Kähler differential algebras for 0-dimensional schemes

Given a 0-dimensional scheme in a projective space \(\mathbb{P}^n\) over a field \(K\), we study the K\"ahler differential algebra \(\Omega_{R/K}\) of its homogeneous coordinate ring \(R\). Using explicit presentations of the modules \(\Omega^m_{R/K}\) of K\"ahler differential \(m\)-forms,...

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Bibliographic Details
Published in:arXiv.org 2017-04
Main Authors: Kreuzer, Martin, Linh, Tran N K, Long, Le Ngoc
Format: Article
Language:English
Subjects:
Functions (mathematics)
Mathematical analysis
Polynomials
Online Access:Get full text
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Summary:Given a 0-dimensional scheme in a projective space \(\mathbb{P}^n\) over a field \(K\), we study the K\"ahler differential algebra \(\Omega_{R/K}\) of its homogeneous coordinate ring \(R\). Using explicit presentations of the modules \(\Omega^m_{R/K}\) of K\"ahler differential \(m\)-forms, we determine many values of their Hilbert functions explicitly and bound their Hilbert polynomials and regularity indices. Detailed results are obtained for subschemes of \(\mathbb{P}^1\), fat point schemes, and subschemes of \(\mathbb{P}^2\) supported on a conic.
ISSN:2331-8422