The regularity of Tor and graded Betti Numbers

By David Eisenbud, Craig Huneke, Bernd Ulrich. 2004

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Let <i>S = K</i>[<i>x</i><sub xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>, . . . , <i>x<sub>n</sub></i>], let <i>A, B</i> be finitely generated graded <i>S</i>-modules, and let <b>m</b> = (<i>x</i><sub xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>, . . . , <i>x<sub>n</sub></i>) ⊂ <i>S</i>. We give bounds for the regularity of the local cohomology of Tor<i><sub>k</sub> (A, B)</i> in terms of the graded Betti numbers of <i>A</i> and <i>B</i>, under the assumption that dim Tor<sub xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (<i>A, B</i>) ≤ 1. We apply the results to syzygies, Gröbner bases, products and powers of ideals, and to the relationship of the Rees and symmetric algebras. For example we show that any homogeneous linearly presented <b>m</b>-primary ideal has some power equal to a power of <b>m</b>; and if the first [(<i>n</i> - 1)/2] steps of the resolution of <i>I</i> are linear, then <i>I</i><sup>2</sup> is a power of <b>m</b>.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bfd842a0c972b645f250f6270eba13c86f9c82b6

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In Algebra and Number Theory

  1. Andrew R. Kustin
  2. C. Polini
  3. Bernd Ulrich

In Computational Theory and Mathematics

  1. Andrew R. Kustin
  2. C. Polini
  3. Bernd Ulrich

In Geometry and Topology

  1. Andrew R. Kustin
  2. C. Polini
  3. Bernd Ulrich

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