Koszul duality and extensions of exponential functors

We study Koszul duality in the category of strict polynomial functors. We compute Koszul duals of various functors and apply these results to the problem of calculating Ext-groups between exponential functors. The main application is a full description of the Ext-groups between twisted exterior and divided powers and between twisted symmetric and divided powers which completes the program started in [V. Franjou, E. Friedlander, A. Scorichenko, A. Suslin, General linear and functor cohomology over finite fields, Ann. of Math. 150 (2) (1999) 663–728].