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Projective dimension of weakly chordal graphic arrangements
Abe, Takuro1; Kühne, Lukas2; Mücksch, Paul3; Mühlherr, Leonie2
1 Rikkyo University Department of Mathematics 3-34-1 Nishi-Ikebukuro Toshima-ku 1718501 Tokyo (Japan)
2 Universität Bielefeld Fakultät für Mathematik D-33501 Bielefeld (Germany)
3 Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik D-30167 Hannover (Germany)
Algebraic Combinatorics, Volume 8 (2025) no. 1, pp. 157-174.

A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.403
Classification: 52C35, 14N20, 32S22
Keywords: arrangement of hyperplanes, graphic arrangement, module of logarithmic derivations, weakly chordal graphs

Abe, Takuro 1; Kühne, Lukas 2; Mücksch, Paul 3; Mühlherr, Leonie 2

1 Rikkyo University Department of Mathematics 3-34-1 Nishi-Ikebukuro Toshima-ku 1718501 Tokyo (Japan)
2 Universität Bielefeld Fakultät für Mathematik D-33501 Bielefeld (Germany)
3 Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik D-30167 Hannover (Germany)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Abe, Takuro; Kühne, Lukas; Mücksch, Paul; Mühlherr, Leonie. Projective dimension of weakly chordal graphic arrangements. Algebraic Combinatorics, Volume 8 (2025) no. 1, pp. 157-174. doi : 10.5802/alco.403. https://alco.centre-mersenne.org/articles/10.5802/alco.403/

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