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Higher Initial Ideals of Homogeneous Ideals
 
Gunnar Fløystad University of Bergen, Bergen, Norway
Front Cover for Higher Initial Ideals of Homogeneous Ideals
Available Formats:
Electronic ISBN: 978-1-4704-0227-3
Product Code: MEMO/134/638.E
List Price: $45.00
MAA Member Price: $40.50
AMS Member Price: $27.00
Front Cover for Higher Initial Ideals of Homogeneous Ideals
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Higher Initial Ideals of Homogeneous Ideals
Gunnar Fløystad University of Bergen, Bergen, Norway
Available Formats:
Electronic ISBN:  978-1-4704-0227-3
Product Code:  MEMO/134/638.E
List Price: $45.00
MAA Member Price: $40.50
AMS Member Price: $27.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1341998; 68 pp
    MSC: Primary 13; 14;



    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
     
     
    • Chapters
    • Introduction
    • 1. Borel-fixed ideals
    • 2. Monomial orders
    • 3. Some algebraic lemmas
    • 4. Defining the higher initial ideals
    • 5. Representing the higher initial ideals
    • 6. Group action on $R^{s+1}(I)$
    • 7. Describing the action on $R^{s+1}(I)$
    • 8. Borel-fixedness
    • 9. Higher initial ideals of hyperplane sections
    • 10. Representing the higher initial ideals of general hyperplane sections
    • 11. Higher initial ideals as combinatorial structures
    • 12. Reading cohomological information
    • 13. Examples: Points and curves in $\mathbf {P}^3$
  • Requests
     
     
    Review Copy – for reviewers who would like to review an AMS book
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1341998; 68 pp
MSC: Primary 13; 14;



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.

  • Chapters
  • Introduction
  • 1. Borel-fixed ideals
  • 2. Monomial orders
  • 3. Some algebraic lemmas
  • 4. Defining the higher initial ideals
  • 5. Representing the higher initial ideals
  • 6. Group action on $R^{s+1}(I)$
  • 7. Describing the action on $R^{s+1}(I)$
  • 8. Borel-fixedness
  • 9. Higher initial ideals of hyperplane sections
  • 10. Representing the higher initial ideals of general hyperplane sections
  • 11. Higher initial ideals as combinatorial structures
  • 12. Reading cohomological information
  • 13. Examples: Points and curves in $\mathbf {P}^3$
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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