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Let $G$ be a semisimple algebraic group over an algebraically closed field$k$ of positive characteristic $p$. Under some restrictions on the size of $p$,the present paper establishes new results on the $G$-module structure of$\Ext^\bullet_{G_1}(V,W)$ when $V,W$ belong to several important classes ofrational $G$-modules, and $G_1$ denotes the first Frobenius kernel of $G$. Forexample, it is proved that, if $L,L'$ are ($p$-regular) irreducible$G_1$-modules, then $\Ext^n_{G_1}(L,L')^{[-1]}$ has a good filtration withcomputable multiplicities. This and many other results depend on the entirelynew technique of using methods of what we call forced gradings in therepresentation theory of $G$, as developed by the authors in recent papers, andextended here.  In addition to providing proofs, these methods lead effectively to a newconceptual framework for the study of rational $G$-modules, and, in thiscontext, to the introduction of a new class of graded finite dimensionalalgebras, which we call Q-Koszul algebras. These algebras are similar to Koszulalgebras, but are quasi-hereditary, rather than semisimple, in grade 0.

New graded methods in the homological algebra of semisimple algebraic groups