  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.
