To every object $X$ of a symmetric tensor category over a field of characteristic $p>0$ we attach $p$-adic integers $\text{Dim}_+(X)$ and $\text{Dim}_-(X)$ whose reduction modulo $p$ is the categorical dimension $\text{dim}(X)$ of $X$, coinciding with the usual dimension when $X$ is a vector space. We study properties of $\text{Dim}_{\pm}(X)$, and in particular show that they don't always coincide with each other, and can take any value in $\mathbb{Z}_p$. We also discuss the connection of $p$-adic dimensions with the theory of $\lambda$-rings and Brauer characters.

中文翻译：

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$p$-特征$p$中对称张量类别中的adic维数

在特征 $p>0$ 的域上对称张量类别的每个对象 $X$ 我们附加 $p$-adic 整数 $\text{Dim}_+(X)$ 和 $\text{Dim}_- (X)$ 的归约模 $p$ 是 $X$ 的分类维度 $\text{dim}(X)$，当 $X$ 是向量空间时，与通常的维度一致。我们研究了 $\text{Dim}_{\pm}(X)$ 的属性，特别是表明它们并不总是彼此重合，并且可以在 $\mathbb{Z}_p$ 中取任何值。我们还讨论了 $p$-adic 维数与 $\lambda$-rings 和 Brauer 字符理论的联系。