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Koszul Graded M\"obius Algebras and Strongly Chordal Graphs

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Pith's one-line read For a graph $G$, the graded Möbius algebra of its cycle matroid is Koszul exactly when $G$ is strongly chordal, and this is equivalent to having a quadratic Gröbner basis and to the cycle matroid being strongly T-chordal.

desk verdict Promising characterization of Koszul Möbius algebras of graphic matroids, undercut by a real but likely repairable gap in one implication. read the letter →

arxiv 2412.18499 v1 pith:7YKATTZV submitted 2024-12-24 math.AC math.CO

Adam LaClair , Matthew Mastroeni , Jason McCullough , Irena Peeva This is my paper · ORCID
classification math.ACmath.CO MSC 16S3713E1005B3513P1005E4005C25
keywords KoszulalgebragradedMöbiuscyclematroidstronglychordalgraphquadraticGröbnerbasisT-chordalMAT-labeling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to determine when the graded Möbius algebra of a matroid is Koszul, and it solves this for cycle matroids of graphs. The result is a perfect match with classical graph theory: for a graph $G$, the algebra $B_G$ is Koszul if and only if $G$ is strongly chordal. The same condition is also equivalent to $B_G$ having a quadratic Gröbner basis and to the cycle matroid $M(G)$ being strongly T-chordal, a new edge-ordering notion. Since every Koszul algebra is quadratic, the paper identifies exactly how far chordality alone gets you: $B_G$ is quadratic for every chordal graph, but Koszulness requires the stronger, trampoline-free condition. A byproduct is a new characterization of strongly chordal graphs by edge orderings rather than vertex orderings.

What carries the argument

The central objects are the graded Möbius algebra $B_M$ of a matroid and, for graphs, the strong edge elimination order. A MAT-triple for a set $S$ is a 3-cycle $\{u,v,w\}$ with $w \succ \min(u,v)$ in a fixed edge order; a circuit is a MAT-circuit if deleting any non-minimal element leaves a set with a MAT-triple; and a matroid is strongly T-chordal when every circuit of size at least four is a MAT-circuit. Theorem 3.9 is the load-bearing equivalence: a quadratic Gröbner basis for $B_M$ exists exactly under strong T-chordality. To connect this to graphs, the proof uses MAT-labelings (edge labelings whose equal-label layers are forests with controlled triangle counts) to construct strong edge elimination orders on strongly chordal graphs, and uses algebra retracts plus Poincaré-series factorizations for large homomorphisms to show that induced trampolines force non-Koszulness.

What would settle it

Take the 3-trampoline $T$ and compute the minimal free resolution of the ground field over its graded Möbius algebra $B_T$; the proof of Theorem 5.2 predicts a non-linear syzygy already in low homological degree, and in the broken 3-trampoline case a minimal quadratic generator in the annihilator $(0:B_B a)$ prevents linearity. If that resolution is linear, or if any graph with an induced trampoline has a Koszul $B_G$ despite not being strongly chordal, the equivalence in Theorem A collapses.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem A: for a graph $G$ with cycle matroid $M(G)$ and graded Möbius algebra $B_G$, the four statements "$M(G)$ is strongly T-chordal", "$B_G$ has a quadratic Gröbner basis", "$B_G$ is Koszul", and "$G$ is strongly chordal" are equivalent. The graded Möbius algebra is the algebra spanned by the flats of the matroid with $y_Fy_G = y_{F\vee G}$ when the ranks add and $0$ otherwise, so it records the lattice of flats in graded form. The forward direction builds a strong edge elimination order on any strongly chordal graph using MAT-labelings, while the reverse direction shows that any graph failing strong chordality contains an induced trampoline $T$ such that $B_T$ is a retract of $B_G$ and is provably not Koszul. This turns Koszulness into a purely graph-theoretic property and yields the edge-ordering characterization of Theorem B.

