Mixed Buchsbaum-Rim Multiplicity Theory

• Mixed Buchsbaum-Rim multiplicity is defined as a family of invariants extending the classical Buchsbaum-Rim multiplicity to collections of ideals or modules by analyzing multivariate polynomial growth.
• The theory employs techniques such as joint reductions, (FC)-sequences, and Rees algebra constructions to derive multiplicities from length functions and symmetric power modules.
• Results link these invariants with Hilbert-Samuel multiplicity and provide geometric and homological interpretations, offering criteria for integral dependence and birationality.

Mixed Buchsbaum-Rim multiplicity is a family of multiplicity invariants attached to collections of ideals or modules that extends the classical Buchsbaum-Rim multiplicity in much the same way that mixed multiplicities extend Hilbert-Samuel multiplicity. In the literature, these invariants are realized in several closely related forms: as leading coefficients of multivariate length polynomials for products of submodules, as associated coefficients of the two-variable Buchsbaum-Rim function for direct sums of cyclic modules, and as intersection numbers on Kleiman-Thorup-type blowups. Across these formulations, the subject connects Rees algebras, reduction theory, Hilbert polynomials, Koszul homology, and geometric criteria for integral dependence and birationality [(Callejas-Bedregal et al., 2011); (Hayasaka, 2018); (Cid-Ruiz, 2023)].

1. Definitions and formal settings

Let $(R,\mathfrak m)$ be a Noetherian local ring. For a module of finite colength in a free module, the classical Buchsbaum-Rim multiplicity is extracted from the leading term of a polynomial growth function on symmetric powers. Mixed Buchsbaum-Rim multiplicities arise when several submodules are allowed to vary simultaneously.

One formulation appears for submodules $E_1,\ldots,E_k\subseteq F$, where $F$ is a free $R$-module of rank $p$. If $N$ is a finitely generated $R$-module and

$$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$

is polynomial for $n_1,\ldots,n_k\gg 0$, then the coefficients of total degree $d+p-1$ define the mixed Buchsbaum-Rim multiplicities

$E_1,\ldots,E_k\subseteq F$0

This recovers ordinary Buchsbaum-Rim multiplicity when $E_1,\ldots,E_k\subseteq F$1, and recovers mixed multiplicity of ideals when the $E_1,\ldots,E_k\subseteq F$2 are of the form $E_1,\ldots,E_k\subseteq F$3 with $E_1,\ldots,E_k\subseteq F$4 $E_1,\ldots,E_k\subseteq F$5-primary (Ferrari et al., 2023).

A second formulation, designed for arbitrary families of modules in a standard graded setting, uses a multigraded length function

$E_1,\ldots,E_k\subseteq F$6

where $E_1,\ldots,E_k\subseteq F$7 has finite colength. The top-degree coefficients define mixed multiplicities, and a central result is that these mixed multiplicities coincide with associated Buchsbaum-Rim multiplicities of suitable quotient modules constructed from $E_1,\ldots,E_k\subseteq F$8-sequences (Callejas-Bedregal et al., 2011).

A third formulation, adapted to several finite-colength modules $E_1,\ldots,E_k\subseteq F$9 with $F$0 free of rank $F$1, uses the joint Buchsbaum-Rim polynomial

$F$2

whose normalized leading coefficient is denoted

$F$3

When each $F$4, this reduces to mixed multiplicity of ideals (Katz et al., 10 Aug 2025).

These definitions are not merely parallel notations. The cited works explicitly relate them through reduction theorems, Rees-algebra constructions, and comparison formulas.

2. Direct sums of cyclic modules and associated multiplicities

A particularly explicit setting is the direct sum of cyclic modules

$F$5

with each $F$6 an $F$7-primary ideal. Here the two-variable Buchsbaum-Rim function is

$F$8

where $F$9 and $R$0 is determined by a presentation of $R$1. For $R$2, $R$3 is a polynomial of total degree $R$4, and its coefficients define the associated Buchsbaum-Rim multiplicities $R$5, $R$6. One has $R$7, the ordinary Buchsbaum-Rim multiplicity, and $R$8 for $R$9 (Hayasaka, 2018).

For the ordinary multiplicity, Kirby and Rees showed that

$p$0

so the classical Buchsbaum-Rim multiplicity is a sum of mixed multiplicities of ideals. This is one of the basic bridges between the Buchsbaum-Rim and mixed-multiplicity theories (Hayasaka, 2018).

Hayasaka obtained a formula for the last positive associated multiplicity: $p$1 This generalizes a Kirby-Rees formula proved earlier in the nested case $p$2, where the right-hand side becomes $p$3. The point is structural: for arbitrary ideals, the last positive associated Buchsbaum-Rim multiplicity is governed by the Hilbert-Samuel multiplicity of the sum of the ideals, not by a mixed multiplicity expression (Hayasaka, 2018).

Hayasaka then computed the second-to-last positive multiplicity: $p$4 where

$p$5

Thus the second-to-last term is expressed through ordinary Buchsbaum-Rim multiplicities of direct sums of two cyclic modules together with a Hilbert-Samuel correction term. In the nested case this specializes to

$p$6

again recovering a Kirby-Rees pattern (Hayasaka, 2018).

