Troesch complexes and extensions of strict polynomial functors

Abstract

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 Ext-computations as well as new results. In particular, we get a cohomological version of the `fundamental theorems' from classical invariant invariant theory for GL_n 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^*(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.