Embeddings of algebras in derived categories of surfaces

Belmans, Pieter; Raedschelders, Theo

doi:10.1090/proc/13497

Embeddings of algebras in derived categories of surfaces

Authors: Pieter Belmans and Theo Raedschelders
Journal: Proc. Amer. Math. Soc. 145 (2017), 2757-2770
MSC (2010): Primary 14F05, 16E35; Secondary 18E30
DOI: https://doi.org/10.1090/proc/13497
Published electronically: February 24, 2017
MathSciNet review: 3637928
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By a result of Orlov there always exists an embedding of the derived category of a finite-dimensional algebra of finite global dimension into the derived category of a high-dimensional smooth projective variety. In this article we give some restrictions on those algebras whose derived categories can be embedded into the bounded derived category of a smooth projective surface. This is then applied to obtain explicit results for hereditary algebras.

[1] Dagmar Baer, Tilting sheaves in representation theory of algebras, Manuscripta Math. 60 (1988), no. 3, 323–347. MR 928291, https://doi.org/10.1007/BF01169343
[2] I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Algebraic vector bundles on 𝑃ⁿ and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67 (Russian). MR 509387
[3] Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
[4] A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25–44 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 23–42. MR 992977, https://doi.org/10.1070/IM1990v034n01ABEH000583
[5] A. I. Bondal and A. E. Polishchuk, Homological properties of associative algebras: the method of helices, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 2, 3–50 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 42 (1994), no. 2, 219–260. MR 1230966, https://doi.org/10.1070/IM1994v042n02ABEH001536
[6] A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1–36, 258 (English, with English and Russian summaries). MR 1996800, https://doi.org/10.17323/1609-4514-2003-3-1-1-36
[7] David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322
[8] Louis de Thanhoffer de Völcsey, Non-commutative projective geometry and Calabi-Yau algebras, Ph.D. thesis, 2015.
[9] William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
[10] Sergey Gorchinskiy and Dmitri Orlov, Geometric phantom categories, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 329–349. MR 3090263, https://doi.org/10.1007/s10240-013-0050-5
[11] Dieter Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389. MR 910167, https://doi.org/10.1007/BF02564452
[12] Dieter Happel, A family of algebras with two simple modules and Fibonacci numbers, Arch. Math. (Basel) 57 (1991), no. 2, 133–139. MR 1116189, https://doi.org/10.1007/BF01189999
[13] Lutz Hille and Markus Perling, Exceptional sequences of invertible sheaves on rational surfaces, Compos. Math. 147 (2011), no. 4, 1230–1280. MR 2822868, https://doi.org/10.1112/S0010437X10005208
[14] Lutz Hille and Markus Perling, Tilting bundles on rational surfaces and quasi-hereditary algebras, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 2, 625–644 (English, with English and French summaries). MR 3330917
[15] Osamu Iyama, Finiteness of representation dimension, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1011–1014. MR 1948089, https://doi.org/10.1090/S0002-9939-02-06616-9
[16] Bernhard Keller, Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123 (1998), no. 1-3, 223–273. MR 1492902, https://doi.org/10.1016/S0022-4049(96)00085-0
[17] A. G. Kuznetsov, Derived categories of Fano threefolds, Tr. Mat. Inst. Steklova 264 (2009), no. Mnogomernaya Algebraicheskaya Geometriya, 116–128 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 264 (2009), no. 1, 110–122. MR 2590842, https://doi.org/10.1134/S0081543809010143
[18] Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, Proceedings of the ICM 2014, 2014.
[19] D. Mumford, Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195–204. MR 249428, https://doi.org/10.1215/kjm/1250523940
[20] Shinnosuke Okawa, Semi-orthogonal decomposability of the derived category of a curve, Adv. Math. 228 (2011), no. 5, 2869–2873. MR 2838062, https://doi.org/10.1016/j.aim.2011.06.043
[21] Dmitri Orlov, Smooth and proper noncommutative schemes and gluing of DG categories, Adv. Math. 302 (2016), 59–105. MR 3545926, https://doi.org/10.1016/j.aim.2016.07.014
[22] Dmitri O. Orlov, Geometric realizations of quiver algebras, Proc. Steklov Inst. Math. 290 (2015), no. 1, 70–83. Published in Russian in Tr. Mat. Inst. Steklova 290 (2015), 80–94. MR 3488782, https://doi.org/10.1134/S0081543815060073
[23] Gonçalo Tabuada and Michel Van den Bergh, Noncommutative motives of Azumaya algebras, J. Inst. Math. Jussieu 14 (2015), no. 2, 379–403. MR 3315059, https://doi.org/10.1017/S147474801400005X
[24] Gonçalo Tabuada and Michel Van den Bergh, Noncommutative motives of Azumaya algebras, J. Inst. Math. Jussieu 14 (2015), no. 2, 379–403. MR 3315059, https://doi.org/10.1017/S147474801400005X