Sublattice resolutions and
Betti preservation in all powers

Caius Folkerts

Abstract. For ideals whose monomial initial ideals come from sublattices of products of chains, equality of Betti numbers in the first power forces equality in every power. We prove this through an integral decomposition of the sublattice Koszul complexes; their relative homology is generated in Rees degree one, so lifting first-power classes lifts all their products. The decomposition commutes with arbitrary base change and retains integral torsion. For closed binomial edge ideals it gives Betti preservation in every power over every field, together with a common integral family of minimal resolutions; the same principle applies to unit interval determinantal facet ideals.

Contents
1.  Introduction
2.  Chain normal forms
2.1.  Crossing blocks
3.  A multiplicative degeneration principle
3.1.  Rees degenerations of homogeneous ideals
4.  Symbolic powers and first-power transfer
5.  Betti formulas and mixed powers
5.1.  Mixed powers
5.2.  Regular-sequence substitutions
6.  Unit interval determinantal ideals
6.1.  Closed binomial edge ideals
6.2.  Higher minors
6.3.  Mixed prefixes
7.  Integral chain decomposition for sublattice powers
8.  Alcove resolutions and stabilization
9.  An integral family for closed graphs
10.  Characteristic dependence for general sublattices
References

1. Introduction

We prove that equality of Betti numbers in a first-power Gröbner degeneration forces equality in every power when the initial ideal is defined by a finite sublattice of a product of chains. The propagation has a multiplicative explanation; every relative Koszul homology class of the Rees algebra is a sum of products of classes from its first power. At the level of complexes, the same structure gives an integral decomposition compatible with every coefficient base change.

For closed binomial edge ideals, this proves the conjecture of Ene, Rinaldo, and Terai [2]; Peeva’s equality of the first-power Betti tables [3] propagates to every power over every field.

A vertex-labeled graph is closed if, at each vertex, its smaller neighbors form a clique and its larger neighbors form a clique. In $𝕜$ , put

Theorem 1.1 (Closed binomial edge ideals). For every closed graph , every field $𝕜$ , every , and all ,

The equality is part of a common integral resolution; for each power, a single complex over specializes to resolutions of the binomial and monomial ideals and is minimal after specialization to either endpoint over any field (Theorem 9.1). A recursion for the coefficient ideals computes these common Betti tables independently of the field. The same recursion covers unit interval determinantal facet ideals; for at least three rows, degree congruences give the first-power comparison directly.

Let be a finite nonempty sublattice of , and set

𝕜

The closed-graph initial ideals above are of this form after an order-preserving relabeling of each component. For and a coordinatewise multichain , define

where the lexicographic order is increasing. We use the fine grading with one degree for each variable. Write and $𝕜$ for the Rees algebra and relative Koszul homology algebra.

Theorem 1.2 (All powers of a sublattice ideal). For every finite sublattice , every field, and every ,

Moreover, is generated in Rees degree one.

The predecessor quotients occur in a monomial degeneration of the Rees algebra. Their minimal generators separate into disjoint variable blocks, each already present at one member of . Tensor products of the block Koszul complexes therefore express every homology class as a sum of products of first-power classes. The first-power filtration is minimal, so these classes lift through the degeneration; multiplication lifts their products.

Consequently, if a homogeneous ideal satisfies and for all , then Theorem 4.3 gives, for every ,

The symbolic and initial identities follow from normal torsion-freeness of , which we prove by realizing its generators as ideal cuts in a finite network. The homological equality then follows from multiplicative lifting. Applied to closed graphs, this propagates Peeva’s comparison and proves Theorem 1.1. The construction also accommodates mixed products of compatible sublattice ideals.

Theorem 7.1 identifies the whole Koszul complex over the integers, up to chain homotopy, with a direct sum of the coefficient complexes in (1.1). Consequently the comparison preserves torsion as well as Betti numbers after passage to a field. The same description locates the possible characteristic dependence; Section 10 realizes finite-complex homology inside a single coefficient quotient, where it persists through all powers. We develop the chain normal forms and multiplicative lifting first, then use their integral form to distinguish the torsion-free families from general sublattices.

2. Chain normal forms

Sorting a product of generators produces a multichain, but it does not yet determine which polynomial coefficients remain independent. The predecessor ideals below record exactly those coefficient relations. Once the chain is fixed, we will separate these relations into disjoint sets of variables.

The meet and join in are coordinatewise minimum and maximum. For , put

Lemma 2.1 (Predecessor ratios). The ideal is generated by for in the coordinatewise partial order. Every variable in these generators has index strictly less than in its own row .

Proof. For a lexicographic predecessor , put . Then , coordinatewise, and . The ratio uses exactly the rows where ; the original ratio may additionally use rows where . This proves both claims. □

Put , with and , and define

For a multichain write and . The presentation gives

Each summand has ordinary internal shift and fine shift . Multiplication is the natural quotient multiplication if the combined support is a chain, and zero otherwise.

Proposition 2.2 (A universal Rees presentation). The Rees algebra has a homogeneous Gröbner degeneration to . A Gröbner basis of its presentation kernel consists of

The leading monomials are the first displayed monomials.

Proof. Place the -variables first, with when , followed by the row variables in row order and with smaller column index first. Use degree reverse lexicographic order. For a sorting relation, is the last variable at which the two terms differ, so the unsorted term leads. For a predecessor relation, the two coefficient monomials have equal degree. In the last differing row, the second term contains the later variable ; hence the first term leads. Lemma 2.1 shows that these leading monomials generate exactly .

Let fix and send to . All displayed relations belong to its kernel. We show that distinct -standard monomials have distinct images. Write one as , where . Its image is , with . The element is the earliest facet in lexicographic order such that . Indeed, if an earlier such existed, set . A variable of has row index smaller than the corresponding coordinate of every , so it must divide . Thus , contrary to standardness because .

The pair therefore determines . Remove this factor and repeat; the remaining standard monomial is a divisor of the original one. Induction recovers every and finally . This proves the claimed injectivity on standard monomials. Division of an element of by the displayed relations has a standard remainder. Its image is zero, so the injectivity forces the remainder to vanish. The displayed relations are consequently a Gröbner basis, as claimed. □

Lemma 2.3 (Minimal first-power filtration). For every and every fine degree ,

Proof. Adjoin the generators in lexicographic order. The corresponding exact sequence of ideals is

By Lemma 2.1, every fine Tor degree of the final module has rowwise maximum-index vector exactly ; the shift supplies in every row, and the colon uses only earlier variables. This follows, for example, by using the Taylor resolution of the colon; unused polynomial variables add no Tor degrees. Inductively, the fine Tor degrees already present have the maximum-index vector of an earlier facet. Therefore no connecting map in the long Tor sequence can be nonzero in any fine degree. The Betti tables add at each step. The last equality is the degree-one instance of (2.2). □

2.1. Crossing blocks. The following elementary observation is the structural point of the argument. It applies to every finite sublattice, without any assumption on linear quotients or on the degrees of the colon generators. As a chain increases, a row variable becomes available for a predecessor ratio when the chain first passes its column index. If a minimal relation used variables first passed at different steps, taking a meet at the earliest step would give a proper relation dividing it. Thus a minimal relation can involve only one first-crossing block.

