Filtrations, 1-parameter Subgroups, and Rational Injectivity

We investigate rational $G$-modules $M$ for a linear algebraic group $G$ over an algebraically closed field $k$ of characteristic $p > 0$ using filtrations by sub-coalgebras of the coordinate algebra $k[G]$ of $G$. Even in the special case of the additive group $\mathbb G_a$, interesting structures and examples are revealed. The "degree" filtration we consider for unipotent algebraic groups leads to a "filtration by exponential degree" applicable to rational $G$ modules for any linear algebraic group $G$ of exponential type; this filtration is defined in terms of 1-parameter subgroups and is related to support varieties introduced recently by the author for such rational $G$-modules. We formulate in terms of this filtration a necessary and sufficient condition for rational injectivity for rational $G$-modules. Our investigation leads to the consideration of two new classes of rational $G$-modules: those that are "mock injective" and those that are "mock trivial".