I'm reading Deligne-Milne's introduction to Tannakian categories, and I noticed a troubling consequence of the definition of rank in a rigid ACU tensor category $(\mathcal{C}, \otimes)$. Specifically, the rank of an object $X$ is defined as the trace of its identity morphism, and the trace of $X$ is defined as the natural map from endomorphsims of $X$ to endomorphisms of the unit object $\mathbb{1}$ induced by evaluation of the internal end object, which is isomorphic to $X^{\vee} \otimes X$.
$$
\mathrm{Tr}_X = \mathrm{Hom}(\mathbb{1}, -)(\underline{Hom} (X, X) \overset{\sim}{\to} X^{\vee} \otimes X \overset{\mathrm{ev}}{\to} \mathbb{1}): \mathrm{End}(X) \to \mathrm{End}(\mathbb{1}).
$$
If $\mathcal{C}$ is the category of $R$-modules for a commutative ring $R$, then the module $R^n$ has rank $n$ (identified with the multiplication action of $n$ on $\mathbb{1} = R$) for all $n$. In characteristic $0$, this is exactly what I would expect. But in the case $\mathrm{char} (R) = n$, this means the rank of $R^n$ is $0$. Moreover, because the rank can be identified with an element of $R$, there is no meaningful notion of rank that can distinguish any object $X$ from $X \oplus R^n$.
Are there ways around this problem working in tensor categories with positive characteristic? Should I consider an alternative framework, or somehow try to define trace as an element of some ring other than $R$, e.g. its $(n)$-adic completion? Or is this just a failure of my intuition that I should work to adjust?
Best Answer
I think the takeaway is that this notion of the categorical dimension or rank of an object really doesn't detect as much in positive characteristic as it does in characteristic zero. Of course you can try to remedy this by looking at more refined invariants, here is a paper that defines a $\mathbb{Z}_p$-valued notion of categorical dimension for rigid ACU tensor categories in characteristic $p$: https://arxiv.org/abs/1510.04339
Still, while the $p$-adic dimension dimension defined there does greatly refine the ordinary categorical dimension, if you look at the paper you'll see it still has some quirks that make it not quite behave how you want a notion of dimension to in all situations. Ultimately, rigid ACU tensor categories in characteristic $p$ are just more complicated than their characteristic zero analogs and there are still lots of things that aren't well understood about them.