---
title: Mixed Buchsbaum-Rim Multiplicity Theory
url: https://www.emergentmind.com/topics/mixed-buchsbaum-rim-multiplicity
type: topic
---

# Mixed Buchsbaum-Rim Multiplicity Theory

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 [1109.5055; 1805.02314; 2311.15105].

## 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
\[
e_{\mathrm{BR}}(E_1^{[d_1]},\ldots,E_k^{[d_k]};N),
\qquad d_1+\cdots+d_k=d+p-1.
\]
This recovers ordinary Buchsbaum-Rim multiplicity when \(k=1\), and recovers mixed multiplicity of ideals when the \(E_i\) are of the form \(I_iF\) with \(I_i\) \(\mathfrak m\)-primary [2310.01216].

A second formulation, designed for arbitrary families of modules in a standard graded setting, uses a multigraded length function
\[
h(n,p,\mathbf r)=\ell\!\left(\frac{I_1^{r_1}\cdots I_q^{r_q}M_{n+p}}{J^n I_1^{r_1}\cdots I_q^{r_q}M_p}\right),
\]
where \(J\) 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 \((FC)\)-sequences [1109.5055].

A third formulation, adapted to several finite-colength modules \(M_i\subseteq F_i\) with \(F_i\) free of rank \(r_i\), uses the joint Buchsbaum-Rim polynomial
\[
p(n_1,\ldots,n_d)=
\lambda\!\left(
\frac{S_{n_1}(F_1)\cdots S_{n_d}(F_d)}
{S_{n_1}(M_1)\cdots S_{n_d}(M_d)}
\right),
\]
whose normalized leading coefficient is denoted
\[
\operatorname{br}(M_1|\cdots|M_d).
\]
When each \(r_i=1\), this reduces to mixed multiplicity of ideals [2508.07437].

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
\[
C=R/I_1\oplus\cdots\oplus R/I_r,
\]
with each \(I_j\) an \(\mathfrak m\)-primary ideal. Here the two-variable Buchsbaum-Rim function is
\[
\Lambda(p,q)=\ell_R\!\left(S_{p+q}/M^pS_q\right),
\]
where \(S=\operatorname{Sym}_R(F)\) and \(M\) is determined by a presentation of \(C\). For \(p,q\gg 0\), \(\Lambda(p,q)\) is a polynomial of total degree \(d+r-1\), and its coefficients define the associated Buchsbaum-Rim multiplicities \(e^j(C)\), \(0\le j\le d+r-1\). One has \(e^0(C)=e(C)\), the ordinary Buchsbaum-Rim multiplicity, and \(e^j(C)=0\) for \(j\ge r\) [1805.02314].

For the ordinary multiplicity, Kirby and Rees showed that
\[
e(R/I_1\oplus\cdots\oplus R/I_r)
=
\sum_{\substack{i_1,\ldots,i_r\ge 0\\ i_1+\cdots+i_r=d}}
e_{i_1\cdots i_r}(I_1,\ldots,I_r),
\]
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 [1804.01707].

Hayasaka obtained a formula for the last positive associated multiplicity:
\[
e^{r-1}(R/I_1\oplus\cdots\oplus R/I_r)=e(I_1+\cdots+I_r).
\]
This generalizes a Kirby-Rees formula proved earlier in the nested case \(I_1\subset\cdots\subset I_r\), where the right-hand side becomes \(e(I_r)\). 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 [1804.01707].

Hayasaka then computed the second-to-last positive multiplicity:
\[
e^{r-2}(R/I_1\oplus\cdots\oplus R/I_r)
=
E_{r-1}(I_1,\ldots,I_r)-(d+1)(r-1)e(I_1+\cdots+I_r),
\]
where
\[
E_{r-1}(I_1,\ldots,I_r)
=
\sum_{j=1}^r
e\!\left(
R/[I_1+\cdots+\widehat{I_j}+\cdots+I_r]
\oplus
R/[I_1+\cdots+I_r]
\right).
\]
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
\[
e^{r-2}(R/I_1\oplus\cdots\oplus R/I_r)=e(R/I_{r-1}\oplus R/I_r),
\]
again recovering a Kirby-Rees pattern [1805.02314].

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 [1804.01707; 1805.02314].

