Abstract. We exhibit cocycles representing certain classes in the cohomology of the algebraic group GLn with coefficients in the representation Γ * (gl (1) n ). These classes' existence was anticipated by van der Kallen, and they intervene in the proof that reductive linear algebraic groups have finitely generated cohomology algebras [18].

We compute Ext-groups between classical exponential functors (i.e. symmetric, exterior or divided powers) and their Frobenius twists. Our method relies on bar constructions, and bridges these Extgroups with the homology of Eilenberg-Mac Lane spaces.Together with [T2], this article provides an alternative approach to classical Ext-computations [FS, FFSS, C1, C2] in the category of strict polynomial functors over fields. We also obtain significant Extcomputations for strict polynomial functors over the integers.1 This follows from the projectivity of divided powers, and the injectivity of symmetric powers, except for the case of the extensions Ext i P k (Λ * , Λ * ). In the latter case, there are many proofs of this vanishing, and we will provide one in remark 7.6.

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study of classical algebraic groups, such as a cohomological stabilization property, the injectivity of external cup products, and the existence of Hopf algebra structures on the (stable) cohomology of a classical algebraic group with coefficients in a Hopf algebra. Our result also opens the way to new explicit cohomology computations. We give an example inspired by recent computations of Djament and Vespa.

We develop a new approach of extension calculus in the category of strict polynomial functors, based on Troesch complexes. We obtain new short elementary proofs of numerous classical Extcomputations as well as new results.In particular, we get a cohomological version of the 'fundamental theorems' from classical invariant invariant theory for GLn for n big enough (and we give a conjecture for smaller values of n).We also study the 'twisting spectral sequence' E s,t (F, G, r) converging to the extension groups Ext * P k (F (r) , G (r) ) between the twisted functors F (r) and G (r) . Many classical Ext computations simply amount to the collapsing of this spectral sequence at the second page (for lacunary reasons), and it is also a convenient tool to study the effect of the Frobenius twist on Ext groups. We prove many cases of collapsing, and we conjecture collapsing is a general fact.