Lemma 2.4 (Crossing blocks). Fix a multichain . For each variable define

with value if the set is empty. Let be the variables with first crossing . There are monomial ideals $𝕜$ such that

In particular, the use pairwise disjoint sets of variables.

Proof. Let be a minimal monomial generator of . By Lemma 2.1, for some and some coordinatewise. Every variable of crosses by time . Let be the least crossing time among these variables and set . We claim that

is precisely the product of the variables of with crossing time . If a row contributes no variable to , then and it contributes none to . Otherwise it contributes to exactly when . This says that its first crossing is at most , hence exactly by the choice of .

Thus is a nonempty divisor of , and . Minimality of forces . Every minimal generator consequently lies in one block and belongs to the corresponding . Let be generated by the minimal generators assigned to block . These ideals have all the stated properties. Variables in do not occur in . □

Proposition 2.5 (Generation of relative Koszul homology). The algebra is generated by its Rees-degree-one part and its unit, over every field.

Proof. In the summand belonging to , Lemma 2.4 gives

𝕜 𝕜 𝕜

The relative Koszul complex is the tensor product of the block Koszul complexes and the complex on the unused polynomial variables. Over a field, its homology is the tensor product of the block homologies; the last polynomial factor contributes only $𝕜$ in degree zero.

Choose cycle representatives of block homology classes, with coefficients in $𝕜$ . The same polynomial and exterior expression is a cycle over ; its differential has coefficients in . Hence is a cycle in the Rees-degree-one part of . Their product is, up to the usual exterior sign, exactly

which represents the chosen tensor class in the -summand. A factor with trivial block homology contributes in homological degree zero. Tensor classes span the homology, so this proves generation on each summand of (2.2), while at Rees degree zero the homology consists only of the unit. □

3. A multiplicative degeneration principle

The block decomposition expresses special-fiber homology classes as sums of products of classes from the first power. A product lifts to a cycle as soon as its factors do. The following proposition turns this observation into a criterion for preserving every Betti table.

Let $𝕜$ . For an -algebra , write

𝕜 𝕜

and write $𝕜$ for its homology algebra. Its product is induced by multiplication of coefficients and exterior multiplication.

Let be a bigraded -algebra with a nonnegative auxiliary grading , with and finite-dimensional components in each fixed internal and auxiliary degree. Give a homogeneous weight filtration, finite, exhaustive, and separated on each such component, restricting to a variable-weight filtration on . Let be its associated graded algebra, viewed over .

Proposition 3.1 (Propagation from Rees degree one). Suppose that

(i): the algebra is generated by its auxiliary-degree-one part, together with the unit;
(ii): the graded Betti numbers of the auxiliary-degree-one -modules and are equal.

Then, for every ,

Moreover, is itself generated in auxiliary degree one.

Proof. Let $𝕜$ and $𝕜$ . Localize the weight-homogenized family over , and form its relative Koszul differential graded algebra , assigning each exterior generator the weight of its corresponding variable. Its special fiber is , and its generic fiber is $𝕜$ after rescaling variables. In every fixed internal and auxiliary degree, is a finite complex of finite free -modules.

In one such degree write

The exact sequence of multiplication by gives

Thus the special and generic homology dimensions differ by . Assumption (ii) forces every in auxiliary degree one to vanish. Every special class in that degree consequently lifts to a cycle in .

By (i), every special homology class is a sum of products of these classes. Multiplying their cycle lifts shows that is surjective in all degrees. Equation (3.1) now gives everywhere, including the top Koszul degree by taking the next, zero, homological degree. All the modules are therefore free over , and their special and generic dimensions agree.

Let be the subalgebra of generated by its auxiliary-degree-one part. The quotient vanishes modulo in every finite homogeneous component. Nakayama’s lemma gives . Passing to and descending along the faithful field extension $𝕜$ proves both assertions. □

Remark 3.2. A term-order degeneration appearing below can be replaced by one integer weight vector realizing the finitely many leading terms of a finite homogeneous Gröbner basis. A positive internal-degree weight can be added if necessary. Thus the finite-filtration hypothesis in this proposition does not require a limiting argument over infinitely many term-order comparisons.

Proof of Theorem 1.2. Apply Proposition 3.1 to the degeneration in Proposition 2.2, using Lemma 2.3 and Proposition 2.5. All constructions preserve the fine grading, so the conclusion holds in that grading. Formula (2.2) then gives (1.1). □

3.1. Rees degenerations of homogeneous ideals.

Lemma 3.3 (A finite weight for an all-powers initial identity). Let be an ideal generated by homogeneous forms of degree , with . If for every , there is a homogeneous weight degeneration preserving internal and Rees degree.

Proof. The all-powers identity says that the finite set consisting of the variables of and the elements is a SAGBI basis; every leading monomial in is a product of their leading monomials. A weight realizing finitely many subduction relations will give the required degeneration.

Let be the toric presentation kernel of in and choose finite binomial generators for it. The images of and in have the same leading monomial, so it cancels in their difference. Subduct that difference using the finite SAGBI basis. The process terminates because the leading monomial strictly decreases in a term order at each step. It produces a relation in the presentation kernel , where the monomial image of each under the monomial presentation is strictly smaller than the common image of . All terms have the same internal and Rees degree.

There are finitely many such relations and comparisons, together with the finitely many comparisons selecting in . One positive integer weight on the variables of realizes them all; this is the usual finite weight realization of a monomial order. Give the weight of . The weight-initial form of is then , so . The two quotient rings have the same Hilbert function in every internal and Rees degree, by the assumed initial identities. A weight degeneration preserves these Hilbert functions. The inclusion is consequently equality. Weight homogenization gives the asserted degeneration, with finite filtrations in each bidegree. □

4. Symbolic powers and first-power transfer

The finite Rees degeneration of Lemma 3.3 requires initial identities in every power. For sublattice ideals these follow from the first power; ordinary powers are already detected by the minimal vertex covers. For a radical ideal , write . A vertex cover of a squarefree monomial ideal is a set of variables meeting the support of every minimal generator; its inclusion-minimal covers index the minimal primes.

Theorem 4.1 (Sublattice ideals are normally torsion-free). For every finite nonempty sublattice and every ,

Proof. We seek a factorization of a given monomial into as many generators as its vertex-cover bounds permit. In the network below, generators become order-ideal cuts and covers become paths. Integer shortest-path distances then supply the required cuts, with each variable used no more often than its prescribed exponent.