Load-bearing premise

The proof that strong chordality forces a quadratic Gröbner basis rests on the external MAT-labeling characterization of strongly chordal graphs—that every such graph has an edge labeling whose equal-label layers are forests with controlled triangle counts, and that these labelings restrict to induced subgraphs, maximal cliques, and unions of overlapping cliques; if that characterization or its restriction property fails, the construction of the strong edge elimination order in Theorem 4.5 collapses.

Editorial extensions

If this is right

  • For any graph $G$, checking whether $B_G$ is Koszul is the same as checking whether $G$ is strongly chordal.
  • Chordal graphs are exactly the graphs for which $B_G$ is quadratic; the additional step from quadratic to Koszul is the absence of induced trampolines.
  • Strongly chordal graphs acquire a new edge-ordering characterization: an edge order in which every cycle of length at least four is a MAT-circuit.
  • Unlike the Orlik-Solomon algebra, which is Koszul for every chordal graph, the graded Möbius algebra sees the finer strongly-chordal boundary.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the characterization is taken as a template, the natural matroid-level question is whether strong T-chordality is equivalent to supersolvability of the matroid; the paper poses this as an open problem, and a positive answer would make the graphic theorem part of a broader dichotomy.
  • The trampoline algebras suggest a family of quadratic algebras whose resolutions stay linear for exactly $n$ steps before breaking; if the paper's computed pattern holds for all $n$, these would be characteristic-independent counterparts to classical non-Koszul examples with arbitrarily long linear resolutions.
  • Because the edge-ordering condition in Theorem B is a local triangle condition on each cycle, it could in principle support a direct algorithmic test for strong chordality, though algorithmic complexity is not discussed in the paper.
  • Example 5.6 shows that outside graphic matroids, Koszulness does not coincide with having a quadratic Gröbner basis, so the clean equivalence of Theorem A is special to the graphic case rather than a universal matroid phenomenon.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies when the graded Möbius algebra of a matroid is Koszul, focusing on cycle matroids of graphs. The main theorem (Theorem A) asserts an equivalence, for a graph G, among strong T-chordality of the cycle matroid M(G), existence of a quadratic Gröbner basis for B_G, Koszulness of B_G, and strong chordality of G. The paper gives a presentation and Gröbner bases for graded Möbius algebras (Proposition 3.1), introduces MAT-triples and strong T-chordality (Definition 3.8), relates these to Tran–Tsujie MAT-labelings (Theorem 4.5), proves a converse edge-ordering characterization (Theorem 4.8), and shows that trampoline graphs have non-Koszul Möbius algebras (Theorem 5.2). A new edge-ordering characterization of strongly chordal graphs is presented as Theorem B.

Significance. If correct, Theorem A fully characterizes the Koszul property for graded Möbius algebras of graphic matroids, identifying it with the classical graph class of strongly chordal graphs. It also yields a new edge-ordering characterization of strongly chordal graphs, and it adds to the short list of settings where the Koszul property is equivalent to the existence of a quadratic Gröbner basis. The paper is careful to state external dependencies, especially the Tran–Tsujie characterization of strongly chordal graphs via MAT-labelings. The explicit, machine-checkable Gröbner basis arguments and the detailed trampoline computation are notable strengths. The main theorem is plausible and the paper contains substantial useful structure, but one load-bearing proof step is currently incomplete.

