Electronic ISBN:  9781470402273 
Product Code:  MEMO/134/638.E 
68 pp 
List Price:  $45.00 
MAA Member Price:  $40.50 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 134; 1998MSC: 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
ReadershipResearch mathematicians in commutative algebra, computer algebra and algebraic geometry.

Table of Contents

Chapters

Introduction

1. Borelfixed 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. Borelfixedness

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 $\mathbf {P}^3$


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

Chapters

Introduction

1. Borelfixed 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. Borelfixedness

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 $\mathbf {P}^3$