Coordinate chains. Let be the poset of join-irreducibles of the finite distributive lattice . The map identifies with the lattice of order ideals; join-irreducibles are join-prime by distributivity, and every element is their join. Compress the distinct attained values of coordinate to . Each nonconstant threshold preserves meets and joins. Its -fiber is a prime filter with a least element . Primality makes a join-irreducible order ideal, hence for a unique . Consequently exactly when . The nested, distinct thresholds give a chain

Every occurs in some coordinate chain; the distinct ideals and must be distinguished by a coordinate. Unattained row variables are unused polynomial variables and may be suppressed.

The network and its covers. Adjoin a bottom vertex and a top vertex to . For each row, use the directed path

labeling its -th arc by the row variable , for . Parallel arcs retain distinct labels. Also include an unlabeled reverse arc for every strict order relation in the extended poset. The outgoing labeled arcs of the cut , for , are exactly the variables of ; each row path crosses once, and no reverse arc leaves the cut.

A set of labeled arcs is a cover precisely when , together with all reverse arcs, contains an – path. Such a path must cross each ideal cut. Conversely, if no such path exists, the vertices reachable from form an order ideal containing and excluding , whose outgoing labeled arcs avoid .

Integer cut packing. Give each labeled arc a nonnegative integer cost and each reverse arc cost zero. The shortest – distance equals

Indeed every cover contains a path, and every path’s labeled arcs contain a minimal cover; paths can be chosen simple because costs are nonnegative. Write for the shortest distance from to . Every vertex is reachable. Reverse arcs imply and whenever . For , therefore,

is an order ideal. A labeled arc occurs in exactly of these cuts, at most by the shortest-path inequality. Hence divides . If every minimal cover has exponent sum at least , then , and any of these cuts give a product of generators dividing the monomial. Conversely, every such product uses at least one variable from each cover in each factor. Thus a monomial belongs to exactly when every minimal cover has exponent sum at least . This is precisely membership in , which is the symbolic power since is squarefree. □

Lemma 4.2 (Initial ideals of symbolic powers over any field). If $𝕜$ has squarefree initial ideal for a term order , then is radical and, over every field,

Proof. If a nonzero standard remainder had , then would imply , a contradiction. Thus is radical. Let be the Hasse derivative, defined on monomials by

Its product rule extends to every localization by expanding , since the constant term of the denominator is a unit there. That rule gives when ; in each product of factors from , at most factors receive a positive derivative order.

Suppose has leading monomial . There is a minimal cover of with . Take on and elsewhere. The preceding localization argument gives . Its leading monomial is ; the leading coefficient is unchanged, since every relevant binomial coefficient is , and dividing surviving monomials by the same monomial preserves their order. This leading monomial avoids , contradicting . No division by a factorial or perfect-field assumption is involved. □

Theorem 4.3 (A first-power criterion). Let be a finite nonempty sublattice and a homogeneous ideal with . Then, for all ,

If also for all , then

and is generated in Rees degree one. The same Betti equality holds in every finer grading respected by the degeneration and refining ordinary internal degree.

Proof. Theorem 4.1 and Lemma 4.2 give the inclusions

Thus all are equal. The contained ideals with the same initial ideal are equal, by division. The degree-one identity supplies homogeneous generators of with leading monomials , and Lemma 3.3 gives the Rees degeneration . Theorem 1.2 gives generation of the special homology in Rees degree one. Under the first-power Betti hypothesis, Proposition 3.1 therefore gives equality in every power and generation of . For a finer grading, generic Betti numbers are at most special Betti numbers in each degree. Ordinary equality makes the sum of these nonnegative differences zero, so each difference is zero. □

5. Betti formulas and mixed powers

The coefficient ideal of a multichain depends only on which lattice elements occur, so varying their multiplicities introduces no new homological data. Grouping by these support chains turns the formula of Theorem 1.2 into a finite expression for all powers. Write and, for a nonempty chain of distinct elements of , put .

Corollary 5.1 (A finite formula for every power). For every finite sublattice and every ,

The same formula holds for under the Betti comparison hypothesis of Theorem 4.3. The binomial coefficient is zero when .

Proof. Group the multichains in (1.1) by their support. A fixed support chain of size occurs times, one for each assignment of positive multiplicities with sum , and every occurrence has the same coefficient ideal. □

In particular,

Thus finitely many coefficient quotients determine the Betti tables of all powers.

Corollary 5.2 (Polynomial behavior from the first power). For each , the normalized entry is, for all positive integers , a polynomial

All its forward differences are nonnegative on the positive integers. Consequently the projective dimensions of the powers are nondecreasing and the depths of their quotient rings are nonincreasing. These conclusions also hold for every satisfying the Betti hypothesis in Theorem 4.3.

Proof. Use (5.1) and , with the usual zero conventions. A nonzero normalized Betti entry therefore remains nonzero at the next power. This gives projective-dimension monotonicity; the depth statement follows from and the Auslander–Buchsbaum formula. □

Corollary 5.3 (First occurrence of a strand). If and is nonzero for some , it is already nonzero for some . If is the minimum element of , multiplication by induces an injection

𝕜 𝕜

with the appropriate degree shift. Under the Betti hypothesis of Theorem 4.3, multiplication by the corresponding generator has the same property for .

Proof. Express a nonzero class as a sum of products of first-Rees-degree classes and select a nonzero product term. At most factors have positive homological degree. The product of those factors is nonzero, since otherwise the full product would be zero. Removing the homological-degree-zero factors removes exactly from the internal degree per factor, preserving the normalized strand . This proves the first assertion.

For the second assertion, and is comparable with every element of . Multiplication by on sends the summand isomorphically onto the summand indexed by ; different multichains have different images. It is therefore injective on relative Koszul homology. The proof of Proposition 3.1 makes the family homology free over the local parameter ring and identifies its reduction with the special homology. A map between finite free modules that is injective modulo the parameter is injective over its fraction field. Apply this first to multiplication by a lift of , and then to multiplication by in the degeneration from . □

Remark 5.4 (Several auxiliary degrees). Proposition 3.1 extends verbatim to an auxiliary grading by . More generally, let be any set of auxiliary degrees which generates the special homology algebra. Equality of Betti tables in the components indexed by makes their special classes lift to cycles. Products then lift every special class, so (3.1) gives equality in all components. We shall apply this form to multi-Rees algebras whose generating degrees are the coordinate units.

5.1. Mixed powers. Adding a color coordinate to the lattice lets the same decomposition distinguish factors from different ideals. We realize a mixed product as the component of an ordinary power with the smallest possible total color degree; that degree leaves no room for exterior color variables in the Koszul complex. The first-power classes on this diagonal are therefore exactly those of the individual ideals. Let be nonempty sets such that

is a sublattice, and put .

Theorem 5.5 (Mixed powers from a colored sublattice). For this colored sublattice, the algebra

𝕜

is generated in the auxiliary degrees . Suppose homogeneous ideals satisfy for every , and one monomial order gives

Then

Proof. Introduce new variables , put , and let . This is exactly the sublattice ideal , with the -variables forming an additional row. Theorem 1.2 says that is generated in Rees degree one.

