Rank formulas for homological shifts
of polymatroidal ideals

1. Introduction

The homological shift ideals of a monomial ideal record the multidegrees occurring in its minimal resolution. For polymatroidal ideals they are again polymatroidal, as proved by Cid-Ruiz, Matherne, and Shapiro [3], so a rank function encodes each set of syzygy degrees. We determine that function explicitly from the rank of the original ideal. For powers, the formula describes exactly when new generators appear in the associated Rees module and when multiplication by the ideal supplies every subsequent piece.

In particular, it proves the stabilization bound conjectured by Ficarra and Lu [2]; the th homological shift module of the powers is generated in Rees degrees at most . The bound is sharp for every , holds over every field, and remains valid with a fixed polymatroidal factor throughout.

Let , where is finite and is any field. For a monomial ideal , set

Write , , and . A polymatroidal ideal is generated in one degree, and its generator exponents satisfy the discrete polymatroid exchange axiom. The zero and unit ideals have no positive homological shifts.

The exact rank formula explains both the uniform bound and the earlier stabilization that can occur for an individual ideal. Recall the rank description of a discrete polymatroid [1, Propositions 1.2–1.4 and Theorem 3.4]. For a polymatroidal ideal of degree , its canonical upper rank is

This function is nonnegative, integral, normalized, monotone, and submodular, with . The generator exponents are exactly

Conversely, each such rank function defines a polymatroidal ideal, denoted .

A coordinate with occurs in no generator. Removing it preserves the minimal resolution up to polynomial extension. For the rank formula we therefore use the active ground set, where

For a nonempty , let be its set partitions into nonempty blocks, and define

The formula comes from viewing a Koszul basis vector as a permitted set of coordinate deletions. In each multidegree these sets form a matroid, whose top homology is nonzero precisely when it has no coloops. This condition gives inequalities on the exponents, and uncrossing them produces the partition minimum .

For , the rank is . Each positive partition defect is therefore a positive integer multiplied by , and truncation at removes its effect once . Tracking the defects determines the exact generation interval in Theorem 4.4. Thus the same formula gives the sharp uniform threshold and the complete stabilization behavior of each homological shift module. It also organizes all homological degrees into a single rank function by adding one coordinate, and its inequalities, viewed as linear weights on the Rees algebra, show that the shift modules are integrally closed. These consequences use the same partition data throughout.

2. The Koszul matroid

In a fixed multidegree, a Koszul basis vector records a set of variables whose exponents are lowered by one. We therefore ask which simultaneous deletions leave a monomial in the ideal. The lower inequalities below express the available room for deletion on every subset of variables. Their submodularity will make the permitted deletions a matroid.

We first record two basic facts in the form used below. Products of polymatroidal ideals correspond to sums of their ranks

Indeed, the integral polymatroid sum theorem states that the Minkowski sum has rank and that every integer point in the sum decomposes into integer points of the two summands [1, Proposition 1.4]. For a point of total degree , both summands must be bases. In particular, .

Every also has a linear resolution, since decreasing reverse lexicographic order gives linear quotients. To verify this, take an earlier generator and a later generator . At their last differing coordinate , . The exchange axiom applied to gives with . The coordinate occurs before , so is earlier than , and divides . Thus the order has linear quotients. The exact sequences for adjoining one generator, together with the Koszul resolutions of the variable-generated colon ideals, give a -linear resolution.

The constraint function in our application need not be monotone; adding a variable increases its exponent sum, but can also increase the required lower bound. The following form of the matroid construction keeps precisely the hypothesis that survives, namely submodularity.

The rank formula for the unit ideal is immediate, so assume . Fix and with . If , the degree- part of the Koszul complex of is zero. Otherwise put and

In homological degree , the degree- Koszul basis consists of for the faces of size . Removing an exterior variable is exactly the oriented simplicial boundary of that face. The empty face corresponds to Koszul degree zero, so the degree shift is . Thus

By Lemma 2.1, is the family in Lemma 2.2 for

This is a nonnegative integer-valued submodular function, because is supermodular. Also and . Consequently is a matroid independence complex of rank at most . Here is the number of deletions allowed inside ; the total-degree condition leaves room for at most deletions in all. This is why only the top homology of the resulting matroid can contribute to .

3. The homological rank

For the matroid , Lemma 3.1 says that (2.5) is nonzero exactly when its rank is and it has no coloops. Since , this is equivalent to

In (2.3), set . For ,

Since the matroid rank is already at most , the minimum in (2.3) equals precisely when every term is at least . For a set meeting , the strongest condition comes from choosing , which contributes the final . Hence (3.1) is equivalent to

Put , so that records the exponents remaining after removing the first copy of every variable in the support. Since for every nonempty , the sets disjoint from satisfy the same inequality automatically. Thus (3.2) becomes

The bounds need not themselves form a rank function; the submodular inequality can fail for disjoint sets. But is additive across the blocks of every partition, so all partition sums are forced bounds as well. Taking their minimum gives the equivalent system

Indeed, (3.3) implies the bound for every partition, and the one-block partition gives for the converse.

Using monotonicity of , (3.4) is equivalent to

One direction follows from . For the other, apply (3.6) to to obtain

Finally, at total degree , Lemma 2.1 says that membership in is equivalent to

Combining these facts gives the following field-independent description, with the inequalities required for every

Here linearity of the resolution has excluded all other total degrees.

3.1. A rank interpolation. The inequality (3.5) is what makes the homological bounds into rank functions. The first interpolation below gives each shift separately, while the second places all shifts in one polymatroid by adding a coordinate for homological degree. Although the minimum of two submodular functions is not usually submodular, (3.5) orders their marginal increments and thus controls the mixed case in which the two sets select different branches.

Apply the lemma to , using Lemma 3.2.

For completeness, an integral polymatroid rank is always attained on every subset by one of its integer bases. Choose a coordinate order placing that subset first, let be its successive initial sets, and take the greedy vector . Monotonicity makes it nonnegative and integral, and submodularity shows for every by summing diminishing marginal increments. It has and attains on each initial set. This proves the asserted canonicality without any additional compatibility hypothesis.

Now . If the inequality is strict, (3.8) has no solutions. If equality holds, its solutions are exactly all integer bases of ; these form the generator set of . This proves Theorem 1.2.

4. Powers and partition defects

For a partition of a nonempty set , write

Both are nonnegative integers; the first measures failure of rank additivity across the blocks, while the second depends only on the partition. In the homological rank they become the slope and intercept of an affine function of the power, so stabilization occurs when every positive-slope term reaches the homological cutoff.

Let and , and use the union of their active coordinates. For , the rank is positive on every nonempty subset. Theorem 1.2 gives

If , every term with is at least . The remaining minimum is independent of , so

The total-rank criterion in Theorem 1.2 makes the two consecutive shifts simultaneously zero or nonzero. In the latter case, the polymatroid sum identity (2.1) gives

This proves both equalities in Theorem 1.1, the ordinary case being .

Lemma 3.4 makes the displayed direct sums Rees modules. Iterating the equality from degree proves their generation bound.

For ordinary powers, delete the inactive coordinates of and put

Since the one-block partition has zero defect, this maximum is always defined. Specializing (4.1) to gives

In particular,

4.1. Sharpness and projective dimension.