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We investigate rational $G$-modules $M$ for a linear algebraic group $G$ overan algebraically closed field $k$ of characteristic $p > 0$ using filtrationsby sub-coalgebras of the coordinate algebra $k[G]$ of $G$. Even in the specialcase of the additive group $\mathbb G_a$, interesting structures and examplesare revealed. The "degree" filtration we consider for unipotent algebraicgroups leads to a "filtration by exponential degree" applicable to rational $G$modules for any linear algebraic group $G$ of exponential type; this filtrationis defined in terms of 1-parameter subgroups and is related to supportvarieties introduced recently by the author for such rational $G$-modules. Weformulate in terms of this filtration a necessary and sufficient condition forrational injectivity for rational $G$-modules. Our investigation leads to theconsideration of two new classes of rational $G$-modules: those that are "mockinjective" and those that are "mock trivial".

Filtrations, 1-parameter Subgroups, and Rational Injectivity