Bar complexes and extensions of classical exponential functors

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 Ext-groups with the homology of Eilenberg-Mac Lane Spaces. Together with the article “Troesch complexes and the twisting spectral sequence” (arXiv 1005.3133), this article provides an alternative approach to classical Ext-computations in the category of strict polynomial functors over fields. We also obtain significant Ext-computations for strict polynomial functors over the integers.