# American Institute of Mathematical Sciences

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] Aslak Bakke Buan, Henning Krause and Oyvind Solberg, Support varieties: An ideal approach, Homology, Homotopy Appl., 9 (2007), 45-74.  Google Scholar [5] Pierre Gabriel, Des catégories abéinnes, Bull. Soc. Math. France, 90 (1962), 323-448.  Google Scholar [6] Yu. I. Manin, "New Dimensions in Geometry," Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Math., 1111, Springer, Berlin, (1985), 59-101.  Google Scholar [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.  Google Scholar [8] Shigeru Mukai, Duality between $D(X)$ and $D(\hatX)$ with its application to Picard sheaves, Nagoya Math. J., 81 (1981), 153-175.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] Alexander L. Rosenberg, Noncommutative schemes, Compositio Math., 112 (1998), 93-125. doi: 10.1023/A:1000479824211.  Google Scholar [12] R. W. Thomason, The classification of triangulated subcategories, Compositio Math., 105 (1997), 1-27. doi: 10.1023/A:1017932514274.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] Aslak Bakke Buan, Henning Krause and Oyvind Solberg, Support varieties: An ideal approach, Homology, Homotopy Appl., 9 (2007), 45-74.  Google Scholar [5] Pierre Gabriel, Des catégories abéinnes, Bull. Soc. Math. France, 90 (1962), 323-448.  Google Scholar [6] Yu. I. Manin, "New Dimensions in Geometry," Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Math., 1111, Springer, Berlin, (1985), 59-101.  Google Scholar [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.  Google Scholar [8] Shigeru Mukai, Duality between $D(X)$ and $D(\hatX)$ with its application to Picard sheaves, Nagoya Math. J., 81 (1981), 153-175.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] Alexander L. Rosenberg, Noncommutative schemes, Compositio Math., 112 (1998), 93-125. doi: 10.1023/A:1000479824211.  Google Scholar [12] R. W. Thomason, The classification of triangulated subcategories, Compositio Math., 105 (1997), 1-27. doi: 10.1023/A:1017932514274.  Google Scholar
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