# Integral structures via lattices

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?

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 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 $$A=\rho^*(\mathcal{O}[\mathrm{GL}(L)]).$$

• Perhaps I should have used right comodules instead, as is the case in Waterhouse's book. I am not sure if this difference is substantial. Aug 27, 2018 at 5:33
• Thank you! Would you also know the answer to a question I had a while back? mathoverflow.net/questions/306153/… Oct 25, 2018 at 6:14
• @Mehta, you are welcome. I will think about your other question. Oct 28, 2018 at 6:58