## 3. Reduction theory, \((FC)\)-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 \(J\) is \(\mathfrak n\)-primary, \(I_1,\ldots,I_d\) are ideals, and \(N\) is a finitely generated module, then
\[
e\!\left(J^{[k_0+1]},\mathbf I^{[\mathbf k]};N\right)
=
\sum_{p\in\Pi}
\ell(N_p)\,
e\!\left(J^{[k_0+1]},\mathbf I^{[\mathbf k]};R/p\right),
\]
where \(\Pi\) is the set of maximal-dimensional minimal primes of \(N\). There is also additivity on short exact sequences in top dimension, as well as recursion formulas obtained from filter-regular or weak-\((FC)\) elements [1208.0233].

For arbitrary modules, \((FC)\)-sequences play the role occupied by superficial sequences or joint reductions in classical mixed multiplicity theory. If \(x_1,\ldots,x_t\) is an \((FC)\)-sequence with \(k_i\) elements from \(I_i\), then
\[
e_j(J^{[k_0]},I_1^{[k_1]},\ldots,I_q^{[k_q]};M)
=
e_j^{BR}(J;M_t),
\]
where
\[
M_t=M/(x_1,\ldots,x_t)M:I^\infty.
\]
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 \((FC)\)-sequence when \(k_0>0\) [1109.5055].

Joint reductions are the other central reduction-theoretic device. For modules \(M_1,\ldots,M_q\) of finite colength in free modules, a collection \((B_1,\ldots,B_q)\) with each \(B_i\subseteq M_i\) 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 [2508.07437].

The converse direction, which is subtler, was established in module form as a converse of Rees’ mixed multiplicity theorem. Under the hypotheses that \((x_1,\ldots,x_k)S\) and the ideals \(I_{E_i}\) have the same height \(k\) and the same radical, and assuming local equality of the Buchsbaum-Rim multiplicity of \((x_1,\ldots,x_k)\) with the mixed Buchsbaum-Rim multiplicity of \((E_1,\ldots,E_k)\) at the relevant minimal primes, the sequence \((x_1,\ldots,x_k)\) is a joint reduction of \((E_1,\ldots,E_k)\) [2310.01216]. 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 \(\mathbb N^p\)-graded algebras \(A\subseteq B\), the multigraded length function
\[
\lambda_t^{A,B}(n_1,\ldots,n_p)
=
\ell_R\!\left(
\frac{[B]_{(n_1,\ldots,n_p)}}
{[A]_{(n_1-t_1+1,\ldots,n_p-t_p+1)}[B]_{(t_1-1,\ldots,t_p-1)}}
\right)
\]
is eventually polynomial, and its top-degree coefficients define relative mixed multiplicities \(e_t(\beta;A,B)\). These are nonincreasing in \(t\), and their stable values satisfy
\[
e_\infty(\beta;A,B)
=
\operatorname{br}_\beta([A]_{e_1},\ldots,[A]_{e_p};B),
\]
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 \(e(\beta;A,B)=e_{(1,\ldots,1)}(\beta;A,B)\) detects finite birational extensions, under equidimensional and catenary hypotheses [2311.15105].

A homological interpretation is given in terms of Euler-Poincaré characteristics. If \((B_1,\ldots,B_d)\) is a joint reduction of \((M_1,\ldots,M_d)\), and \(\phi_i:F_i\to F_i\) are endomorphisms with image \(B_i\), then
\[
\operatorname{br}(M_1|\cdots|M_d)
=
\chi\bigl(K_\bullet(\phi_1,\ldots,\phi_d)\bigr),
\]
where \(K_\bullet(\phi_1,\ldots,\phi_d)\) is the tensor product of the two-term Koszul-like complexes \(0\to F_i\xrightarrow{\phi_i}F_i\to 0\). A comparison theorem identifies this Euler characteristic with the Euler characteristic of the determinant Koszul complex, yielding
\[
\operatorname{br}(M_1|\cdots|M_d)=e(I_1|\cdots|I_d),
\]
where \(I_i\) is the ideal of maximal minors of a presentation matrix of \(M_i\). In this sense, the module-theoretic mixed Buchsbaum-Rim multiplicity is identified with the classical mixed multiplicity of determinant ideals [2508.07437].

