Highest weight categories and recollements
[Catégories de plus haut poids et recollements]
Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2679-2701.

Nous donnons plusieurs descriptions équivalentes des catégories de plus haut poids au moyen de recollements de catégories abéliennes. En outre, nous expliquons la relation entre suites d’objets standards et exceptionnels.

We provide several equivalent descriptions of a highest weight category using recollements of abelian categories. Also, we explain the connection between sequences of standard and exceptional objects.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3147
Classification : 16G10, 16D90, 16E65, 18E30
Keywords: Highest weight category, quasi-hereditary algebra, recollement, exceptional sequence, derived category
Mot clés : Catégorie de plus haut poids, algèbre quasi-héréditaire, recollement, suite exceptionelle, catégorie dérivée
Krause, Henning 1

1 Fakultät für Mathematik Universität Bielefeld 33501 Bielefeld (Germany)
@article{AIF_2017__67_6_2679_0,
     author = {Krause, Henning},
     title = {Highest weight categories and recollements},
     journal = {Annales de l'Institut Fourier},
     pages = {2679--2701},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     doi = {10.5802/aif.3147},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.3147/}
}
TY  - JOUR
AU  - Krause, Henning
TI  - Highest weight categories and recollements
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 2679
EP  - 2701
VL  - 67
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.3147/
DO  - 10.5802/aif.3147
LA  - en
ID  - AIF_2017__67_6_2679_0
ER  - 
%0 Journal Article
%A Krause, Henning
%T Highest weight categories and recollements
%J Annales de l'Institut Fourier
%D 2017
%P 2679-2701
%V 67
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.3147/
%R 10.5802/aif.3147
%G en
%F AIF_2017__67_6_2679_0
Krause, Henning. Highest weight categories and recollements. Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2679-2701. doi : 10.5802/aif.3147. http://www.numdam.org/articles/10.5802/aif.3147/

[1] Auslander, Maurice Representation theory of Artin algebras. I, Commun. Algebra, Volume 1 (1974), pp. 177-268 | DOI | Zbl

[2] Beĭlinson, Alexander A.; Bernstein, Joseph N.; Deligne, Pierre René Faisceaux pervers, Analysis and topology on singular spaces (Astérisque), Volume 100, Société Mathématique de France, 1982, pp. 5-171 | Zbl

[3] Bondal, Alexey I. Representation of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat., Volume 53 (1989) no. 1, pp. 25-44 translation in Math. USSR-Izv. 34 (1990), no. 1, 23–42 | Zbl

[4] Buchweitz, Ragnar-Olaf; Leuschke, Graham J.; Van den Bergh, Michel On the derived category of Grassmannians in arbitrary characteristic, Compos. Math., Volume 151 (2015) no. 7, pp. 1242-1264 | DOI | Zbl

[5] Cline, Edward T.; Parshall, Brian J.; Scott, Leonard L. Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math., Volume 391 (1988), pp. 85-99 | Zbl

[6] Dlab, Vlastimil; Ringel, Claus Michael Quasi-hereditary algebras, Ill. J. Math., Volume 33 (1989) no. 2, pp. 280-291 | Zbl

[7] Dlab, Vlastimil; Ringel, Claus Michael The module theoretical approach to quasi-hereditary algebras, Representations of algebras and related topics (Kyoto, 1990) (London Mathematical Society Lecture Note Series), Volume 168, Cambridge University Press, 1992, pp. 200-224 | Zbl

[8] Efimov, Alexander I. Derived categories of Grassmannians over integers and modular representation theory, Adv. Math., Volume 304 (2017), pp. 179-226 | DOI | Zbl

[9] Gabriel, Pierre Des catégories abéliennes, Bull. Soc. Math. Fr., Volume 90 (1962), pp. 323-448 | DOI | Zbl

[10] Gabriel, Pierre Indecomposable representations. II, Algebra commut., Geometria (Convegni 1971/1972) (Symposia Mathematica), Volume 11, Academic Press, 1973, pp. 81-104 | Zbl

[11] Gabriel, Pierre; Zisman, Michel Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 85, Springer, 1967, x+168 pages | Zbl

[12] Geigle, Werner; Lenzing, Helmut Perpendicular categories with applications to representations and sheaves, J. Algebra, Volume 144 (1991) no. 2, pp. 273-343 | DOI | Zbl

[13] Gorodentsev, Alexey L. Exceptional bundles on surfaces with a moving anticanonical class, Izv. Akad. Nauk SSSR Ser. Mat., Volume 52 (1988) no. 4, pp. 740-757 translation in Math. USSR-Izv. 33 (1989), no. 1, 67–83 | Zbl

[14] Gorodentsev, Alexey L.; Rudakov, Alexei Nikolaevich Exceptional vector bundles on projective spaces, Duke Math. J., Volume 54 (1987) no. 1, pp. 115-130 | DOI | Zbl

[15] Hartshorne, Robin Residues and duality, Lecture Notes in Mathematics, 20, Springer, 1966, 423 pages | Zbl

[16] Hille, Lutz; Perling, Markus Tilting bundles on rational surfaces and quasi-hereditary algebras, Ann. Inst. Fourier, Volume 64 (2014) no. 2, pp. 625-644 | DOI | Zbl

[17] Illusie, Luc Existence de résolutions globales, Théorie des Intersections et théorème de Riemann-Roch (SGA 6, 1966/67) (Lecture Notes in Mathematics), Volume 225, Springer, 1971, pp. 160-221 | Zbl

[18] Iyama, Osamu Finiteness of representation dimension, Proc. Am. Math. Soc., Volume 131 (2003) no. 4, pp. 1011-1014 | DOI | Zbl

[19] Keller, Bernhard Derived categories and their use, Handbook of algebra. Volume 1, North-Holland, 1996, pp. 671-701 | Zbl

[20] Krause, Henning Highest weight categories and strict polynomial functors, Representation theory - Current trends and perspectives, European Mathematical Society, 2017, pp. 331-373 | Zbl

[21] Parshall, Brian J.; Scott, Leonard L. Derived categories, quasi-hereditary algebras, and algebraic groups, Proceedings of the Ottawa-Moosonee Workshop in Algebra (Carleton University Notes), Volume 3, Carleton University, 988, pp. 1-105 | Zbl

[22] Psaroudakis, Chrysostomos Homological theory of recollements of abelian categories, J. Algebra, Volume 398 (2014), pp. 63-110 | DOI | Zbl

[23] Rouquier, Raphaël q-Schur algebras and complex reflection groups, Mosc. Math. J., Volume 8 (2008) no. 1, pp. 119-158 | Zbl

[24] Helices and vector bundles: Seminaire Rudakov (Rudakov, Alexei Nikolaevich, ed.), London Mathematical Society Lecture Note Series, 148, Cambridge University Press, 1990, 143 pages | Zbl

[25] Scott, Leonard L. Simulating algebraic geometry with algebra. I. The algebraic theory of derived categories, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) (Proceedings of Symposia in Pure Mathematics), Volume 47, American Mathematical Society, 1987, pp. 271-281 | Zbl

[26] Serre, Jean-Pierre Groupes proalgébriques, Publ. Math., Inst. Hautes Étud. Sci., Volume 7 (1961), pp. 341-403 | Zbl

[27] Verdier, Jean-Louis Des catégories dérivées des catégories abéliennes, Astérisque, 239, Société Mathématique de France, 1996, ix+253 pages | Zbl

Cité par Sources :