A recurring misconception is that all higher associated Buchsbaum-Rim multiplicities should themselves be mixed multiplicities. The direct-sum formulas show that this is false in general: the last positive term is explicitly an ordinary Hilbert-Samuel multiplicity of a sum ideal, while the next term mixes ordinary Buchsbaum-Rim and Hilbert-Samuel contributions rather than a single mixed-multiplicity datum (Hayasaka, 2018, Hayasaka, 2018).

3. Reduction theory, $p$7-sequences, and additivity

The computational core of mixed Buchsbaum-Rim theory is reduction to better-behaved modules or ideals. In the multigraded setting, additivity takes the same form familiar from Samuel multiplicity. If $p$8 is $p$9-primary, $N$0 are ideals, and $N$1 is a finitely generated module, then

$N$2

where $N$3 is the set of maximal-dimensional minimal primes of $N$4. There is also additivity on short exact sequences in top dimension, as well as recursion formulas obtained from filter-regular or weak-$N$5 elements (Viet et al., 2012).

For arbitrary modules, $N$6-sequences play the role occupied by superficial sequences or joint reductions in classical mixed multiplicity theory. If $N$7 is an $N$8-sequence with $N$9 elements from $R$0, then

$R$1

where

$R$2

This realizes mixed multiplicities as associated Buchsbaum-Rim multiplicities of a quotient module. Positivity of the mixed multiplicity is characterized by the existence of such an $R$3-sequence when $R$4 (Callejas-Bedregal et al., 2011).

Joint reductions are the other central reduction-theoretic device. For modules $R$5 of finite colength in free modules, a collection $R$6 with each $R$7 minimally generated by the rank of the ambient free module is a joint reduction if the corresponding symmetric powers satisfy a reduction identity for all sufficiently large degrees. In this setting, joint reductions admit valuative and determinantal characterizations, and exist when the number of modules is at least the dimension (Katz et al., 10 Aug 2025).

The converse direction, which is subtler, was established in module form as a converse of Rees’ mixed multiplicity theorem. Under the hypotheses that $R$8 and the ideals $R$9 have the same height $$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$0 and the same radical, and assuming local equality of the Buchsbaum-Rim multiplicity of $$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$1 with the mixed Buchsbaum-Rim multiplicity of $$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$2 at the relevant minimal primes, the sequence $$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$3 is a joint reduction of $$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$4 (Ferrari et al., 2023). This places mixed Buchsbaum-Rim multiplicity alongside reduction theory as a numerical test for algebraic generation phenomena.

4. Geometric and homological interpretations

The geometric formulation of mixed Buchsbaum-Rim multiplicity goes back to Kleiman-Thorup and is extended in multigraded form by relative mixed multiplicities. For an inclusion of standard $$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$5-graded algebras $$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$6, the multigraded length function

$$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$7

is eventually polynomial, and its top-degree coefficients define relative mixed multiplicities $$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$8. These are nonincreasing in $$Q(n_1,\ldots,n_k;N)=\ell_R(E_1^{n_1}\cdots E_k^{n_k}\otimes_R N)$$9, and their stable values satisfy

$n_1,\ldots,n_k\gg 0$0

where the right-hand side is the mixed Buchsbaum-Rim multiplicity defined as an intersection number on a Kleiman-Thorup-type blowup. Vanishing of all these stable values detects finite integral extensions, and vanishing of the distinguished values $n_1,\ldots,n_k\gg 0$1 detects finite birational extensions, under equidimensional and catenary hypotheses (Cid-Ruiz, 2023).

A homological interpretation is given in terms of Euler-Poincaré characteristics. If $n_1,\ldots,n_k\gg 0$2 is a joint reduction of $n_1,\ldots,n_k\gg 0$3, and $n_1,\ldots,n_k\gg 0$4 are endomorphisms with image $n_1,\ldots,n_k\gg 0$5, then

$n_1,\ldots,n_k\gg 0$6

where $n_1,\ldots,n_k\gg 0$7 is the tensor product of the two-term Koszul-like complexes $n_1,\ldots,n_k\gg 0$8. A comparison theorem identifies this Euler characteristic with the Euler characteristic of the determinant Koszul complex, yielding

$n_1,\ldots,n_k\gg 0$9

where $d+p-1$0 is the ideal of maximal minors of a presentation matrix of $d+p-1$1. In this sense, the module-theoretic mixed Buchsbaum-Rim multiplicity is identified with the classical mixed multiplicity of determinant ideals (Katz et al., 10 Aug 2025).

For ordinary Buchsbaum-Rim multiplicity, intersection theory also yields projection and expansion formulas. If $d+p-1$2 is module-finite over $d+p-1$3 of pure degree $d+p-1$4, then

$d+p-1$5

and there is also an expansion over minimal primes,

$d+p-1$6

Although these formulas are stated for ordinary relative Buchsbaum-Rim multiplicity, they provide the intersection-theoretic template on which mixed theories are built (Kleiman, 2015).