major comments (2)
  1. [3.3, Theorem 3.9] In the proof of (d)=>(c), the authors show only that y_{C\i} is not a minimal generator of in_{>lex}(Q) for each circuit C of size four, and then conclude that in_{>lex}(Q) has no minimal generators of degree three and, citing [34, 34.13], that the quadratic generators of Q form a Gröbner basis. However, the Gröbner basis of Proposition 3.1(b) also contains monomials y_{C\i} for circuits of size at least five, which have degree at least four. An ideal generated by quadrics can have an initial ideal with no minimal generators of degree three and yet have minimal generators of higher degree, so [34, 34.13] does not justify the conclusion. This gap is load-bearing: Theorem 4.5 invokes (d)=>(c) to conclude that a chordal graph with every 4-cycle a MAT-circuit has a quadratic Gröbner basis. The proof must either handle circuits of size at least five, or Theorem 3.9(d) must be restricted to a setting where such an additional argument is supplied.
  2. [4.2, Theorem 4.5] In the induction step, the authors assert that λ restricts to a MAT-labeling of G' = G \setminus π_{ω(G)-1} and that G' is strongly chordal, saying this is 'clear' from Proposition 4.4. Proposition 4.4 describes the intersection of π_k with maximal cliques and does not address edge-deleted subgraphs; G' is not an induced subgraph, so the restriction results [44, 4.9] do not directly apply. Removing the maximum-label edges could in principle destroy chordality by deleting all chords of a cycle, so this step needs an explicit proof or a precise reference.
minor comments (4)
  1. [Introduction, Theorem B] The displayed statement of Theorem B contains a corrupted expression 'C /i⋉tegerdividemin≺ C'; it should read C \setminus {min_≺ C}. This appears to be an OCR artifact but should be fixed.
  2. [3.3, Theorem 3.9] The citation [34, 34.13] is used for a crucial conclusion, but the statement of that result is not given. Please state explicitly what [34, 34.13] says so the reader can verify that it applies to the situation at hand.
  3. [5, Lemma 5.3] The proof of Lemma 5.3 is quite long and dense; a short overview of the strategy before the technical details would improve readability.
  4. [6] The Betti tables for n=5 and n=6 use asterisks for some entries; please provide the actual numbers or explicitly say they are omitted for space.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Koszul characterization is derived from explicit Gröbner-basis arguments and external MAT-labeling theorems, not from its own conclusion.

full rationale

The paper's derivation chain is self-contained in the relevant sense: there are no fitted parameters or normalization choices that force the conclusion, and no central definition is given in terms of the target property. Theorem 3.9 proves the equivalence between quadratic Gröbner bases and strong T-chordality directly from the universal Gröbner basis of Proposition 3.1, by checking divisibility of broken-circuit monomials by quadratic leading terms; neither direction assumes the conclusion. The bridge from strong T-chordality of cycle matroids to classical strong chordality of graphs uses the external Tran–Tsujie MAT-labeling theorem [44], which is not authored by the present authors, and Theorem 4.8 supplies an independent direct converse. The Koszul-to-strongly-chordal direction in Theorem 5.2 is established via algebra retracts and Poincaré-series obstructions from trampolines, again without importing the claimed equivalence. The self-citations that occur ([26], [29], [34]) are motivational, standard textbook facts, or auxiliary technical references; none is the sole load-bearing justification for the main theorem. In particular, [34, 34.13] is a standard parameter-free Gröbner-basis criterion whose assumptions do not include the target result. The reviewer concern about Theorem 3.9(d)=>(c) omitting circuits of size at least five is a potential proof gap or correctness risk, not a circularity, because it concerns whether the argument covers all degrees, not whether the conclusion is assumed in an input equation or defined into existence.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claims require no fitted constants or ad hoc numerical inputs. They rest on published theorems in matroid theory, graph theory, and homological algebra. The only specialized external load is the Tran-Tsujie MAT-labeling characterization.

