# Algebra Counterexample Request: Linear Quotients

A result of Herzog, Hibi, and Zheng in "Monomial ideals whose powers have a linear resolution" states that:

Theorem: Let $I\subseteq\Bbbk[x_1,\ldots,x_n]$ be a monomial ideal generated in degree 2. The following conditions are equivalent:

1. I has a linear resolution.
2. I has linear quotients.
3. Each power of I has a linear resolution.

This, when combined with (a slightly extended version of) Froberg's characterization of degree 2 (squarefree) ideals with a linear resolution, gives a complete description of all degree 2 (monomial) ideals with linear quotients.

However, this still leaves open a characterization of monomial ideals generated in higher degree with linear resolutions/quotients/linear resolutions of their powers. Sturmfels [in "Four Counterexamples in Combinatorial Algebraic Geometry"] and Conca [in "Regularity jumps for powers of ideals"] have a number of examples of ideals $I$ with linear quotients and linear quotients of their powers $I, I^2,\ldots,I^{k-1}$, which fail to have a linear resolution for $I^k$.

The counterexample request then is this: Are there any (monomial) homogeneous ideals which have characteristic independent linear resolutions but no linear quotients under any order of the generators? The theorem above implies that such an example would have to be generated (at least partially) in degree 3 or higher.

Edit: As Professor Conca pointed out, triangulations of the projective plane provide such an example. I was hoping to find a characteristic independent counterexample, but forgot to include this in the description. The request was largely because my research partner A. Hoefel and I were searching for such characteristic independent examples to test a conjecture - but were unable to locate any in the literature (or on a few targeted searches with M2 and GAP.)

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Take a monomial ideal with a linear resolution in characteristic 0 and a non-linear resolution in characteristic 2. Then it cannot have linear quotients (with respect to the monomial generators) because having linear quotients is a combinatorial propriety. The standard triangulation of the real projective plane gives such an example.

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Welcome, Professor Conca! It's good to see a new commutative algebraist here. – Hailong Dao Jun 23 '11 at 21:46
Ah, yes. I should have clarified. I did mean characteristic independent! – Gwyn Whieldon Jun 24 '11 at 4:24

If you take square-free monomials ideals and consider the Alexander duality, then linear resolution is dual to Cohen-Macaulay and linear quotient is dual to shellable. Hence you are looking for a non-shellable simplicial complex that is CM in all the characteristic.

Volkmar Welker mentioned to me that the EG models web page (http://www.eg-models.de/) lists many examples of simplicial complexes with unusual or difficult to find proprieties. There you find what you are looking for. For example, EG Model No. 2003.05.003 is a nonshellable but constructible 2-dimensional simplicial complex.

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Thank you very much! That's a great resource to have. – Gwyn Whieldon Jun 24 '11 at 17:21