Give and its Koszul exterior generator the fine -degree . In Rees degree , every coefficient in has total -degree at least . Fix with . A Koszul term of fine -degree containing any exterior -generator would require a coefficient of total -degree less than , so it is zero. The whole subcomplex in these degrees is therefore literally

These subcomplexes, over all , form a differential graded subalgebra naturally isomorphic to the Koszul complex of the multi-Rees algebra. Fine -degree is preserved by the differential, so taking this diagonal commutes with homology.

Choose homogeneous Rees-degree-one generators of the full homology algebra. Each has total -degree at least one. A product of such generators contributing to the diagonal of total -degree must use only generators of total -degree one. Their fine -degrees are the coordinate units , and the identified complexes show that these are precisely the components $𝕜$ . The Rees-degree-zero homology is just $𝕜$ . This proves the asserted generation for the multi-Rees algebra.

The mixed initial identities provide a finite SAGBI degeneration

The finite weight construction of Lemma 3.3 applies unchanged, retaining the multicolored Rees degree. The multigraded form of Proposition 3.1 now gives Betti equality and homology generation in every multidegree. □

For example, take any finite sublattice and any chain in it, and set . These sets satisfy the colored-sublattice hypothesis. Thus the theorem applies to all mixed products of principal lattice prefixes, without assuming that their individual resolutions are linear.

The same diagonal retains the coefficient decomposition, giving an explicit formula for mixed Betti numbers. Set . For a multichain in , let count its colors and put . Then

For , use the unique empty chain and ; the formula then states the identity for . For positive total degree, apply the fine formula for and take the same minimal total -degree diagonal. Only chains with the indicated colors can contribute; the fine -degree-zero Koszul homology of a coefficient quotient is the homology of its restriction to . Cross-color predecessor ratios contain a -variable, so that restriction is exactly . For for every , grouping by strict colored chains gives a polynomial in for each normalized ordinary Betti entry, with nonnegative coefficients in the basis , where counts color in the strict chain. The same ordinary graded specialization holds for the products of the in Theorem 5.5.

The hypothesis on all mixed initial ideals can itself be checked by a single ordinary initial identity. The colored sublattice is what allows the first-power criterion to enforce every product identity.

Corollary 5.6 (A finite condition for mixed transfer). Under the colored-sublattice hypothesis of Theorem 5.5, let be homogeneous ideals and put . Suppose one term order on satisfies

Then its restriction to satisfies

If for every , all the mixed products have equal graded Betti tables, and the multi-Rees Koszul homology for the is generated in the coordinate unit degrees.

Proof. Theorem 4.3, applied to without a Betti hypothesis, gives . Both ideals are fine -graded, and taking initial ideals commutes with their fine -components. For , their components are respectively and . Theorem 5.5 now applies; lifting the associated-graded algebra generators gives its generation conclusion as in Proposition 3.1. □

The network also gives a finite primary description of every monomial mixed product

Here an exponent zero means ; the intersection may be redundant. Indeed take the fine -degree component of and subtract the exponent contribution of the -variables.

For nested families, this single initial identity follows from the individual ones; in each fine color degree, the coefficient ideal is the largest participating member. The network description then also expresses a mixed product as an intersection of ordinary powers.

Corollary 5.7 (Nested families and initial prefixes). Let be homogeneous ideals with for one term order, and suppose the satisfy the colored-sublattice hypothesis of Theorem 5.5. For , put . Then

An exponent zero means the whole ring. If in addition for every , every mixed product has the same graded Betti table as its monomial initial product, and the multi-Rees Koszul homology is generated in coordinate unit degrees.

Proof. Put and . For a nonzero fine -degree , the coefficient ideal of is , because the family is nested. The analogous assertion holds for . Therefore any extension of to has . Corollary 5.6 supplies the mixed initial identities and the assertions under the Betti hypothesis.

For the monomial intersection formula, one may use all vertex covers in the primary formula for . Every such cover contains one of the form , where covers ; take to be the largest color whose is absent. A cover containing every gives only an exponent-zero condition. Conversely every displayed set is a cover, since for . The primary formula thus reduces to

using Theorem 4.1 for each color. Nestedness gives . The reverse equality follows from the initial-ideal sandwich

Contained ideals with equal initial ideals are equal. □

5.2. Regular-sequence substitutions. The construction also survives replacing the row variables by a homogeneous regular sequence. Flat base change carries both the resolutions and their Koszul products, while positive degrees preserve minimality.

Corollary 5.8 (Homogeneous regular-sequence substitutions). Replace the variables of the sublattice ideal by any homogeneous regular sequence of positive degrees in a polynomial ring over $𝕜$ , and put

The minimal multigraded resolutions of all powers of base change to minimal free resolutions of the corresponding powers of . Their fine shifts are regraded by . Moreover, $𝕜$ is generated in Rees degree one. Thus the same first-power propagation theorem applies to any homogeneous Rees degeneration with this substituted target. Disjoint monomial substitutions are a particular case.

Proof. The homomorphism , , is graded free; lifts of a homogeneous basis of generate over , and the regular-sequence Hilbert series makes the resulting free-module surjection an isomorphism. Base change therefore preserves resolutions and gives . Positive degrees keep every differential entry in the homogeneous maximal ideal, preserving minimality. Derived base change identifies the Tor algebras

𝕜 𝕜

with the stated regrading, and preserves their products. Apply Theorem 1.2 and Proposition 3.1. □

6. Unit interval determinantal ideals

For unit interval complexes, the coefficient ideals have an additional recursive form; each is a sum of linear variables and a smaller restriction of the original ideal. This recursion computes the Betti numbers over every field. For at least three rows it also separates the possible homological degrees, so the determinantal degeneration cannot cancel any Betti entries.

A pure simplicial complex with -element facets is unit interval if, whenever is a facet, every -subset of is a facet. Equivalently, its facets are the union of the complete -uniform families on its maximal interval cliques. Throughout this section . Let be its facet set and its diagonal monomial ideal. Coordinatewise meet and join preserve , so the preceding results apply.

List the maximal interval cliques with increasing left and right endpoints, and let be the left endpoint of the first clique whose right endpoint is at least . For a facet , that clique contains , and . Set

For a set of retained variables, a superscript denotes intersection with $𝕜$ . For an ideal in its indicated polynomial ring, write for the quotient Betti polynomial.

Lemma 6.1 (Coefficient ideals and their recursion). The predecessor ideal of is

The variables generating are disjoint from those of , and when . For every variable restriction , its quotient Betti polynomial satisfies

Each residual is a variable restriction of with fewer generators. In particular, all the fine-graded Betti numbers are independent of the field, and

Proof. By Lemma 2.1, it suffices to consider ratios from coordinatewise. Each generator of comes from decreasing one coordinate of inside a clique containing it. Suppose a ratio avoids these variables. If , avoidance forces , then successively for every , a contradiction. Thus . For , avoidance gives either or . If and , strict increase of the facet gives ; avoiding the next interval of linear variables therefore forces . Repeating gives . But no facet can start below and end at . Consequently for every , and its ratio is . The converse inclusions are immediate. The displayed bounds also prove disjointness and nesting.

