# Higher Initial Ideals of Homogeneous Ideals

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*Gunnar Fløystad*

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

#### Table of Contents

# Table of Contents

## Higher Initial Ideals of Homogeneous Ideals

- Contents vii8 free
- Introduction 110 free
- Section 1. Borel-fixed ideals 615 free
- Section 2. Monomial orders 1120
- Section 3. Some algebraic lemmas 2029
- Section 4. Defining the higher initial ideals 2231
- Section 5. Representing the higher initial ideals 3039
- Section 6. Group action on R[sup(s+1)](I) 3342
- Section 7. Describing the action on R[sup(s+1)](I) 3847
- Section 8. Borel-fixedness 4352
- Section 9. Higher initial ideals of hyperplane sections 4756
- Section 10. Representing the higher initial ideals of general hyperplane sections 5160
- Section 11. Higher initial ideals as combinatorial structures 5261
- Section 12. Reading cohomological information 5463
- Section 13. Examples : Points and curves in P[sup(3)] 5968
- References 6776