assumptions (5)
  • standard math Matroid restriction, contraction, simplification, and flat lattice facts, including closure identities used in Lemma 5.3 and Section 5.
    Used throughout; the standard references are Oxley [32, 3.1.8, 3.1.12, 1.7.5].
  • domain assumption Embedding of B_M into the augmented Chow ring and the two lemmas from Braden, Huh, Matherne, Wang that give the presentation of B_M.
    This is the starting point of Proposition 3.1 and is cited from [3, 2.15, Lemma 2.9, Proposition 2.15].
  • domain assumption Farber's forbidden-subgraph characterization: strongly chordal graphs are exactly chordal graphs with no induced n-trampoline.
    Used in Theorem 4.8 and Theorem 5.2 to translate non-strong-chordality into the presence of a trampoline [15, 3.3, 4.1].
  • domain assumption Tran-Tsujie MAT-labeling theorem: every strongly chordal graph admits a MAT-labeling satisfying (ML1)-(ML3), and such labelings restrict to induced subgraphs, maximal cliques, and certain unions.
    Load-bearing for Theorem 4.5 and Proposition 4.4; cited as [44, 4.9, 4.10, 5.8, 5.12].
  • standard math Poincaré series tools: large homomorphisms of Levin [25] and Herzog [22], and the Conca-De Negri-Rossi criterion that a quotient of a Koszul algebra by an ideal with linear resolution is Koszul [8, Theorem 2].
    Used in Lemma 5.4 and Theorem 5.2 to prove trampoline Möbius algebras are not Koszul.

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Pith. "Pith review of Koszul Graded M\"obius Algebras and Strongly Chordal Graphs." pith.science (2026). https://pith.science/paper/7YKATTZV

@misc{pith2026241218499,
  author       = {Pith},
  title        = {Pith review of: Koszul Graded M\"obius Algebras and Strongly Chordal Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7YKATTZV}},
  note         = {Machine review of arXiv:2412.18499}
}
read the original abstract

The graded M\"{o}bius algebra of a matroid is a commutative graded algebra which encodes the combinatorics of the lattice of flats of the matroid. As a special subalgebra of the augmented Chow ring of the matroid, it plays an important role in the recent proof of the Dowling-Wilson Top Heavy Conjecture. Recently, Mastroeni and McCullough proved that the Chow ring and the augmented Chow ring of a matroid are Koszul. We study when graded M\"obius algebras are Koszul. We characterize the Koszul graded M\"obius algebras of cycle matroids of graphs in terms of properties of the graphs. Our results yield a new characterization of strongly chordal graphs via edge orderings.

Figures

Figures reproduced from arXiv: 2412.18499 by the authors.

Figure 3
Figure 3. below summarizes the connections between the various co [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 3.1
Figure 3.1. below summarizes the connections between the various combinatorial properties of M and quadracity properties of BM and AM discussed. We conclude this section with a few examples related to certain implications in the figure. supersolvable binary and supersolvable strongly T-chordal AM quadratic GB C-chordal BM quadratic GB AM quadratic BM quadratic 3-independent line-closed T-chordal [5, 2.8] [9, 2.2] Thm 3.9 [33, 3… view at source ↗
Figure 3.2
Figure 3.2. The Betsy Ross matroid In the remaining sections of the paper, we specialize to studying the graded M¨obius algebras of graphic matroids. Below we observe that there exist non-graphic matroids which are strongly T-chordal and, hence, have Koszul graded M¨obius algebras. Example 3.13. The Fano matroid F7 is the projective geometry of points in the projective plane over Z/2Z shown below with points being represented b… view at source ↗
Figures from the paper (2 more)
Figure 4.1
Figure 4.1. Figure 4.1: The 4-trampoline Trampoline graphs are chordal, since it is easily seen that w1, . . . , wn, v1, . . . , vn is a perfect elimination order for the n-trampoline. On the other hand, setting wi+n = wi , we note that wi ∈ N[vi+1] r N[vi+2] and wi+2 ∈ N[vi+2] r N[vi+1] fo…
Figure 5.1
Figure 5.1. Figure 5.1: Chordality and Quadracity Properties for Graded M¨obius and Orlik-Solomon Algebras of Graphic Matroids [PITH_FULL_IMAGE:figures/full_fig_p024_5_1.png]

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Reference graph

Works this paper leans on

47 extracted references · 46 canonical work pages

  1. [1]

