Componentwise polymatroidality is not preserved by homological shifts
Henry Zweiman
September 8, 2026
Abstract
We give a negative answer to a question of Ficarra on homological shifts of componentwise polymatroidal ideals. Over any field, the ideal is componentwise polymatroidal for every and , but each of its intermediate homological shift ideals fails polymatroidality in every sufficiently large homogeneous component. A tensor product of two elementary resolutions gives all the shifts, and a single exchange obstruction proves the failure uniformly. The smallest example also contradicts the first-shift assertion in Theorem 10.1 of Ficarra–Lu, arXiv:2509.11977v1; we identify the invalid step in its proof.
1. Statement and context
Let be a polynomial ring over a field , with its standard grading. For a monomial ideal , write for the ideal generated by its degree- monomials. The ideal is componentwise polymatroidal if each nonzero is polymatroidal. Recall that a monomial ideal generated in one degree is polymatroidal if, whenever are minimal generators and , there is a variable with such that is a generator.
The th homological shift ideal is generated by the monomials corresponding to the multigraded shifts in homological position of a minimal multigraded free resolution of . Thus ; the indexing refers to the ideal, rather than its quotient.
Ficarra [2, Question 9] asks whether every homological shift of a componentwise polymatroidal ideal is again componentwise polymatroidal. For equigenerated polymatroidal ideals, preservation of polymatroidality has been proved by Cid-Ruiz, Matherne, and Shapiro [1, Theorem A]. Ficarra–Lu [3, Theorem 10.1] assert the first-shift case of the componentwise question. The following family answers the question negatively and shows that this latter assertion, as stated in the cited version, is false.
Theorem 1.
Let be any field, , and . In
the ideal is componentwise polymatroidal and . For every and every , the ideal is not polymatroidal. The top shift is the principal ideal
In particular, failure occurs in arbitrarily many homological positions simultaneously, and it persists in all sufficiently large components of each intermediate shift. The proof does not depend on the characteristic of .
2. Proof
Put . For an integer , let be the ideal generated by the squarefree monomials of degree in , with and for or .
First we verify the hypothesis of Theorem 1. The degree-one component of is . For , a degree- monomial is in precisely when it involves a variable of or both and . Therefore
For , the only degree- monomial outside is , so
Both descriptions are bounded Veronese sets and satisfy exchange directly: if , equality of total degrees supplies a variable with , and increasing by one respects every upper bound since . Thus is componentwise polymatroidal.
Proposition 2.
For every ,
Proof.
Write and . There is an isomorphism
The first factor has its minimal Koszul resolution, with squarefree -multidegrees of degree in position . The second has the minimal multigraded resolution
The notation records the multidegree of the monomial . Exactness follows because a relation gives , and the coprime monomials divide , respectively. All matrix entries have positive degree. Tensoring these resolutions over the field gives a minimal free resolution over : exactness follows from the tensor product over a field, and minimality follows because all differential entries remain in the homogeneous maximal ideal. Multiplying the multidegrees and adding homological positions gives the shifts of . Position of that resolution is position of the resolution of , yielding (1).
For , only the last summand of (1) survives and is nonzero; for all summands vanish. This proves the projective-dimension and top-shift assertions.
Fix , choose with , and choose . Write . For , set
These monomials have degree and belong to : the second summand of (1) contains , and the fourth contains . They are consequently minimal generators of . The exponent of in exceeds that in , and the only variable with smaller exponent in than in is . Exchange would therefore require
to belong to . It does not: its -support has size , excluding the first three summands of (1), while its -exponent is smaller than , excluding the fourth. This proves Theorem 1.
3. The smallest example and the first-shift assertion
For and , the example is
Its degree-four component contains and but not the exchange monomial . Meanwhile , and for every the degree- monomials of are all such monomials except .
for a componentwise polymatroidal ideal . This equality already fails for the principal ideal in : at , the left side is zero, whereas
The argument for (2) passes from least common multiples of distinct generators of a homogeneous component to least common multiples of distinct minimal generators of . The former two monomials may instead be multiples of the same minimal generator of . Theorem 1 shows that the final first-shift assertion itself fails, in addition to this intermediate equality. These observations concern the componentwise assertion only and do not address the equigenerated or asymptotic results of [3].
References
[1] Y. Cid-Ruiz, J. P. Matherne, and A. Shapiro, Syzygies of polymatroidal ideals, arXiv:2507.13153v1, July 17, 2025. https://arxiv.org/abs/2507.13153.
[3] A. Ficarra and D. Lu, Polymatroidal ideals and their asymptotic syzygies, arXiv:2509.11977v1, September 15, 2025. https://arxiv.org/abs/2509.11977v1.