Comptes Rendus
no. 4
Homological algebra/Algebraic geometry
Upper bounds for dimensions of singularity categories
[Bornes supérieures pour les dimensions des catégories de singularités]

Présenté par : Claire Voisin

Hailong Dao1 ; Ryo Takahashi2, 3
1 Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523, USA
2 Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
3 Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130, USA
Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 297-301.

Cet article donne des bornes supérieures pour la dimension de la catégorie de singularité d'un anneau local Cohen–Macaulay à singularité isolée. L'une de nos estimations redonne une borne fournie par Ballard, Favero et Katzarkov dans le cas des hypersurfaces.

This paper gives upper bounds for the dimension of the singularity category of a Cohen–Macaulay local ring with an isolated singularity. One of them recovers an upper bound given by Ballard, Favero and Katzarkov in the case of a hypersurface.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.01.012
Hailong Dao 1 ; Ryo Takahashi 2, 3

1 Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523, USA
2 Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
3 Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130, USA 
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     title = {Upper bounds for dimensions of singularity categories},
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     pages = {297--301},
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     doi = {10.1016/j.crma.2015.01.012},
     language = {en},
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Hailong Dao; Ryo Takahashi. Upper bounds for dimensions of singularity categories. Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 297-301. doi : 10.1016/j.crma.2015.01.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.01.012/

[1] T. Aihara; R. Takahashi Generators and dimensions of derived categories, Commun. Algebra (2015) (in press) | arXiv

[2] M. Ballard; D. Favero; L. Katzarkov Orlov spectra: bounds and gaps, Invent. Math., Volume 189 (2012) no. 2, pp. 359-430

[3] P.A. Bergh; S.B. Iyengar; H. Krause; S. Oppermann Dimensions of triangulated categories via Koszul objects, Math. Z., Volume 265 (2010) no. 4, pp. 849-864

[4] A. Bondal; M. van den Bergh Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J., Volume 3 (2003) no. 1, pp. 1-36 (258)

[5] W. Bruns; J. Herzog Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, UK, 1998

[6] R.-O. Buchweitz Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings, 1986 http://hdl.handle.net/1807/16682 (Preprint)

[7] J.D. Christensen Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math., Volume 136 (1998) no. 2, pp. 284-339

[8] H. Krause; D. Kussin Rouquier's theorem on representation dimension, Trends in Representation Theory of Algebras and Related Topics, Contemp. Math., vol. 406, 2006, pp. 95-103

[9] D.O. Orlov Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Proc. Steklov Inst. Math., Volume 246 (2004) no. 3, pp. 227-248

[10] D.O. Orlov Triangulated categories of singularities, and equivalences between Landau–Ginzburg models, Sb. Math., Volume 197 (2006) no. 11–12, pp. 1827-1840

[11] D. Orlov Derived categories of coherent sheaves and triangulated categories of singularities, Algebra, Arithmetic, and Geometry: In Honor of Yu.I. Manin. Vol. II, Progress in Mathematics, vol. 270, Birkhäuser Boston, Inc., Boston, MA, USA, 2009, pp. 503-531

[12] D. Orlov Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math., Volume 226 (2011) no. 1, pp. 206-217

[13] R. Rouquier Dimensions of triangulated categories, J. K-Theory, Volume 1 (2008), pp. 193-256

[14] H.-J. Wang On the Fitting ideals in free resolutions, Mich. Math. J., Volume 41 (1994) no. 3, pp. 587-608

[15] Y. Yoshino Cohen–Macaulay Modules over Cohen–Macaulay Rings, London Mathematical Society Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990

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