    The closure of a linear space in a product of lines

    Federico Ardila and Adam Boocher. The closure of a linear space in a product of lines. J. Algebraic Combin. , 43(1):199–235, 2016

    work page 2016

  2. [2]

    Matherne, Nicholas Proudf oot, and Botong W ang

    Tom Braden, June Huh, Jacob P. Matherne, Nicholas Proudf oot, and Botong W ang. Singular Hodge theory for combinatori al geometries, 2020. arXiv.CO 2010.06088

    arXiv 2020

  3. [3]

    Matherne, Nicholas Proudf oot, and Botong W ang

    Tom Braden, June Huh, Jacob P. Matherne, Nicholas Proudf oot, and Botong W ang. A semi-small decomposition of the Chow ring of a matroid. Adv. Math. , 409(part A):Paper No. 108646, 49, 2022

    work page 2022

  4. [4]

    The homology and shellability of matro ids and geometric lattices

    Anders Bj¨ orner. The homology and shellability of matro ids and geometric lattices. In Matroid applications , volume 40 of Encyclopedia Math. Appl. , pages 226–283. Cambridge Univ. Press, Cambridge, 1992. KOSZUL GRADED M ¨OBIUS ALGEBRAS 27

    work page 1992

  5. [5]

    Anders Bj¨ orner and G¨ unter M. Ziegler. Broken circuit c omplexes: factorizations and generalizations. J. Combin. Theory Ser. B, 51(1):96–126, 1991

    work page 1991

  6. [6]

    The pinched Veronese is Koszul

    Giulio Caviglia. The pinched Veronese is Koszul. J. Algebraic Combin. , 30(4):539–548, 2009

    work page 2009

  7. [7]

    Koszul property of proje ctions of the Veronese cubic surface

    Giulio Caviglia and Aldo Conca. Koszul property of proje ctions of the Veronese cubic surface. Adv. Math. , 234:404–413, 2013

    work page 2013

  8. [8]

    K oszul algebras and regularity

    Aldo Conca, Emanuela De Negri, and Maria Evelina Rossi. K oszul algebras and regularity. In Commutative algebra , pages 285–315. Springer, New York, 2013

    work page 2013

Show all 47 references
  1. [9]

    How is a c hordal graph like a supersolvable binary matroid? Discrete Math

    Raul Cordovil, David Forge, and Sulamita Klein. How is a c hordal graph like a supersolvable binary matroid? Discrete Math. , 288(1-3):167–172, 2004

    2004

  2. [10]

    Koszul algebras and their syzygies

    Aldo Conca. Koszul algebras and their syzygies. In Combinatorial algebraic geometry , volume 2108 of Lecture Notes in Math. , pages 1–31. Springer, Cham, 2014

    2014

  3. [11]

    Supersolvability of built lattices and K oszulness of generalized Chow rings, 2023

    Basile Coron. Supersolvability of built lattices and K oszulness of generalized Chow rings, 2023. arXiv:2302.130 72

    2023

  4. [12]

    Gr ¨ obner flags and Gorenstein algebras

    Aldo Conca, Maria Evelina Rossi, and Giuseppe Valla. Gr ¨ obner flags and Gorenstein algebras. Compositio Math. , 129(1):95– 121, 2001

    2001

  5. [13]

    Homotopy invariants for M 0,n via Koszul duality

    Vladimir Dotsenko. Homotopy invariants for M 0,n via Koszul duality. Invent. Math. , 228(1):77–106, 2022

    2022

  6. [14]

    Line-closed matroids, quadratic algebr as, and formal arrangements

    Michael Falk. Line-closed matroids, quadratic algebr as, and formal arrangements. Adv. in Appl. Math. , 28(2):250–271, 2002

    2002

  7. [15]

    Characterizations of strongly chordal graphs

    Martin Farber. Characterizations of strongly chordal graphs. Discrete Math. , 43(2-3):173–189, 1983

    1983

  8. [16]

