Higher initial ideals of homogeneous ideals
Author(s)
Bibliographic Information
Higher initial ideals of homogeneous ideals
（Memoirs of the American Mathematical Society, no. 638）
American Mathematical Society, 1998
Available at 19 libraries
Note
"July 1998, volume 134, number 638 (fourth of 6 numbers)"
Includes bibliographical references
Description and Table of Contents
Description
Given a homogeneous ideal $I$ and a monomial order, one may form the initial ideal $\textnormal{in}(I)$. The initial ideal gives information about $I$, for instance $I$ and $\textnormal{in}(I)$ 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 $\textnormal{in}(I)$. This work remedies this by defining a series of higher initial ideals $\textnormal{in}_s(I)$ for $s\geq0$. Each cohomological dimension $h^i({\mathbf P}^n, \mathcal I(\nu))$ may be read from the $\textnormal{in}_s(I)$. The $\textnormal{in}_s(I)$ 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 $\textnormal{in}_0(I)$ and the higher initial ideal $\textnormal{in}_1(I)$ have very simple representations in the form of plane diagrams. It enables one to visualize cohomology of projective schemes in ${\mathbf P}^n$. It provides an algebraic approach to studying projective schemes. It gives structures which are generalizations of initial ideals.
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.
by "Nielsen BookData"