# Linearity of free resolutions of monomial ideals

Linearity of free resolutions of monomial ideals We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except for the last step (the “almost linear” case). We also give sharp bounds on Castelnuovo–Mumford regularity and numbers of generators in some cases. It is a basic observation that linearity properties are inherited by the restriction of an ideal to a subset of variables, and we study when the converse holds. We construct fractal examples of almost linear primary ideals with relatively few generators related to the Sierpiński triangle. Our results also lead to classes of highly connected simplicial complexes Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta$$\end{document} that cannot be extended to the complete dimΔ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathrm{dim}\Delta$$\end{document}-skeleton of the simplex on the same variables by shelling. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

# Linearity of free resolutions of monomial ideals

, Volume 9 (2) – Jun 1, 2022
15 pages

/lp/springer-journals/linearity-of-free-resolutions-of-monomial-ideals-gHR2PKo0bi
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
eISSN
2197-9847
DOI
10.1007/s40687-022-00330-6
Publisher site
See Article on Publisher Site

### Abstract

We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except for the last step (the “almost linear” case). We also give sharp bounds on Castelnuovo–Mumford regularity and numbers of generators in some cases. It is a basic observation that linearity properties are inherited by the restriction of an ideal to a subset of variables, and we study when the converse holds. We construct fractal examples of almost linear primary ideals with relatively few generators related to the Sierpiński triangle. Our results also lead to classes of highly connected simplicial complexes Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta$$\end{document} that cannot be extended to the complete dimΔ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathrm{dim}\Delta$$\end{document}-skeleton of the simplex on the same variables by shelling.

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Jun 1, 2022

Keywords: Monomial ideals; Ndp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{d p}$$\end{document} conditions; Linear syzygies; Fractals; Shelling; 13F55; 13D02; 13C05

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