Troesch complexes and extensions of strict polynomial functors

TL;DR

This paper introduces a new extension calculus for strict polynomial functors using Troesch complexes, providing simplified proofs of classical Ext-computations and new results, including a cohomological version of fundamental theorems in invariant theory.

Contribution

It develops a novel approach based on Troesch complexes, offering new proofs and results in the extension theory of strict polynomial functors, and analyzes the collapsing of the twisting spectral sequence.

Findings

New elementary proofs of classical Ext-computations

Cohomological version of fundamental theorems for GL_n

Many cases of spectral sequence collapsing

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.