    Matherne, Matthew Stevens, and L orenzo Vecchi

    Luis Ferroni, Jacob P. Matherne, Matthew Stevens, and L orenzo Vecchi. Hilbert-Poincar´ e series of matroid Chow ri ngs and intersection cohomology. Adv. Math. , 449:Paper No. 109733, 55, 2024

    2024

  9. [17]

    Koszul algebras

    Ralph Fr¨ oberg. Koszul algebras. In Advances in commutative ring theory (Fez, 1997) , volume 205 of Lecture Notes in Pure and Appl. Math. , pages 337–350. Dekker, New York, 1999

    1997

  10. [18]

    Chow rings of toric varieties defined by atomic lattices

    Eva Maria Feichtner and Sergey Yuzvinsky. Chow rings of toric varieties defined by atomic lattices. Invent. Math. , 155(3):515– 536, 2004

    2004

  11. [19]

    On the M¨ obius algebra of a partially ord ered set

    Curtis Greene. On the M¨ obius algebra of a partially ord ered set. Advances in Math. , 10:177–187, 1973

    1973

  12. [20]

    Grayson and Michael E

    Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Availa ble at http://www.math.uiuc.edu/Macaulay2/

  13. [21]

    Gulliksen

    Tor H. Gulliksen. Massey operations and the Poincar´ e s eries of certain local rings. J. Algebra, 22:223–232, 1972

    1972

  14. [22]

    Algebra retracts and Poincar´ e-series

    J¨ urgen Herzog. Algebra retracts and Poincar´ e-series. Manuscripta Math. , 21(4):307–314, 1977

    1977

  15. [23]

    Distributive lattices, affine semigroup rings and algebras with straightening laws

    Takayuki Hibi. Distributive lattices, affine semigroup rings and algebras with straightening laws. In Commutative algebra and combinatorics (Kyoto, 1985) , volume 11 of Adv. Stud. Pure Math. , pages 93–109. North-Holland, Amsterdam, 1987

    1985

  16. [24]

    Enumeration of points, lines, planes, etc

    June Huh and Botong W ang. Enumeration of points, lines, planes, etc. Acta Math. , 218(2):297–317, 2017

    2017

  17. [25]

    Large homomorphisms of local rings

    Gerson Levin. Large homomorphisms of local rings. Math. Scand. , 46(2):209–215, 1980

    1980

  18. [26]

    Chow rings of m atroids are Koszul

    Matthew Mastroeni and Jason McCullough. Chow rings of m atroids are Koszul. Math. Ann. , 2022

    2022

  19. [27]

    Sperner property a nd finite-dimensional Gorenstein algebras associated to ma troids

    Toshiaki Maeno and Yasuhide Numata. Sperner property a nd finite-dimensional Gorenstein algebras associated to ma troids. J. Commut. Algebra , 8(4):549–570, 2016

    2016

  20. [28]

    Supersolvable satur ated matroids and chordal graphs

    Dillon Mayhew and Andrew Probert. Supersolvable satur ated matroids and chordal graphs. Adv. in Appl. Math. , 153:Paper No. 102616, 36, 2024

    2024

  21. [29]

    Qua dratic Gorenstein rings and the Koszul property I

    Matthew Mastroeni, Hal Schenck, and Mike Stillman. Qua dratic Gorenstein rings and the Koszul property I. Trans. Amer. Math. Soc. , 374(2):1077–1093, 2021

    2021

  22. [30]

    Koszul bipartite gr aphs

    Hidefumi Ohsugi and Takayuki Hibi. Koszul bipartite gr aphs. Adv. in Appl. Math. , 22(1):25–28, 1999

    1999

  23. [31]

    Combinatorics and topology of complements of hyperplanes, Invent

    Peter Orlik and Louis Solomon. Combinatorics and topology of complements of hyperplanes, Invent. Math. , 56(2):167–189, 1980

    1980

  24. [32]

