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Sep 15, 2020Q:

Isomorphism $\mathcal{O}(L)\otimes \mathbb{F}_p \cong \mathbb{F}_{p^2}$

Is the following statement true:
For every lattice $L$, we have
$$\mathcal{O}(L)\otimes \mathbb{F}_p \cong \mathbb{F}_{p^2}$$
Furthermore, I have seen this question asked on MSE. Now, this question is much easier to ask than to answer. So I do not care about the answer to my question, but I want to know if the statement is really true. If it is not, then what is an example of a lattice $L$ and an $\mathbb{F}$-algebra $\mathcal{O}$ such that we have an isomorphism above?

A:

I do not care about the answer to my question, but I want to know if the statement is really true.

It’s not true. Consider the root lattice $\mathrm{Q}$ with a $1$-dimensional root space. This is the root lattice of $\mathrm{E}_6$. Consider $\mathcal O_\mathrm{Q}$ the regular representation of $Q$. Then we have

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