# Troesch complexes and extensions of strict polynomial functors

Annales scientifiques de l'École Normale Supérieure (2012)

- Volume: 45, Issue: 1, page 53-99
- ISSN: 0012-9593

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topTouzé, Antoine. "Troesch complexes and extensions of strict polynomial functors." Annales scientifiques de l'École Normale Supérieure 45.1 (2012): 53-99. <http://eudml.org/doc/272134>.

@article{Touzé2012,

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 $\{\mathrm \{Ext\}\}$-computations as well as new results.
In particular, we get a cohomological version of the “fundamental theorems” from classical 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 $\{\mathrm \{Ext\}\}^*_\{\mathcal \{P\} _\mathbb \{k\} \}(F^\{(r)\}, G^\{(r)\})$ between the twisted functors $F^\{(r)\}$ and $G^\{(r)\}$. Many classical $\{\mathrm \{Ext\}\}$ 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 $\{\mathrm \{Ext\}\}$ groups. We prove many cases of collapsing, and we conjecture collapsing is a general fact.},

author = {Touzé, Antoine},

journal = {Annales scientifiques de l'École Normale Supérieure},

keywords = {strict polynomial functors; extensions; Frobenius twist; general linear group; cohomology algebras; functor homology; rational cohomology},

language = {eng},

number = {1},

pages = {53-99},

publisher = {Société mathématique de France},

title = {Troesch complexes and extensions of strict polynomial functors},

url = {http://eudml.org/doc/272134},

volume = {45},

year = {2012},

}

TY - JOUR

AU - Touzé, Antoine

TI - Troesch complexes and extensions of strict polynomial functors

JO - Annales scientifiques de l'École Normale Supérieure

PY - 2012

PB - Société mathématique de France

VL - 45

IS - 1

SP - 53

EP - 99

AB - 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 ${\mathrm {Ext}}$-computations as well as new results.
In particular, we get a cohomological version of the “fundamental theorems” from classical 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 ${\mathrm {Ext}}^*_{\mathcal {P} _\mathbb {k} }(F^{(r)}, G^{(r)})$ between the twisted functors $F^{(r)}$ and $G^{(r)}$. Many classical ${\mathrm {Ext}}$ 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 ${\mathrm {Ext}}$ groups. We prove many cases of collapsing, and we conjecture collapsing is a general fact.

LA - eng

KW - strict polynomial functors; extensions; Frobenius twist; general linear group; cohomology algebras; functor homology; rational cohomology

UR - http://eudml.org/doc/272134

ER -

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