For a retained generator, taking its predecessor colon commutes with intersection with ; a variable outside in a preceding generator cannot cancel against . The resulting colon is . Lemma 2.3 makes this generator filtration minimal in every fine degree. Splitting off the linear variables gives the factor ; the generator shift and passage from an ideal to its quotient give . This proves the recursion.

The residual retains only columns below in row and columns at most in row , in addition to the restriction . Every residual generator precedes , so induction terminates at the empty ideal. Its coefficients are nonnegative integers. The same recursion, with monomial degree weights in place of powers of , proves fine-graded characteristic independence. A linear Koszul factor changes and equally; the factor changes by . This proves the congruence. □

For a nonempty support chain , let be the variables generating . Nesting gives . In the ring on the remaining variables, put . Then

The chain formula therefore proves characteristic independence for every power of , together with

6.1. Closed binomial edge ideals. We first arrange that each connected component occupies a consecutive block of labels and every edge spans a clique. In a closed labeling, a shortest path has no local maximum or minimum of its labels, since either would give a shortcut; every shortest path is therefore label-monotone. Consecutive vertices in the induced order of a component must be adjacent, and these consecutive edges, together with closedness, force all vertices between the endpoints of any edge to lie in one clique.

Concatenating the components while preserving their internal orders gives the required interval labeling. The corresponding permutation of the pairs carries both and its specified monomial initial ideal to the new ideals, preserving their Betti numbers. Isolated vertices contribute only polynomial variables, so they can be omitted and reinserted without affecting the comparison. If there are no edges, both ideals and all their positive powers are zero.

Henceforth assume this interval labeling. In particular, for ,

For a vertex with a smaller neighbor, put

put if there is no smaller neighbor. The function is nondecreasing. Indeed, if and , equation (6.6) gives and hence ; if , the same inequality is immediate.

Let . It is a sublattice of . For incomparable edges , say and , we have . The outer edge spans a clique, so and are edges. These are the coordinatewise meet and join. Comparable pairs need no argument. The ideal is therefore exactly the sublattice ideal of Section 2, with and .

Proof of Theorem 1.1. For a graph with an edge, use the edge sublattice just described. The ordinary initial identity and Peeva’s first-power theorem verify the hypotheses of Theorem 4.3. It yields the displayed equality for every over the arbitrary field $𝕜$ . The labeling reduction and the zero-ideal case above include disconnected graphs, isolated vertices, and the empty edge set. □

In row size two, for every edge ending at , and the coefficient ideals in Lemma 6.1 become

Thus the recursion and (6.4) give finite formulas for all graph powers.

The degeneration respects the vertex and row gradings, while the coefficient recursion is independent of the field. These two features refine the ordinary Betti equality as follows.

Corollary 6.2 (Natural multigraded preservation). Give the grading

Then for every and every natural multidegree ,

For a disconnected graph, the number of -variables can be recorded separately in each connected component, with the same conclusion.

Proof. The homogeneous degeneration respects these gradings. In each such degree, the finite filtered Koszul complex gives the usual inequality from generic to special Betti numbers. Theorem 1.1 says that the sums of those nonnegative differences at every fixed vanish. Each difference must therefore vanish. The relations are also homogeneous for separate row counts in each component. □

Corollary 6.3 (Independence of the field). For every closed graph, all powers of have field-independent Betti numbers in the finest monomial grading. All powers of have field-independent Betti numbers in the natural grading of Corollary 6.2, and hence in the ordinary grading.

Proof. For a support chain , killing the variables in leaves , an induced variable restriction of . Its Betti table is field-independent by Lemma 6.1. The exterior factor and the chain formula in Corollary 5.1 therefore give the same assertion for every . The proof of that formula respects the finest monomial grading. Corollary 6.2 transfers the assertion to . □

For -free closed graphs, the linear-strand comparison was proved by Dohadwala et al. [4]. The coefficient formula gives the following expression for every closed graph.

Corollary 6.4 (Linear strands for arbitrary closed graphs). For every and ,

Proof. The ideal is generated by quadrics. Its quotient has no positive-homological-degree Tor in internal degree equal to homological degree. Hence the entries on this diagonal in (6.4) come entirely from the exterior factor . □

6.2. Higher minors. Let be the ideal of maximal minors associated with the facets of a unit interval complex in a generic matrix. Its diagonal initial ideal is [1, Theorem 2.7].

Theorem 6.5. If , then, over every field and for every ,

These numbers are independent of the field, and a nonzero entry satisfies .

Proof. The assertion is immediate for the zero ideal. Otherwise, Theorem 4.3 gives . By (6.5), the latter has no two nonzero Betti entries in adjacent homological degrees and the same internal degree, since . In the corresponding weight degeneration, a torsion summand of the family homology would contribute to both such entries by (3.1). There is therefore no torsion, and the Betti numbers agree. Lemma 6.1 and the chain formula supply characteristic independence and the congruence. □

For the congruence is vacuous. Peeva’s first-power theorem and multiplicative transfer give precisely the comparison needed in that case.

6.3. Mixed prefixes. Passing to an initial-column prefix does not change the predecessor colon of any retained facet. Consequently the mixed chain formula uses the same coefficient quotients as ordinary powers, and the same recursion computes its Betti tables.

Closed graphs.. Use the interval labeling of Section 6.1, choose , and let be the nonzero binomial edge ideal of the induced graph on , extended to the common ring . Its edge set is . These are nested sublattices, and their colored set is a sublattice; the meet belongs to the smaller prefix and the join to the larger. Each induced graph is closed, so Peeva’s theorem supplies every first-power Betti equality. Thus (5.3), all mixed Betti equalities, and multi-Rees homology generation hold over every field. Moreover these mixed Betti tables are independent of the field. For , every coordinatewise predecessor belongs to the same prefix. Lemma 2.1 therefore gives , the colon in the full edge lattice. The coefficient ideals in (5.2) are consequently the same sums as for ordinary graph powers. The positive recursion of Lemma 6.1 makes their fine Betti tables characteristic independent. The mixed chain formula gives fine independence for the monomial products, and the mixed comparison gives ordinary independence for the binomial ones.

Higher-row prefixes.. The same conclusions hold for nonzero initial-column prefixes of a unit interval determinantal facet ideal. Retaining facets with last column at most again gives nested colored sublattices, and their prefix colons equal those in the full facet lattice. Theorem 6.5 supplies the individual first-power comparison; its positive recursion for coefficient quotients supplies characteristic independence. Thus every such mixed product satisfies (5.3) and the same Betti-preservation and field-independence conclusions.

7. Integral chain decomposition for sublattice powers

Over a field, the block decomposition expresses homology classes as products of first-power classes. Over the integers, tensor products can also contribute torsion, so lifting homology classes alone no longer suffices. We retain that information by lifting each entire block complex to Rees degree one and then multiplying the chain maps. The separation of first-power strands by their rowwise maximum indices makes these stronger lifts possible.

