Memoirs of the American Mathematical Society 1998; 68 pp; softcover Volume: 134 ISBN10: 0821808532 ISBN13: 9780821808535 List Price: US$45 Individual Members: US$27 Institutional Members: US$36 Order Code: MEMO/134/638
 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
 Borelfixed 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 \(R^{s+1}(I)\)
 Borelfixedness
 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
