Homology, Homotopy and Applications

The Horrocks correspondence for coherent sheaves on projective spaces

Iustin Coandă

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We establish an equivalence between the stable category of coherent sheaves (satisfying a mild restriction) on a projective space and the homotopy category of a certain class of minimal complexes of free modules over the exterior algebra Koszul dual to the homogeneous coordinate algebra of the projective space. We also relate these complexes to the Tate resolutions of the respective sheaves. In this way, we extend from vector bundles to coherent sheaves the results of G. Trautmann and the author (2005), which interpret in terms of the BGG correspondence the results of Trautmann (1978) about the correspondence of Horrocks (1964), (1977). We also give direct proofs of the BGG correspondences for graded modules and for coherent sheaves and of the theorem of Eisenbud, Fløystad and Schreyer (2003) describing the linear part of the Tate resolution associated to a coherent sheaf. Moreover, we provide an explicit description of the quotient of the Tate resolution by its linear strand corresponding to the module of global sections of the various twists of the sheaf.

Article information

Homology Homotopy Appl., Volume 12, Number 1 (2010), 327-353.

First available in Project Euclid: 28 January 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 13A02: Graded rings [See also 16W50] 13D25 18E30: Derived categories, triangulated categories

Coherent sheaf projective space stable category derived category


Coandă, Iustin. The Horrocks correspondence for coherent sheaves on projective spaces. Homology Homotopy Appl. 12 (2010), no. 1, 327--353. https://projecteuclid.org/euclid.hha/1296223832

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