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Higher Initial Ideals of Homogeneous Ideals
Gunnar Fløystad, University of Bergen, Norway
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Memoirs of the American Mathematical Society
1998; 68 pp; softcover
Volume: 134
ISBN-10: 0-8218-0853-2
ISBN-13: 978-0-8218-0853-5
List Price: US$42
Individual Members: US$25.20
Institutional Members: US$33.60
Order Code: MEMO/134/638
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Given a homogeneous ideal \(I\) and a monomial order, one may form the initial ideal \(\mathrm{in}(I)\). The initial ideal gives information about \(I\), for instance \(I\) and \(\mathrm{in}(I)\) have the same Hilbert function. However, if \(\mathcal I\) is the sheafification of \(I\) one cannot read the higher cohomological dimensions \(h^i({\mathbf P}^n, \mathcal I(\nu))\) from \(\mathrm{in}(I)\). This work remedies this by defining a series of higher initial ideals \(\mathrm{ in}_s(I)\) for \(s\geq0\). Each cohomological dimension \(h^i({\mathbf P}^n, \mathcal I(\nu))\) may be read from the \(\mathrm{in}_s(I)\). The \(\mathrm{in}_s(I)\) are however more refined invariants and contain considerably more information about the ideal \(I\).

This work considers in particular the case where \(I\) is the homogeneous ideal of a curve in \({\mathbf P}^3\) and the monomial order is reverse lexicographic. Then the ordinary initial ideal \(\mathrm{in}_0(I)\) and the higher initial ideal \(\mathrm{in}_1(I)\) have very simple representations in the form of plane diagrams.

Features:

  • enables one to visualize cohomology of projective schemes in \({\mathbf P}^n\)
  • provides an algebraic approach to studying projective schemes
  • gives structures which are generalizations of initial ideals

Readership

Research mathematicians in commutative algebra, computer algebra and algebraic geometry.

Table of Contents

  • Introduction
  • Borel-fixed ideals
  • Monomial orders
  • Some algebraic lemmas
  • Defining the higher initial ideals
  • Representing the higher initial ideals
  • Group action on \(R^{s+1}(I)\)
  • Describing the action on \(R^{s+1}(I)\)
  • Borel-fixedness
  • Higher initial ideals of hyperplane sections
  • Representing the higher initial ideals of general hyperplane sections
  • Higher initial ideals as combinatorial structures
  • Reading cohomological information
  • Examples: Points and curves in \(\mathbf P^3\)
  • References
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