All monomial ideals in this section are defined by the same generators over the indicated coefficient ring. Write , and retain the notation of Sections 2 and 2.1.

Theorem 7.1 (Integral chain decomposition). For every finite sublattice , every , and every fine degree , there is a chain homotopy equivalence over

More precisely, for a monic integer-weight Rees degeneration to , each fixed fine/auxiliary component of its relative Koszul complex is chain homotopy equivalent over to its constant special fiber complex.

Consequently, for every commutative coefficient ring , with ,

In particular, over this preserves every torsion summand and its exponent. The equivalences are noncanonical; no -linearity or multiplicative coherence between different powers is asserted. Without an additional torsion-freeness hypothesis, homology-algebra generation in Rees degree one is asserted only over fields.

Proof. The integral family. The relations in Proposition 2.2 are monic with coefficients , and its normal-form injectivity proof works over . Choose a positive integer weight realizing their leading terms, and homogenize them with a parameter of weight one. Their standard monomials form a -basis of the resulting algebra ; division gives spanning, and after inverting their images are distinct monomials of the original Rees algebra, proving independence. Thus and . Give each exterior generator the weight of its base variable. For fixed fine degree and Rees degree , let be the resulting relative Koszul complex over , and let be its special fiber. These are bounded complexes; the former has finite free -terms. Their weight gradings are bounded below.

Chain sections in the first power. At fine degree , the summand of has the single total Koszul weight

independent of homological degree. Its homology can be nonzero only when is the rowwise maximum-index vector of . Indeed a different nonzero summand has a higher row variable in its residual fine degree, unused by ; the corresponding polynomial Koszul strand is contractible over . Concretely, that variable contributes the two-term complex in its positive fine degree. The contraction therefore uses a unit and survives every coefficient change. Thus, when this maximum vector is , all first-power special homology is concentrated at the one weight .

The weight- exact sequence is

Induction from the lower weight bound shows that is acyclic for . At , the kernel is therefore a contractible bounded complex of finite free abelian groups. The surjection onto admits a chain section; choose a degreewise section and correct its differential defect with a contracting homotopy of the kernel. Explicitly, if is the chosen section and contracts the kernel, put . Then takes values in that kernel and , so

The correction does not change the projection of . Hence every chain map from a finite free abelian complex into this special component lifts to the family.

Lifting block complexes. Fix a multichain and put . A negative component gives zero. Use the crossing blocks of Lemma 2.4. Their disjoint supports and give

If has positive degree in , the corresponding special strand is contractible and can be omitted. Otherwise, writing , that strand is exactly the tensor complex

The -th block complex maps to the first-power special complex by the quotient map and multiplication by , since . Its fine degree is . Every variable of lies below in its row, so the maximum vector of is exactly . The chain section just constructed therefore lifts this entire block map, not merely its homology classes. Extend the lifts -linearly, tensor them, and multiply in the family Koszul differential graded algebra. This gives a homogeneous chain map

Its reduction modulo is the usual block decomposition on every retained summand. The omitted summands are integrally contractible, so this reduction is a quasi-isomorphism. Tensoring whole complexes retains the integral Künneth torsion classes.

The cone and arbitrary base change. Assign the weight . The cone of is weight graded and bounded below, has finite free -terms, and is acyclic modulo . Its weight-component exact sequences, as above, prove its acyclicity by induction on the weight. A bounded acyclic complex of projective modules is contractible, splitting successively from its lowest degree. Thus is a chain homotopy equivalence over . Reinstating the contractible omitted summands identifies its source up to homotopy with the constant complex .

Specializing proves (7.1). Tensoring the homotopies with any commutative ring preserves them. Monomial ideals, their powers, and monomial quotients base change literally. The polynomial variables are a regular sequence over , so these relative Koszul complexes compute Tor also when is not a field. This proves (7.2). □

Corollary 7.2 (Integral mixed formula). Under the colored-sublattice hypothesis of Theorem 5.5, for every commutative coefficient ring the mixed formula holds as an isomorphism of fine-graded -modules

where as in (5.2).

Proof. If , the formula is the identity for . Otherwise apply Theorem 7.1 to over and take fine -degree in Rees degree . At the chain level this diagonal has no exterior -terms. Only chains with the prescribed colors contribute, and their residual -degree is zero. Cross-color predecessor ratios contain a -variable; restricting the coefficient ideals to therefore gives precisely . This proves the assertion without any fieldwise Künneth argument. □

8. Alcove resolutions and stabilization

The chain decomposition determines the homology in each power, but its possible homological degrees are more easily bounded geometrically. Products of generators correspond to the integer points of a dilated order polytope, whose dimension bounds the length of a cellular resolution. A maximal lattice chain supplies a complete intersection attaining that bound; together with first-power generation, this also determines when the normalized Betti support stabilizes.

Let be the poset of join-irreducibles of , let , and write for the number of polynomial variables, including unused ones. Retain the coordinate chains obtained in the proof of Theorem 4.1. Compress their attained values to , so row corresponds to .

Theorem 8.1 (An integral cellular resolution). For every , the ideal has an explicit, generally nonminimal, cellular free resolution over of length at most . Its base change to any commutative coefficient ring is still a free resolution. Over every field, the analytic spread of is and

For each fixed field, the ordinary normalized Betti support

is nondecreasing with and is constant for . Put $𝕜$ . The depth identity below assumes a positive row count. Consequently, for ,

𝕜

Proof. Vertices and labels. The minimal generators of correspond bijectively to the integer order-reversing maps . For a product of generators, let count the order ideals containing ; conversely, its level sets form a multichain of order ideals. The row exponents of its monomial label are

with exponent for a constant row. These exponents determine because every belongs to some coordinate chain. All labels have degree , so distinct labels are minimal generators.

Let . Triangulate by the affine braid hyperplanes

Equivalently, subdivide each integer unit cube by ordering the fractional coordinates. The resulting simplices have integer vertices. The polytope is a union of these simplices; denote its finite triangulation by . Label its vertex by and each face by the least common multiple of its vertex labels.

Exactness in each fine degree. Use one free generator of degree in homological degree , with differential

for the augmentation to . For a monomial , the fine strand is the augmented simplicial chain complex of the subcomplex induced by vertices whose labels divide . The divisor conditions are

Together with these define a convex polytope , possibly empty. A constant row either imposes no additional condition or makes it empty. Every bounding hyperplane belongs to the displayed affine arrangement. Thus is a union of its simplices, and the induced divisor subcomplex is exactly this polytope’s triangulation. It is contractible when nonempty. Its augmented integral chain complex is therefore exact; if empty, the whole augmented fine strand is zero. This proves exactness over , and the same argument works over every coefficient ring. Since , the length bound follows. The resolution need not be minimal; a face and a facet may have the same monomial label. Thus its existence does not preclude torsion in the Koszul homology discussed in Section 10.

