I wonder about the basic facts in the theory of algebraic groups and Lie algebras that are wrong in characteristic $p$. For example,

  1. The trace of $1$, that is the dimension of a representation, may be zero, therefore there is no Cartan criterions.

  2. The derivative of non-constant polynomial may be zero, therefore there are non-smooth algebraic group schemes.

  3. There is no exponential map. Of course, even in characteristic $0$, there is an exponential map only for unipotent algebraic groups, and it is boring, but there is one for both usual and $p$-adic Lie groups (see Schneider, p-Adic Lie Groups, p. 153).

What are the other basic problems? For example, what is a fundamental reason why $\operatorname{GL}(\mathbb F_p)$ is not linearly reductive, that is there is no direct complements in finite-dimensional representations?


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