  We consider rational representations of a connected linear algebraic group
$\mathbb G$ over a field $k$ of positive characteristic $p > 0$. We introduce a
natural extension $M \mapsto \Pi(\mathbb G)_M$ to $\mathbb G$-modules of the
$\pi$-point support theory for modules $M$ for a finite group scheme $G$ and
show that this theory is essentially equivalent to the more "intrinsic" and
"explicit" theory $M \mapsto \mathbb P\mathfrak C(\mathbb G)_M$ of supports for
an algebraic group of exponential type, a theory which uses 1-parameter
subgroups $\mathbb G_a \to \mathbb G$. We extend our support theory to bounded
complexes of $\mathbb G$-modules, $C^\bullet \mapsto \Pi(\mathbb
G)_{C^\bullet}$. We introduce the tensor triangulated category $StMod(\mathbb
G)$, the Verdier quotient of the bounded derived category $D^b(Mod(\mathbb G))$
by the thick subcategory of mock injective modules. Our support theory
satisfies all the standard properties" for a theory of supports for
$StMod(\mathbb G)$. As an application, we employ $C^\bullet \mapsto \Pi(\mathbb
G)_{C^\bullet}$ to establish the classification of $(r)$-complete, thick tensor
ideals of $stmod(\mathbb G)$ in terms of $stmod(\mathbb G)$-realizable subsets
of $\Pi(\mathbb G)$ and the classification of $(r)$-complete, localizing
subcategories of $StMod(\mathbb G)$ in terms of $StMod(\mathbb G)$-realizable
subsets of $\Pi(\mathbb G)$.