The lower bound. Suppose , choose a linear extension of , and let correspond to . In the chain , the newly crossed variables at step are the old state variables in rows whose coordinate chain contains . Their product is

The crossing block created at step is precisely . Indeed, a block generator can be taken as by Lemma 2.4. If the order ideal of omitted an earlier , choose the first such and a row containing . Its state variable has first crossing , a contradiction. Thus contains and differs from only by . Each is nonconstant, and their supports are disjoint. Therefore

is a complete intersection of length . Since , omitting leaves the same coefficient ideal. Its top Koszul homology is nonzero, so (1.1) gives . Repeating chain factors gives the same lower bound for every . When , the ideal is principal and all assertions are immediate.

Support, depth, and spread. Corollary 5.3 says that any nonzero normalized strand in homological degree has appeared by power . Since , every such strand has appeared by . Corollary 5.2 makes it persist thereafter; in degree zero the only normalized internal degree is zero, present from the first power. This proves support stabilization. The regularity formula follows by taking the maximum of on that finite support. The depth formula follows from and Auslander–Buchsbaum.

Finally the fiber algebra is the monomial algebra on the labels . Its exponent semigroup is the image of the pairs under the injective linear map given by the row-difference formula. The vectors of the bottom and a maximal ideal chain span a group of rank ; successive differences recover the coordinate directions of , in addition to . The monomial algebra therefore has dimension , which is the analytic spread. □

9. An integral family for closed graphs

For closed graphs, the positive recursion removes integral torsion from the coefficient homology. We can therefore lift bases through the degeneration and assemble them into one free complex whose fibers resolve both the monomial and binomial powers. In this section is a deformation parameter of ordinary internal degree zero.

Let be a closed graph with the interval labeling used above, and set

Theorem 9.1 (A universal minimal complex). For each there is a finite graded free resolution of whose matrices have entries in . Every base change from preserves its exactness. In particular, specialization to and over every field gives minimal free resolutions of and , with the same ranks and shifts.

To construct this complex, we first prove that the relative Tor modules are free over the parameter ring. A positive parameter weight allows integral cycle lifts to be chosen as a basis, after which the resolution is built by lifting generators of successive syzygies.

Lemma 9.2 (Integral multiplicative transfer). Put and . Let be an -flat graded -algebra, with auxiliary degree and ordinary internal degree , whose fixed components are finite free over . Suppose it has an additional bounded-below integer weight grading with , and every fixed component is finite free over . Let . Assume

(i): is free as a graded abelian group and is generated as a -algebra by auxiliary degree one and its unit;
(ii): over every field the first-degree special and generic Betti tables agree, and after inverting the family and its Koszul complex are obtained from a constant fiber by diagonal rescaling.

Then every fixed part of is a finite free graded -module, its reduction modulo is , and it is generated over in auxiliary degree one. Relative Koszul homology commutes with arbitrary base change from .

Proof. Write , assigning each exterior variable the weight of its coefficient variable. First work in auxiliary degree one over a field . Over the DVR , the universal-coefficient formula is

where counts the cyclic -primary torsion summands. Equality of special and generic Betti numbers forces all to vanish. The diagonal rescaling away from zero rules out torsion supported at any other irreducible polynomial. Thus first-degree homology over is free. A homogeneous basis has the weights of its basis modulo , whose dimensions are independent of ; by assumption (i), the special integral homology is free, so universal coefficients commute with passage to every field in each retained weight. Thus the first-degree family homology has field-independent dimensions in every fixed weight.

Now apply universal coefficients over to each finite free complex . Comparison of with each shows that no integral homology group in auxiliary degree one has integer torsion; the difference of the dimensions is the sum of two nonnegative adjacent torsion counts. There is also no -torsion in that degree. A class killed by vanishes after tensoring with , where is regular, and is therefore integer torsion.

The long exact sequence of consequently lifts every integral special class of auxiliary degree one to a family cycle. Products of these cycles lift all special classes by assumption (i). Hence is surjective in every degree, and the same long exact sequence gives

The next, zero, term of the sequence gives the assertion in top homological degree as well.

Fix , choose a homogeneous integral basis of the special homology, and lift it to . Let be the map from the graded free -module on these lifts. Its cokernel is zero modulo . Because the grading is bounded below and has positive weight, induction on weight makes the cokernel zero. If is the kernel, regularity of on gives an injection ; its image is zero because the chosen bases identify with . Thus , and the same bounded-below weight induction gives . This proves freeness. Applying that induction to the quotient by the algebra generated in degree one proves the generation claim.

Finally, a bounded complex of flat -modules with flat homology commutes with arbitrary base change. In the present finite free complexes this also follows by induction from the bottom; the exact sequences defining boundaries and cycles split whenever their quotient is projective, so every homology summand survives tensoring. This proves the last assertion. □

Proof of Theorem 9.1. Every fine Koszul homology group of is free; the recursion of Lemma 6.1 makes its dimensions independent of the field, and universal coefficients exclude integral torsion. Theorem 7.1 therefore makes free over . Its proof also gives integral generation in Rees degree one. Indeed each crossing-block complex is a fine strand of a coefficient quotient with all other block degrees zero, so its homology is free. Integral Künneth expresses every block-tensor class as a product. The chain equivalence in that proof is defined by multiplying lifts of the whole block complexes into Rees degree one. After specializing , every homology class is therefore a sum of products of first-power classes. This generation assertion requires no compatibility between the choices for different multichains.

For the displayed binomial family assign

Every generator is homogeneous. After inverting , substituting transforms its generator into , so the family and its Koszul complex are diagonal rescalings of the constant binomial fiber.

For each , the products of displayed generators form a monic Gröbner basis over . Over this follows from the all-powers initial identity and the diagonal rescaling. Division is monic and therefore stays in ; its remainder vanishes over , hence already over . Standard monomials give a free -basis of the quotient. The ideal itself has a free basis indexed by the monomials of ; for each such monomial choose a monic multiple of a basis element with that leading monomial. Triangular division proves spanning and independence. These choices are weight homogeneous. Fixed ordinary degree components are finite free over , and fixing the weight as well gives finite free abelian groups, bounded below in weight.

The special fiber of is therefore exactly , and arbitrary field base changes give the intended powers. The first paragraph supplies integral freeness and generation of its homology. The fine first-power recursion supplies weightwise characteristic independence, and Peeva’s theorem supplies first-power equality in every field. Lemma 9.2 now applies. In particular, for , all modules are finite free over .

Lift a homogeneous -basis of to a map ; graded induction in ordinary internal degree makes it surjective. Its kernel is -flat because and are -flat, and its quotient modulo is the free relative module. Lifting a basis of this quotient repeats the construction. At every step, the syzygy remains -flat, every matrix entry lies in , and the next quotient modulo is the corresponding free relative Tor module. The construction terminates after at most steps, the length of the Koszul resolution of over ; when the next relative Tor is zero, graded induction makes the next syzygy zero.

