Troesch complexes and extensions of strict polynomial functors

Antoine Touzé

Annales scientifiques de l'École Normale Supérieure (2012)

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

Abstract

top
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 theory for  G L 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 𝒫 𝕜 * ( 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.

How to cite

top

Touzé, 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 -

References

top
  1. [1] D. J. Benson, Representations and cohomology. I, second éd., Cambridge Studies in Advanced Math. 30, Cambridge Univ. Press, 1998. Zbl0908.20001MR1644252
  2. [2] D. J. Benson, Representations and cohomology. II, second éd., Cambridge Studies in Advanced Math. 31, Cambridge Univ. Press, 1998. Zbl0908.20002MR1634407
  3. [3] M. Chałupnik, Extensions of strict polynomial functors, Ann. Sci. École Norm. Sup.38 (2005), 773–792. Zbl1089.20029MR2195259
  4. [4] M. Chałupnik, Koszul duality and extensions of exponential functors, Adv. Math.218 (2008), 969–982. Zbl1148.18008MR2414328
  5. [5] E. Cline, B. Parshall, L. Scott & W. van der Kallen, Rational and generic cohomology, Invent. Math.39 (1977), 143–163. Zbl0336.20036MR439856
  6. [6] C. de Concini & C. Procesi, A characteristic free approach to invariant theory, Adv. Math.21 (1976), 330–354. Zbl0347.20025MR422314
  7. [7] V. Franjou & E. M. Friedlander, Cohomology of bifunctors, Proc. Lond. Math. Soc.97 (2008), 514–544. Zbl1153.20042MR2439671
  8. [8] V. Franjou, E. M. Friedlander, A. Scorichenko & A. Suslin, General linear and functor cohomology over finite fields, Ann. of Math.150 (1999), 663–728. Zbl0952.20035MR1726705
  9. [9] V. Franjou, J. Lannes & L. Schwartz, Autour de la cohomologie de Mac Lane des corps finis, Invent. Math.115 (1994), 513–538. Zbl0798.18009MR1262942
  10. [10] V. Franjou & T. Pirashvili, Strict polynomial functors and coherent functors, Manuscripta Math.127 (2008), 23–53. Zbl1168.18001MR2429912
  11. [11] E. M. Friedlander & A. Suslin, Cohomology of finite group schemes over a field, Invent. Math.127 (1997), 209–270. Zbl0945.14028MR1427618
  12. [12] J. C. Jantzen, Representations of algebraic groups, second éd., Mathematical Surveys and Monographs 107, Amer. Math. Soc., 2003. Zbl1034.20041
  13. [13] W. van der Kallen, Cohomology with Grosshans graded coefficients, in Invariant theory in all characteristics, CRM Proc. Lecture Notes 35, Amer. Math. Soc., 2004, 127–138. Zbl1080.20039MR2066461
  14. [14] B. Totaro, Projective resolutions of representations of GL ( n ) , J. reine angew. Math. 482 (1997), 1–13. Zbl0859.20034MR1427655
  15. [15] A. Touzé, Cohomology of classical algebraic groups from the functorial viewpoint, Adv. Math.225 (2010), 33–68. Zbl1208.20043MR2669348
  16. [16] A. Touzé, Universal classes for algebraic groups, Duke Math. J.151 (2010), 219–249. Zbl1196.20052MR2598377
  17. [17] A. Touzé & W. van der Kallen, Bifunctor cohomology and cohomological finite generation for reductive groups, Duke Math. J.151 (2010), 251–278. Zbl1196.20053MR2598378
  18. [18] A. Troesch, Une résolution injective des puissances symétriques tordues, Ann. Inst. Fourier (Grenoble) 55 (2005), 1587–1634. Zbl1077.18009MR2172274

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.