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We investigate fundamental properties of adjoint functors to the precomposition functor in the category of strict polynomial functors. In particular, we obtain the proof of Touzé's Collapsing Conjecture on Ext-groups between twisted functors and refinement of a general program of computing Ext-groups between strict polynomial functors important in representation theory.Mathematics Subject Classification (2010) 18A25, 18A40, 18G15. 20G15.

We give some basics about homological algebra of difference representations. We consider both the difference discrete and the difference rational case. We define the corresponding cohomology theories and show the existence of spectral sequences relating these cohomology theories with the standard ones.