained in this way. 


How is (3) proven?



Integral structures via lattices
Here is a proof of (3). Hopefully there are no gaps.
I will write $K$ instead of $k$, which I usually reserve for the residue field.


Just to set the terminology: 


* $G$ is an algebraic group over $K$.
* An $\mathcal{O}$-structure for $G$ is a finitely generated Hopf $\mathcal{O}$-subalgebra $A$ of $K[G]$ such that $K\cdot A=K[G]$. (This is the same as saying that $\mathcal{G}:=\mathrm{Spec}\, A$ is a flat group $\mathcal{O}$-scheme of finite type with generic fiber $G$.)
* If $\rho:G\to\mathrm{GL}(V)$ is a faithful representation and $L\subseteq V$
is an $\mathcal{O}$-lattice, then the $\mathcal{O}$-structure of $G$ induced by $\rho$ is the $\rho^\*(\mathcal{O}[\mathrm{GL}(L)])$, where $\rho^\*:K[\mathrm{GL}(V)]\to K[G]$ is the $K$-algebra homomorphism adjoint to $\rho$.


Also, recall that giving a representation $\rho:G\to \mathrm{GL}(V)$ is the same as endowing $V$ with a (left) $K[G]$-comodule structure $\Delta\_\rho:V\to K[G]\otimes V$.


**And now to the proof.** We are given an $\mathcal{O}$-structure $A\subseteq K[G]$ for $G$ and we wish to show that it is induced by some faithful
representation $\rho:G\to \mathrm{GL}(V)$ and some $\mathcal{O}$-lattice $L\subseteq V$.


**Claim.** There exists a finitely generated $\mathcal{O}$-submodule $L$ of $A$ such that $L$ generates $A$ as an algebra and $\Delta(L)\subseteq A\otimes L$ ($\Delta$ is the comultiplication).


*Sketch of proof.* The analogous statement for fields is well-known, e.g. see Section 3.3 in Waterhouse's "Introduction to Affine Group Schemes". If $A$ is projective over $\mathcal{O}$, then $A$ has an $\mathcal{O}$-basis (because $\mathcal{O}$ is local) and the argument in Waterhouse applies without change. I think that $A$ must always be projective over $\mathcal{O}$, but in case it is not, one can use Corollary 1.5 in [this paper](https://doi.org/10.1016/0001-8708(87)90016-8) by Thomason. $\square$


Now, since $L$ is finitely generated torsion-free and since $\mathcal{O}$ is a valutation ring, $L$ is free.
Write $V=L\cdot K$. Then $\Delta (V)\subseteq K[G]\otimes V$ and thus $V$ is naturally a $K[G]$-comdule, corresponding to a representation $\rho :G\to \mathrm{GL}(V)$. This representation is faithful because $V$ generates $K[G]$. I claim that $\rho$ and $L$ induce the $\mathcal{O}$-structure $A$.


Note first that since $\Delta(L)\subseteq A\otimes L$, the morphism $\rho$ extends to a representation of the group $\mathcal{O}$-scheme $\mathrm{Spec}\, A$ into $\mathrm{GL}(L)$, hence $\rho^\*(\mathcal{O}[\mathrm{GL}(L)])\subseteq A$.


To see the converse, fix an $\mathcal{O}$-basis $\{v\_1,\dots,v\_t\}$ to $L$
and
write
$$
\Delta v\_i=\sum\_j a\_{ij} \otimes v\_j
$$
with $a\_{ij}\in A$; the $a\_{ij}$ are uniquely determined by the $v\_i$. The Hopf algebra axioms imply
that $v\_i=\sum\_j a\_{ij}\varepsilon(v\_j)$ (where $\varepsilon:A\to\mathcal{O}$ is the counit), and so $A$ is generated by the $a\_{ij}$.


We use the basis $\{v\_i\}$ to identify $\mathcal{O}[\mathrm{GL}(L)]$
with $\mathcal{O}[x\_{11},x\_{12},\dots,x\_{tt},y]/(\det(x\_{ij})y-1)$
and $K[\mathrm{GL}(V)]$ with $K[x\_{11},x\_{12},\dots,x\_{tt},y]/(\det(x\_{ij})y-1)$.
The tautological representation of $\mathrm{GL}(V)$, denoted $\tau$, corresponds to the $K[\mathrm{GL}(V)]$-comodule structure on $V$ determined by
$$\Delta\_\tau(v\_i)=\sum\_j x\_{ij}\otimes v\_j.$$


Now, the fact that $\rho=\tau\circ\rho$
implies that $\Delta\_\rho=(\rho^\*\otimes \mathrm{id}\_V)\circ \Delta\_\tau$.
This, together with the previous equations, imply that $\rho^\*(x\_{ij})=a\_{ij}$.
Since the $a\_{ij}$ generate $A$, we see that 
$
A\subseteq \rho^\*(\mathcal{O}[\mathrm{GL}(L)])
$.
We conclude that
$$
A=\rho^\*(\mathcal{O}[\mathrm{GL}(L)]).
$$