For ordinary Buchsbaum-Rim multiplicity, intersection theory also yields projection and expansion formulas. If \(R'\) is module-finite over \(R\) of pure degree \(\delta\), then
\[
\delta\cdot e(M,N)
=
\sum_{\mathfrak m\in\Phi}
o_{\mathfrak m}\,e(M_{\mathfrak m},N_{\mathfrak m}),
\]
and there is also an expansion over minimal primes,
\[
e(M,N)=\sum_{\mathfrak p\in\Lambda}\ell(R_{\mathfrak p})\,e(M(\mathfrak p),N(\mathfrak p)).
\]
Although these formulas are stated for ordinary relative Buchsbaum-Rim multiplicity, they provide the intersection-theoretic template on which mixed theories are built [1507.08865].

## 5. Bounds, equalities, and low-dimensional phenomena

Numerical inequalities for Buchsbaum-Rim multiplicity frequently reflect mixed-multiplicity behavior. In dimension at least \(4\), Lech-type estimates extend to both ordinary Buchsbaum-Rim multiplicity and mixed multiplicities. For \(E\subseteq F=R^r\) with \(\lambda(F/E)<\infty\),
\[
br(mE)<\frac{(d+r-1)!}{r!}\lambda(F/E)e(R),
\]
and for \(\mathfrak m\)-primary ideals \(I_1,\ldots,I_d\),
\[
e(mI_1,\ldots,mI_d)
<
(d-1)!\sum_{i=1}^d \lambda(R/I_i)e(R).
\]
A key identity in the direct-sum case is
\[
br(mE)=
\sum_{a_1+\cdots+a_r=d}
e(mI_1^{[a_1]},\ldots,mI_r^{[a_r]}),
\]
which reduces a Buchsbaum-Rim problem to mixed multiplicities of ideals [1912.01073].

In two-dimensional regular local rings, torsion-free modules exhibit sharp bounds involving adjoint ideals. If \(M\) has rank \(r\), double dual \(F=M^{**}\), and \(I(M)\) is the ideal of maximal minors of a presentation matrix, then
\[
e(I)-\lambda(R/\operatorname{adj}(I))
\le e(M)\le
\lambda(F/M)+\lambda(R/\operatorname{adj}(I)).
\]
The upper bound is attained exactly for integrally closed modules. In the special case \(M=I\oplus J\), the difference \(e(I(M))-e(M)\) is exactly the mixed multiplicity \(e_1(I|J)\), and for integrally closed ideals one has
\[
e_1(I|J)=\lambda(R/IJ)-\lambda(R/I)-\lambda(R/J).
\]
This supplies a concrete two-dimensional bridge between module-theoretic Buchsbaum-Rim invariants and mixed multiplicities of ideals [2510.07779].

Related coefficient inequalities arise for fiber multiplicity. If \(M\subseteq F\) has finite colength and rank \(r\), with Buchsbaum-Rim coefficients \(br_i(M)\), then over a two-dimensional Cohen-Macaulay local ring,
\[
f_0(M)\le br_1(M)-br_0(M)+\ell(F/M)+\mu(M)-r.
\]
For direct sums \(M=I^{\oplus u}\oplus J^{\oplus v}\), explicit binomial expressions for \(br_0(M)\), \(br_1(M)\), and \(f_0(M)\) show that these coefficients behave as mixed Buchsbaum-Rim data in concrete families of modules [1804.01255].

## 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 \(e^j(C)\), 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 [1804.01707].

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 \((FC)\)-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 [1109.5055; 2311.15105; 2508.07437].

Third, numerical criteria have become increasingly precise. Equality of local Buchsbaum-Rim and mixed Buchsbaum-Rim multiplicities can force joint reduction [2310.01216]. Vanishing of relative mixed multiplicities detects integral dependence and birationality [2311.15105]. In two-dimensional regular local rings, joint reductions of integrally closed modules satisfy a joint-reduction-number-zero theorem:
\[
M_1M_2=B_1M_2+M_1B_2,
\]
for any joint reduction \((B_1,B_2)\) of integrally closed modules \(M_1,M_2\) of finite colength [2508.07437].

A current theme is explicit higher-coefficient theory. Hayasaka’s formula for \(e^{r-2}\) and the conjectural linear-combination pattern for further \(e^{r-j}(C)\) 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 [1805.02314]. 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.

Source: https://www.emergentmind.com/topics/mixed-buchsbaum-rim-multiplicity