Higher Initial Ideals Of Homogeneous Ideals - new book

ISBN: 9780821808535

Given a homogeneous ideal $I$ and a monomial order, one may form the initial ideal $ extnormal{in}(I)$. The initial ideal gives information about $I$, for instance $I$ and $ extnormal{in}… More...

Given a homogeneous ideal $I$ and a monomial order, one may form the initial ideal $ extnormal{in}(I)$. The initial ideal gives information about $I$, for instance $I$ and $ extnormal{in}(I)$ have the same Hilbert function. However, if $mathcal I$ is the sheafification of $I$ one cannot read the higher cohomological dimensions $h^i({mathbf P}^n, mathcal I(; u)$ from $ extnormal{in}(I)$. This. Given a homogeneous ideal $I$ and a monomial order, one may form the initial ideal $ extnormal{in}(I)$. The initial ideal gives information about $I$, for instance $I$ and $ extnormal{in}(I)$ have the same Hilbert function. However, if $mathcal I$ is the sheafification of $I$ one cannot read the higher cohomological dimensions $h^i({mathbf P}^n, mathcal I(; u)$ from $ extnormal{in}(I)$. This work remedies this by defining a series of higher initial ideals $ extnormal{in} s(I)$ for $sgeq0$. Each cohomological dimension $h^i({mathbf P}^n, mathcal I(; u))$ may be read from the $ extnormal{in} s(I)$. The $ extnormal{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 $ extnormal{in} 0(I)$ and the higher initial ideal $ extnormal{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. Books, Science and Geography~~Mathematics~~Algebra, Higher Initial Ideals Of Homogeneous Ideals~~Book~~9780821808535~~Gunnar Floystad, , , , , , , , , ,, [PU: American Mathematical Society]<

1998, ISBN: 0821808532

[EAN: 9780821808535], Neubuch, [PU: American Mathematical Society], Mathematics|Algebra|General, Mathematics|Group Theory, Mathematics|Topology, Brand new. We distribute directly for the … More...

[EAN: 9780821808535], Neubuch, [PU: American Mathematical Society], Mathematics|Algebra|General, Mathematics|Group Theory, Mathematics|Topology, Brand new. We distribute directly for the publisher. 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 o<