    Matroid theory, volume 21 of Oxford Graduate Texts in Mathematics

    James Oxley. Matroid theory, volume 21 of Oxford Graduate Texts in Mathematics . Oxford University Press, Oxford, second edition, 2011

    2011

  25. [33]

    Hyperplane arrangements and linear stran ds in resolutions

    Irena Peeva. Hyperplane arrangements and linear stran ds in resolutions. Trans. Amer. Math. Soc. , 355(2):609–618, 2003

    2003

  26. [34]

    Graded syzygies, volume 14 of Algebra and Applications

    Irena Peeva. Graded syzygies, volume 14 of Algebra and Applications . Springer-Verlag London, Ltd., London, 2011

    2011

  27. [35]

    Quadratic algebras , volume 37 of University Lecture Series

    Alexander Polishchuk and Leonid Positselski. Quadratic algebras , volume 37 of University Lecture Series . American Mathe- matical Society, Providence, RI, 2005

    2005

  28. [36]

    Stewart B. Priddy. Koszul resolutions. Trans. Amer. Math. Soc. , 152:39–60, 1970

    1970

  29. [37]

    Andrew M. Probert. Chordality in Matroids . PhD thesis, Victoria University of W ellington, 2018. In Se arch of The Converse to Hlinˇ en´ y ’s Theorem

    2018

  30. [38]

    On rational K[π, 1]-spaces and Koszul algebras

    Stefan Papadima and Sergey Yuzvinsky. On rational K[π, 1]-spaces and Koszul algebras. J. Pure Appl. Algebra , 144(2):157–167, 1999. 28 A. LACLAIR, M. MASTROENI, J. MCCULLOUGH, AND I. PEEV A

    1999

  31. [39]

    Commutative non-Koszul algebras havin g a linear resolution of arbitrarily high order

    Jan-Erik Roos. Commutative non-Koszul algebras havin g a linear resolution of arbitrarily high order. Applicatio ns to torsion in loop space homology. C. R. Acad. Sci. Paris S´ er. I Math. , 316(11):1123–1128, 1993

    1993

  32. [40]

    The Burnside algebra of a finite group

    Louis Solomon. The Burnside algebra of a finite group. J. Combinatorial Theory , 2:603–615, 1967

    1967

  33. [41]

    Schenck and Alexander I

    Henry K. Schenck and Alexander I. Suciu. Lower central s eries and free resolutions of hyperplane arrangements. Trans. Amer. Math. Soc. , 354(9):3409–3433, 2002

    2002

  34. [42]

    R. P. Stanley. Supersolvable lattices. Algebra Universalis , 2:197–217, 1972

    1972

  35. [43]

    Koszul algebras fro m graphs and hyperplane arrangements

    Brad Shelton and Sergey Yuzvinsky. Koszul algebras fro m graphs and hyperplane arrangements. J. London Math. Soc. (2) , 56(3):477–490, 1997

    1997

  36. [44]

    MAT-free graphic arran gements and a characterization of strongly chordal graphs b y edge-labeling

    Tan Nhat Tran and Shuhei Tsujie. MAT-free graphic arran gements and a characterization of strongly chordal graphs b y edge-labeling. Algebraic Combinatorics , 6(6):1447–1467, 2023

    2023

  37. [45]

    D. J. A. W elsh. Matroid theory. L. M. S. Monographs, No. 8. Academic Press [Harcourt Brace J ovanovich, Publishers], London- New York, 1976

    1976

  38. [46]

    Douglas B. W est. Introduction to graph theory . Prentice Hall, Inc., Upper Saddle River, NJ, 1996

    1996

  39. [47]

    Yuzvinsky

    S. Yuzvinsky. Orlik-Solomon algebras in algebra and to pology. Russian Math. Surveys , 56(2):293–364, 2001. Purdue University, Department of Mathematics, West Lafayet te, IN, USA Email address : alaclair@purdue.edu SUNY Polytechnic Institute, Utica, NY, USA Email address : mas...

    2001

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