2011, 18: 50-53. doi: 10.3934/era.2011.18.50

Spectrum of some triangulated categories

1. 

Institute of Mathematical Sciences, IV Cross Road, CIT Campus, Taramani, Chennai 600 113, Tamil Nadu, India

2. 

Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain

Received  March 2011 Published  July 2011

In this note we announce the computation of the triangular spectrum (as defined by P. Balmer) of two classes of tensor triangulated categories which are quite common in algebraic geometry. One of them is the derived category of $G$-equivariant sheaves on a smooth quasi projective scheme $X$ for a finite group $G$ which acts on $X$. The other class is the derived category of split super-schemes.
Citation: Umesh V. Dubey, Vivek M. Mallick. Spectrum of some triangulated categories. Electronic Research Announcements, 2011, 18: 50-53. doi: 10.3934/era.2011.18.50
References:
[1]

Paul Balmer, Presheaves of triangulated categories and reconstruction of schemes, Math. Ann., 324 (2002), 557-580. doi: 10.1007/s00208-002-0353-1.

[2]

Paul Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math., 588 (2005), 149-168. doi: 10.1515/crll.2005.2005.588.149.

[3]

Alexei Bondal and Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math., 125 (2001), 327-344. doi: 10.1023/A:1002470302976.

[4]

Aslak Bakke Buan, Henning Krause and Oyvind Solberg, Support varieties: An ideal approach, Homology, Homotopy Appl., 9 (2007), 45-74.

[5]

Pierre Gabriel, Des catégories abéinnes, Bull. Soc. Math. France, 90 (1962), 323-448.

[6]

Yu. I. Manin, "New Dimensions in Geometry," Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Math., 1111, Springer, Berlin, (1985), 59-101.

[7]

Yuri I. Manin, "Gauge field theory and complex geometry," Grundlehren der Mathematischen Wissenschaften, (Russian) [Fundamental Principles of Mathematical Sciences]," 289, Translated from Russian by N. Koblitz and J. R. King, Springer-Verlag, Berlin, 1988.

[8]

Shigeru Mukai, Duality between $D(X)$ and $D(\hatX)$ with its application to Picard sheaves, Nagoya Math. J., 81 (1981), 153-175.

[9]

David Mumford, "Abelian varieties," with appendices by C. P. Ramanujam and Yuri Manin, corrected reprint of the 2nd (1974) edition, Tata Institute of Fundamental Research Studies in Mathematics, 5, Published for the Tata Institute of Fundamental Research, Bombay, by Hindustan Book Agency, New Delhi, 2008.

[10]

Amnon Neeman, The Grothendick duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc., 9 (1996), 205-236. doi: 10.1090/S0894-0347-96-00174-9.

[11]

Alexander L. Rosenberg, Noncommutative schemes, Compositio Math., 112 (1998), 93-125. doi: 10.1023/A:1000479824211.

[12]

R. W. Thomason, The classification of triangulated subcategories, Compositio Math., 105 (1997), 1-27. doi: 10.1023/A:1017932514274.

show all references

References:
[1]

Paul Balmer, Presheaves of triangulated categories and reconstruction of schemes, Math. Ann., 324 (2002), 557-580. doi: 10.1007/s00208-002-0353-1.

[2]

Paul Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math., 588 (2005), 149-168. doi: 10.1515/crll.2005.2005.588.149.

[3]

Alexei Bondal and Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math., 125 (2001), 327-344. doi: 10.1023/A:1002470302976.

[4]

Aslak Bakke Buan, Henning Krause and Oyvind Solberg, Support varieties: An ideal approach, Homology, Homotopy Appl., 9 (2007), 45-74.

[5]

Pierre Gabriel, Des catégories abéinnes, Bull. Soc. Math. France, 90 (1962), 323-448.

[6]

Yu. I. Manin, "New Dimensions in Geometry," Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Math., 1111, Springer, Berlin, (1985), 59-101.

[7]

Yuri I. Manin, "Gauge field theory and complex geometry," Grundlehren der Mathematischen Wissenschaften, (Russian) [Fundamental Principles of Mathematical Sciences]," 289, Translated from Russian by N. Koblitz and J. R. King, Springer-Verlag, Berlin, 1988.

[8]

Shigeru Mukai, Duality between $D(X)$ and $D(\hatX)$ with its application to Picard sheaves, Nagoya Math. J., 81 (1981), 153-175.

[9]

David Mumford, "Abelian varieties," with appendices by C. P. Ramanujam and Yuri Manin, corrected reprint of the 2nd (1974) edition, Tata Institute of Fundamental Research Studies in Mathematics, 5, Published for the Tata Institute of Fundamental Research, Bombay, by Hindustan Book Agency, New Delhi, 2008.

[10]

Amnon Neeman, The Grothendick duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc., 9 (1996), 205-236. doi: 10.1090/S0894-0347-96-00174-9.

[11]

Alexander L. Rosenberg, Noncommutative schemes, Compositio Math., 112 (1998), 93-125. doi: 10.1023/A:1000479824211.

[12]

R. W. Thomason, The classification of triangulated subcategories, Compositio Math., 105 (1997), 1-27. doi: 10.1023/A:1017932514274.

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