5. Bounds, equalities, and low-dimensional phenomena

Numerical inequalities for Buchsbaum-Rim multiplicity frequently reflect mixed-multiplicity behavior. In dimension at least $d+p-1$7, Lech-type estimates extend to both ordinary Buchsbaum-Rim multiplicity and mixed multiplicities. For $d+p-1$8 with $d+p-1$9,

$E_1,\ldots,E_k\subseteq F$00

and for $E_1,\ldots,E_k\subseteq F$01-primary ideals $E_1,\ldots,E_k\subseteq F$02,

$E_1,\ldots,E_k\subseteq F$03

A key identity in the direct-sum case is

$E_1,\ldots,E_k\subseteq F$04

which reduces a Buchsbaum-Rim problem to mixed multiplicities of ideals (Nguyen et al., 2019).

In two-dimensional regular local rings, torsion-free modules exhibit sharp bounds involving adjoint ideals. If $E_1,\ldots,E_k\subseteq F$05 has rank $E_1,\ldots,E_k\subseteq F$06, double dual $E_1,\ldots,E_k\subseteq F$07, and $E_1,\ldots,E_k\subseteq F$08 is the ideal of maximal minors of a presentation matrix, then

$E_1,\ldots,E_k\subseteq F$09

The upper bound is attained exactly for integrally closed modules. In the special case $E_1,\ldots,E_k\subseteq F$10, the difference $E_1,\ldots,E_k\subseteq F$11 is exactly the mixed multiplicity $E_1,\ldots,E_k\subseteq F$12, and for integrally closed ideals one has

$E_1,\ldots,E_k\subseteq F$13

This supplies a concrete two-dimensional bridge between module-theoretic Buchsbaum-Rim invariants and mixed multiplicities of ideals (Hayasaka et al., 9 Oct 2025).

Related coefficient inequalities arise for fiber multiplicity. If $E_1,\ldots,E_k\subseteq F$14 has finite colength and rank $E_1,\ldots,E_k\subseteq F$15, with Buchsbaum-Rim coefficients $E_1,\ldots,E_k\subseteq F$16, then over a two-dimensional Cohen-Macaulay local ring,

$E_1,\ldots,E_k\subseteq F$17

For direct sums $E_1,\ldots,E_k\subseteq F$18, explicit binomial expressions for $E_1,\ldots,E_k\subseteq F$19, $E_1,\ldots,E_k\subseteq F$20, and $E_1,\ldots,E_k\subseteq F$21 show that these coefficients behave as mixed Buchsbaum-Rim data in concrete families of modules (Balakrishnan et al., 2018).

6. Scope, distinctions, and current directions

The modern theory has clarified several distinctions that were not transparent in the older literature. First, ordinary Buchsbaum-Rim multiplicity, associated Buchsbaum-Rim multiplicities $E_1,\ldots,E_k\subseteq F$22, and mixed Buchsbaum-Rim multiplicities are related but not interchangeable notions. The direct-sum formulas of Hayasaka show that even for cyclic summands the last positive associated term is not a mixed multiplicity formula, but the Hilbert-Samuel multiplicity of the sum ideal (Hayasaka, 2018).

Second, the theory is no longer confined to ideals or to finite-colength submodules in a single free module. It now includes arbitrary families of modules via graded-algebra methods and $E_1,\ldots,E_k\subseteq F$23-sequences, multigraded relative multiplicities with stable values equal to Kleiman-Thorup mixed Buchsbaum-Rim multiplicities, and joint reductions for collections of modules with determinantal and valuative characterizations [(Callejas-Bedregal et al., 2011); (Cid-Ruiz, 2023); (Katz et al., 10 Aug 2025)].

Third, numerical criteria have become increasingly precise. Equality of local Buchsbaum-Rim and mixed Buchsbaum-Rim multiplicities can force joint reduction (Ferrari et al., 2023). Vanishing of relative mixed multiplicities detects integral dependence and birationality (Cid-Ruiz, 2023). In two-dimensional regular local rings, joint reductions of integrally closed modules satisfy a joint-reduction-number-zero theorem: $E_1,\ldots,E_k\subseteq F$24 for any joint reduction $E_1,\ldots,E_k\subseteq F$25 of integrally closed modules $E_1,\ldots,E_k\subseteq F$26 of finite colength (Katz et al., 10 Aug 2025).

A current theme is explicit higher-coefficient theory. Hayasaka’s formula for $E_1,\ldots,E_k\subseteq F$27 and the conjectural linear-combination pattern for further $E_1,\ldots,E_k\subseteq F$28 indicate that higher associated Buchsbaum-Rim multiplicities may admit systematic expressions in terms of ordinary Buchsbaum-Rim multiplicities of smaller direct sums and Hilbert-Samuel terms (Hayasaka, 2018). This suggests a layered structure: mixed multiplicities dominate the ordinary Buchsbaum-Rim term, Hilbert-Samuel multiplicity governs the last associated term, and intermediate associated multiplicities interpolate between them through increasingly intricate reduction formulas.