The syzygies are -flat, so base change preserves exactness. Over a field the entries remain in the homogeneous maximal ideal, giving minimality. □

10. Characteristic dependence for general sublattices

The integral decomposition locates all characteristic dependence in the coefficient quotients. For general sublattices these can carry finite-complex homology, including torsion; we first encode a complex by a bipartite incidence graph and a squarefree ideal, then realize that ideal as a single predecessor colon. The fine grading isolates its homology, and repeating the top lattice element carries the same group into every power.

Lemma 10.1. Let be a bipartite graph with parts , no isolated vertices, and . In , put

Let , assume has a vertex, and let be the product of all variables of . Then

The same assertion holds over every field.

Proof. Let be the Stanley–Reisner complex of , on the full variable set , and . Write and . Directly from the generators,

Here each factor of consists of an isolated vertex and a disjoint simplex. Thus , while is a cone on every . For nonempty , set . Its -th join factor has leaves indexed by . Consequently is a sphere up to homotopy if , and is contractible otherwise. Every contains the simplex .

The relative Mayer–Vietoris spectral sequence for , relative to , has just one nonzero row; for . Orient its generator as the ordered join of the zero-cycles , choosing . Different leaf choices are homologous inside their leaf simplex, so all nonzero inclusion maps are . The surviving row is exactly the ordinary simplicial chain complex of . Hence

For the second identity, use excision and the triple ; the map is augmentation. All statements hold integrally, since the spectral sequence has only one nonzero row. Finally, with , a basis vector in the squarefree Koszul strand is , and it survives precisely when is a face of . Thus homological degree corresponds to cochain degree . The strand is the augmented cochain complex of , with this reversed indexing. Integral combinatorial Alexander duality gives

which proves the formula. The cohomology convention is essential for placing integral torsion in the correct degree. □

Proposition 10.2. There is a sublattice with elements such that, writing $𝕜$ ,

𝕜 𝕜

Over , its fine Koszul homology in the product of all variables is in degree and zero otherwise.

Proof. Take the complex on with facets and minimal nonfaces

Every four-element set contains a listed nonface. Its one-skeleton is complete. Collapse the spanning tree with edges , . The first five facets kill the cycle-basis elements . For the remaining elements , the last five boundaries impose

Thus and all other reduced integral homology vanishes; these ten boundaries have full rational rank, with determinant on the cycle basis. Over characteristic , reduced homology has dimension one in degrees ; over every other field it vanishes.

Let index the ten nonfaces, and set equal to the indexed triple. Then has edges, with degrees on and on . Put every element of below every element of in a height-two poset ; the edges of select only the coordinate chains. For each edge , record the intersection of an order ideal with the chain . Its size is a coordinate in . The image is the required sublattice; it consists of the vectors

The stages meet only at the all-one vector, so . The coordinate map is injective and preserves meet and join, since every poset element occurs in a selected chain.

At the all-two top element, the exact colon is obtained from in Lemma 10.1 by

and adjoining the unused variables . Indeed the lower stage gives the product of the two-generated row-block ideals, and the upper stage gives the . This disjoint monomial substitution is free; in each new block, divide a monomial by the largest power of the complete block product; the remaining monomials form a basis over the old polynomial ring. It therefore preserves resolutions and their minimality over fields, and identifies integral Tor after augmentation. The full old degree becomes . Equation (10.1) gives integral Tor in degree at .

To isolate this group in , use the exact prefix sequences over . If , then contains some with , while uses only variables below in that row by Lemma 2.1. A polynomial extension has no Koszul homology of positive fine degree in an unused variable. The long exact sequences therefore identify with . Universal coefficients give the asserted field dimensions. Since is squarefree, is the only possible fine Betti degree of ordinary degree . □

The construction depends only on the incidence between vertices and minimal nonfaces. Replacing the ten listed nonfaces by those of any finite complex gives the following general form; the same top-colon summand survives in every power.

Corollary 10.3 (Realization of finite-complex homology). Let be a finite simplicial complex on its nonempty vertex set , with no cone vertices, and let be its set of minimal nonfaces. Put and . There is a sublattice with elements such that, over ,

for every . In particular is the full squarefree degree. Over every field,

𝕜 𝕜 𝕜

Proof. Use and . The graph has no isolated vertices; each minimal nonface is nonempty, and a vertex absent from every minimal nonface would be a cone vertex. Apply the same two-stage sublattice construction, free substitution, and integral prefix argument as in Proposition 10.2, using Lemma 10.1. No coefficient colon contains any . Hence the exponents of all these variables force every factor of the contributing multichain to be the top element. The integral chain formula of Theorem 7.1 therefore consists of the single top-colon summand in degree . The field assertion follows by universal coefficients, or by the coefficient-ring version of that theorem. □

For every , Proposition 10.2 therefore gives a characteristic- excess of at least one at each of and . In fact the fine degree has Betti number one in these two homological degrees in characteristic and zero otherwise. Integrally, its Tor group is exactly in homological degree and zero in every other degree.

Characteristic dependence thus persists under passage to higher powers. Conversely, the crossing blocks cannot create torsion when all first-power coefficient quotients are torsion-free. These two facts give a second first-power criterion, this time for independence of the characteristic.

Corollary 10.4. For a finite sublattice , characteristic independence of the Betti table of one positive power of is equivalent to characteristic independence of all its powers. More precisely, for a prime , fixed , and row size , the difference

has nonnegative integer coefficients . In particular all its forward differences are nonnegative for .

Proof. Every fine component of the integral Koszul complex of a monomial quotient is free over , so universal coefficients make its characteristic- Betti excess nonnegative, both finely and ordinarily. Grouping the chain formula by strict supports proves the displayed expansion; is the sum of the excesses of over strict chains of length . Pascal’s identity gives the forward-difference assertion.

If one power is independent, its constant-chain summands force every to be independent, by nonnegativity. Ordinary independence implies fine independence by the same argument degree by degree. Conversely, the crossing decomposition has $𝕜 𝕜$ . The fine Koszul strands supported on are precisely those of the corresponding variable restriction of . Hence independence of all , Künneth on the disjoint blocks, and the chain formula give independence of every power. □

References

[1]: A. Almousa, K.-N. Lin, and W. Liske, Rees algebras of unit interval determinantal facet ideals, Journal of Pure and Applied Algebra 228 (2024), no. 5, 107601. DOI 10.1016/j.jpaa.2023.107601. arXiv 2008.10950v4.
[2]: V. Ene, G. Rinaldo, and N. Terai, Powers of binomial edge ideals with quadratic Gröbner bases, Nagoya Mathematical Journal 246 (2022), 233–255. DOI 10.1017/nmj.2021.1. arXiv 2009.08341v2.
[3]: I. Peeva, Closed binomial edge ideals, Journal für die reine und angewandte Mathematik 803 (2023), 1–33. DOI 10.1515/crelle-2023-0048.
[4]: A. Dohadwala, B. Flores-Silva, A. Orozco-Moya, and Z. Siegelnickel, Linear strands of powers of certain binomial edge ideals, arXiv 2601.10842 (2026).

Sublattice resolutions and Betti preservation in all powers

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