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2 results

  • The resolution property of algebraic surfaces

  • Part of
  • Philipp Gross
  • Journal:
    Compositio Mathematica / Volume 148 / Issue 1 / January 2012
    Published online by Cambridge University Press:
    09 November 2011, pp. 209-226
    Print publication:
    January 2012
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  • We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective nor embeddable into toric varieties. Our methods extend to arbitrary two-dimensional schemes that are proper over an excellent ring.

  • Existence of vector bundles and global resolutions for singular surfaces

  • Stefan Schröer, Gabriele Vezzosi
  • Journal:
    Compositio Mathematica / Volume 140 / Issue 3 / May 2004
    Published online by Cambridge University Press:
    04 December 2007, pp. 717-728
    Print publication:
    May 2004
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  • We prove two results about vector bundles on singular algebraic surfaces. First, on proper surfaces there are vector bundles of rank two with arbitrarily large second Chern number and fixed determinant. Second, on separated normal surfaces any coherent sheaf is the quotient of a vector bundle. As a consequence, for such surfaces the Quillen K-theory of vector bundles coincides with the Waldhausen K-theory of perfect complexes. Examples show that, on non-separated schemes, usually many coherent sheaves are not quotients of vector bundles.