Highest weight categories and recollements
Annales de l'Institut Fourier, Volume 67 (2017) no. 6, p. 2679-2701

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.

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.

Received : 2015-12-15
Accepted : 2017-03-01
Published online : 2017-12-14
DOI : https://doi.org/10.5802/aif.3147
Classification:  16G10,  16D90,  16E65,  18E30
Keywords: Highest weight category, quasi-hereditary algebra, recollement, exceptional sequence, derived category
@article{AIF_2017__67_6_2679_0,
     author = {Krause, Henning},
     title = {Highest weight categories and recollements},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     pages = {2679-2701},
     doi = {10.5802/aif.3147},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_6_2679_0}
}
Krause, Henning. Highest weight categories and recollements. Annales de l'Institut Fourier, Volume 67 (2017) no. 6, pp. 2679-2701. doi : 10.5802/aif.3147. http://www.numdam.org/item/AIF_2017__67_6_2679_0/

[1] Auslander, Maurice Representation theory of Artin algebras. I, Commun. Algebra, Tome 1 (1974), pp. 177-268 | Article | Zbl 0285.16028

[2] Beĭlinson, Alexander A.; Bernstein, Joseph N.; Deligne, Pierre René Faisceaux pervers, Analysis and topology on singular spaces, Société Mathématique de France (Astérisque) Tome 100 (1982), pp. 5-171 | Zbl 0536.14011

[3] Bondal, Alexey I. Representation of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat., Tome 53 (1989) no. 1, pp. 25-44 (translation in Math. USSR-Izv. 34 (1990), no. 1, 23–42) | Zbl 0692.18002

[4] Buchweitz, Ragnar-Olaf; Leuschke, Graham J.; Van Den Bergh, Michel On the derived category of Grassmannians in arbitrary characteristic, Compos. Math., Tome 151 (2015) no. 7, pp. 1242-1264 | Article | Zbl 1333.14017

[5] Cline, Edward T.; Parshall, Brian J.; Scott, Leonard L. Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math., Tome 391 (1988), pp. 85-99 | Zbl 0657.18005

[6] Dlab, Vlastimil; Ringel, Claus Michael Quasi-hereditary algebras, Ill. J. Math., Tome 33 (1989) no. 2, pp. 280-291 | Zbl 0666.16014

[7] Dlab, Vlastimil; Ringel, Claus Michael The module theoretical approach to quasi-hereditary algebras, Representations of algebras and related topics (Kyoto, 1990), Cambridge University Press (London Mathematical Society Lecture Note Series) Tome 168 (1992), pp. 200-224 | Zbl 0793.16006

[8] Efimov, Alexander I. Derived categories of Grassmannians over integers and modular representation theory, Adv. Math., Tome 304 (2017), pp. 179-226 | Article | Zbl 1365.14020

[9] Gabriel, Pierre Des catégories abéliennes, Bull. Soc. Math. Fr., Tome 90 (1962), pp. 323-448 | Article | Zbl 0201.35602

[10] Gabriel, Pierre Indecomposable representations. II, Algebra commut., Geometria (Convegni 1971/1972), Academic Press (Symposia Mathematica) Tome 11 (1973), pp. 81-104 | Zbl 0276.16001

[11] Gabriel, Pierre; Zisman, Michel Calculus of fractions and homotopy theory, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete, Tome 85 (1967), x+168 pages | Zbl 0186.56802

[12] Geigle, Werner; Lenzing, Helmut Perpendicular categories with applications to representations and sheaves, J. Algebra, Tome 144 (1991) no. 2, pp. 273-343 | Article | Zbl 0748.18007

[13] Gorodentsev, Alexey L. Exceptional bundles on surfaces with a moving anticanonical class, Izv. Akad. Nauk SSSR Ser. Mat., Tome 52 (1988) no. 4, pp. 740-757 (translation in Math. USSR-Izv. 33 (1989), no. 1, 67–83) | Zbl 0736.14003

[14] Gorodentsev, Alexey L.; Rudakov, Alexei Nikolaevich Exceptional vector bundles on projective spaces, Duke Math. J., Tome 54 (1987) no. 1, pp. 115-130 | Article | Zbl 0646.14014

[15] Hartshorne, Robin Residues and duality, Springer, Lecture Notes in Mathematics, Tome 20 (1966), 423 pages | Zbl 0212.26101

[16] Hille, Lutz; Perling, Markus Tilting bundles on rational surfaces and quasi-hereditary algebras, Ann. Inst. Fourier, Tome 64 (2014) no. 2, pp. 625-644 | Article | Zbl 1348.14100

[17] Illusie, Luc Existence de résolutions globales, Théorie des Intersections et théorème de Riemann-Roch (SGA 6, 1966/67), Springer (Lecture Notes in Mathematics) Tome 225 (1971), pp. 160-221 | Zbl 0241.14002

[18] Iyama, Osamu Finiteness of representation dimension, Proc. Am. Math. Soc., Tome 131 (2003) no. 4, pp. 1011-1014 | Article | Zbl 1018.16010

[19] Keller, Bernhard Derived categories and their use, Handbook of algebra. Volume 1, North-Holland (1996), pp. 671-701 | Zbl 0862.18001

[20] Krause, Henning Highest weight categories and strict polynomial functors, Representation theory - Current trends and perspectives, European Mathematical Society (2017), pp. 331-373 | Zbl 06751586

[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 (Carleton University Notes) Tome 3 (988), pp. 1-105 | Zbl 0711.18002

[22] Psaroudakis, Chrysostomos Homological theory of recollements of abelian categories, J. Algebra, Tome 398 (2014), pp. 63-110 | Article | Zbl 1314.18016

[23] Rouquier, Raphaël q-Schur algebras and complex reflection groups, Mosc. Math. J., Tome 8 (2008) no. 1, pp. 119-158 | Zbl 1213.20007

[24] Rudakov, Alexei Nikolaevich Helices and vector bundles: Seminaire Rudakov, Cambridge University Press, London Mathematical Society Lecture Note Series, Tome 148 (1990), 143 pages | Zbl 0727.00022

[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), American Mathematical Society (Proceedings of Symposia in Pure Mathematics) Tome 47 (1987), pp. 271-281 | Zbl 0659.20038

[26] Serre, Jean-Pierre Groupes proalgébriques, Publ. Math., Inst. Hautes Étud. Sci., Tome 7 (1961), pp. 341-403 | Zbl 0097.35901

[27] Verdier, Jean-Louis Des catégories dérivées des catégories abéliennes, Société Mathématique de France, Astérisque, Tome 239 (1996), ix+253 pages | Zbl 0882.18010