Transactions of the American Mathematical Society

Author(s): I. Coanda; G. Trautmann
Journal: Trans. Amer. Math. Soc. 358 (2006), 1015-1031.
MSC (2000): Primary 14F05, 15A75, 16E05
Posted: March 31, 2005
Retrieve article in: PDF DVI PostScript

Abstract: We construct an explicit equivalence between a category of complexes over the exterior algebra, which we call HT-complexes, and the stable category of vector bundles on the corresponding projective space, essentially translating into more fancy terms the results of Trautmann (1978) which, in turn, were influenced by ideas of Horrocks (1964), (1980). However, the result expressed by Theorem 5.1 and its corollary, which establishes a relation between the Tate resolutions over the exterior algebra (described in a paper by Eisenbud, Fløystad, and Schreyer) and HT-complexes, might be new, although, perhaps, not a surprise to experts.

1.: H. Abo, W. Decker, N. Sasakura, An elliptic conic bundle in $\mathbb{P} ^4$ arising from a stable rank-3 vector bundle, Math. Z. 229 (1998), 725-741. MR 1664785 (2001b:14071)
2.: I.N. Bernstein, I.M. Gel'fand, S.I. Gel'fand, Algebraic bundles over $\mathbb{P} ^n$ and problems of linear algebra, Funktsional'nyi Analyz i Ego Prilozheniya 12, No. 3 (1978), 66-67. MR 0509387 (80c:14010a)
3.: I. Coanda, On the Bernstein-Gel'fand-Gel'fand correspondence and a result of Eisenbud, Fløystad, and Schreyer, Journ. Math. Kyoto Univ. 43 (2003), 429-439. MR 2051032
4.: D. Eisenbud, G. Fløystad, F.-O. Schreyer, Sheaf cohomology and free resolutions over exterior algebras, Transactions of the AMS 355 (2003), 4397-4426. MR 1990756 (2004f:14031)
5.: G. Fløystad, Describing coherent sheaves on projective spaces via Koszul duality, preprint math.AG/0012263.
6.: G. Fløystad, Koszul duality and equivalences of categories, preprint math.RA/0012264.
7.: S.I. Gel'fand, Sheaves on $\mathbb{P} ^n$ and problems of linear algebra, appendix to the Russian edition of the book Ch. Okonek, M. Schneider, H. Spindler: Vector Bundles on Complex Projective Spaces, Birkhäuser 1980, Mir, Moscow 1984. MR 0561910 (81b:14001); MR 0778380 (86i:14005)
8.: Ph. Griffiths, J. Harris, Principles of algebraic geometry, John Wiley & Sons, Inc., New York 1978. MR 0507725 (80b:14001)
9.: G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. 14 (1964), 689-713. MR 0169877 (30:120)
10.: G. Horrocks, Construction of bundles on $\mathbb{P} ^n$ , In: A. Douady and J.-L. Verdier (eds.) Les équations de Yang-Mills, Séminaire E.N.S. (1977-1978), Astérisque 71-72, Soc. Math. de France (1980), 197-203. MR 0589891 (84j:14026)
11.: G. Horrocks, Examples of rank three vector bundles on five-dimensional projective space, J. London Math. Soc. 18 (1978), 15-27.MR 0502651 (80d:14011)
12.: G. Horrocks, D. Mumford, A rank 2 vector bundle on $\mathbb{P} ^4$ with 15,000 symmetries, Topology 12 (1973), 63-81. MR 0382279 (52:3164)
13.: H. Tango, An example of indecomposable vector bundle of rank on $\mathbb{P} ^n$ , J. Math. Kyoto Univ. 16, No. 1 (1976), 201-207. MR 0401766 (53:5593)
14.: G. Trautmann, Darstellung von Vektorraumbündeln über ${\mathbb C}^n\smallsetminus\{0\}$ , Archiv der Mathematik (Basel) 24 (1973), 303-313. MR 0352523 (50:5010)
15.: G. Trautmann, Moduli of Vector Bundles on $\mathbb{P} _n(\mathbb C)$ , Math. Ann. 237 (1978), 167-186. MR 0507912 (80a:14008)
16.: U. Vetter, Zu einem Satz von G. Trautmann über den Rang gewisser kohärenter analytischer Moduln, Archiv der Mathematik (Basel) 24 (1973), 158-161. MR 0344518 (49:9257)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14F05, 15A75, 16E05

I. Coanda
Affiliation: Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO--70700 Bucharest, Romania
Email: Iustin.Coanda@imar.ro

G. Trautmann
Affiliation: Fachbereich Mathematik, Universität Kaiserslautern, Erwin-Schrödinger-Straße, D-67663 Kaiserslautern, Germany
Email: trm@mathematik.uni-kl.de

DOI: 10.1090/S0002-9947-05-03755-4
PII: S 0002-9947(05)03755-4
Received by editor(s): December 11, 2003
Received by editor(s) in revised form: March 24, 2004
Posted: March 31, 2005
Additional Notes: The first author was partially supported by DFG and by CERES grant 152/2001 of the Romanian Ministry of Education and Research
The research of the second author was supported by the DFG-Schwerpunktprogramm 1094
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS