Futoshi Hayasaka, Eero Hyry
In this article we prove that the Buchsbaum-Rim multiplicity $e(F/N)$ of a parameter module $N$ in a free module $F=A^r$ is bounded above by the colength $\ell_A(F/N)$. Moreover, we prove that once the equality $\ell_A(F/N)=e(F/N)$ holds true for some parameter module $N$ in $F$, then the base ring $A$ is Cohen-Macaulay.
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In this article, we compute the Buchsbaum-Rim function of two variables associated to a direct sum of cyclic modules and give a formula for the last positive associated Buchsbaum-Rim multiplicity in terms of the ordinary Hilbert-Samuel multiplicity of an ideal. This is a generalization of a formula for the last positive Buchsbaum-Rim multiplicity given by Kirby and Rees.
The associated Buchsbaum-Rim multiplicities of a module are a descending sequence of non-negative integers. These invariants of a module are a generalization of the classical Hilbert-Samuel multiplicity of an ideal. In this article, we compute the associated Buchsbaum-Rim multiplicity of a direct sum of cyclic modules and give a formula for the second to last positive Buchsbaum-Rim multiplicity in terms of the ordinary Buchsbaum-Rim and Hilbert-Samuel multiplicities. This is a natural generalization of a formula given by Kirby and Rees.
In this article, we prove that the Buchsbaum-Rim function $\ell_A(§_{ν+1}(F)/N^{ν+1})$ of a parameter module $N$ in $F$ is bounded above by $e(F/N) \binom{ν+d+r-1}{d+r-1}$ for every integer $ν\geq 0$. Moreover, it turns out that the base ring $A$ is Cohen-Macaulay once the equality holds for some integer $ν$. As a direct consequence, we observe that the first Buchsbaum-Rim coefficient $e_1(F/N)$ of a parameter module $N$ is always non-positive.
Joint reductions and mixed Buchsbaum-Rim multiplicities of modules and a joint-reduction-number-zero theorem
本文提出了 Noetherian 局部环上模的联合约化与混合 Buchsbaum-Rim 重数的新定义，证明了二维正则局部环上整闭模的联合约化数零定理，并揭示了模的混合重数与 Koszul 复形特征及理想重数间的联系。
We generalize an improved Lech bound, due to Huneke, Smirnov, and Validashti, for the Buchsbaum-Rim multiplicity and mixed multiplicity. We reduce the problem to the graded case and then to the polynomial ring case. There we use complete reductions, studied by Rees, to prove sharper bounds for the mixed multiplicity in low dimensions before proving the general case.
Over a regular local ring of dimension two with maximal ideal m, we study the Buchsbaum-Rim multiplicity of a finitely generated module M of finite colength in a free module F. First, we investigate the colength of an m-primary ideal and its Hilbert-Samuel multiplicity using linkage theory. As applications, we establish several multiplicity formula that express the Buchsbaum-Rim multiplicity of M in terms of the Hilbert-Samuel multiplicities of ideals related to a certain minimal reduction U of M. Moreover, there exists m-primary Bourbaki ideals I and J of the modules F and M respectively such…
In 1960, D.G. Northcott proved that if $e_0(I)$ and $e_1(I)$ denote zeroth and first Hilbert-Samuel coefficients of an $\mathfrak m$-primary ideal $I$ in a Cohen-Macaulay local ring $(R,\mathfrak m)$, then $e_0(I)-e_1(I)\le \ell (R/I)$. In this article, we study an analogue of this inequality for Buchsbaum-Rim coefficients. We prove that if $(R,\mathfrak m)$ is a two dimensional Cohen-Macaulay local ring and $M$ is a finitely generated $R$-module contained in a free module $F$ with finite co-length, then $br_0(M)-br_1(M)\le \ell (F/M)$, where $br_0(M)$ and $br_1(M$) denote zeroth and first Buc…
We study the Buchsbaum-Rim multiplicity, ab inicio, through a Koszul-Čech spectral sequence. We show that the Buchsbaum-Rim multiplicity is the arithmetic genus (Euler characteristic) of Koszul homology sheaves on a projective space over the base scheme.
Counterexamples to Minkowski-type inequalities and the Rees converse for modules via mixed Buchsbaum-Rim Multiplicities
本文构造混合Buchsbaum-Rim重数，证明其系数乘积公式，给出Minkowski型不等式等失效的反例，明确一维情形下Rees逆命题不成立，仅在特定条件下成立，并提出高维逆命题问题。
Relative mixed multiplicities and mixed Buchsbaum-Rim multiplicities
本文定义并研究了由 Simis 等人引入的相对重数的自然多重分级扩展，即相对混合重数，证明其稳定值等于混合 Buchsbaum-Rim 重数，并展示了可通过相对混合重数的消失来检测整依赖性与双有理性。
Let A be a Buchsbaum local ring with the maximal ideal m and let e(A) denote the multiplicity of A. Let Q be a parameter ideal in A and put I=Q:m. Then the equality I^2=QI holds true, if e(A)=2 and depthA>0. The assertion is no longer true, unless e(A)=2. Counterexamples are given.
Mapping free resolutions of length three II -- Module formats
本文研究投射维数为3的完美模的极小自由分解格式，借助泛型环定义高阶结构映射，并提出基于 Buchsbaum–Rim 复形的模链接推广，给出特定格式下的链接类判据。
In this survey article we want to discuss a way of constructing arithmetically Gorenstein varieties of high codimension. Consider kernel sheaves B_G of general, generically surjective morphisms G between decomposable bundles on P^n. The top-dimensional part of the zero-locus of a regular section of B_G is Gorenstein if B_G has odd rank. We show how to implement the construction method in the computer algebra system Singular and produce some examples of Gorenstein curves and threefolds in P^6. Finally, we investigate a class of sheaves B_G, where the degeneracy locus of G does not have the expe…
We prove the results about mixed Buchsbaum--Rim multiplicities announced in (9.10)(ii) on p.224 of our recent paper [J.Alg.(1994)], including a general mixed-multiplicity formula. In addition, we identify these multiplicities as the coefficients of the ``leading form'' of the appropriate Buchsbaum-Rim polynomial in three variables, and we prove a positivity theorem. In fact, we define the multiplicities as the degrees of certain zero-dimensional ``mixed twisted'' Segre classes, and we develop an encompassing general theory of these new rational equivalence classes in all dimensions. In paralle…
This paper begins by introducing and characterizing Buchsbaum-Rim sheaves on $Z = \Proj R$ where $R$ is a graded Gorenstein K-algebra. They are reflexive sheaves arising as the sheafification of kernels of sufficiently general maps between free R-modules. Then we study multiple sections of a Buchsbaum-Rim sheaf $\cBf$, i.e, we consider morphisms $ψ: \cP \to \cBf$ of sheaves on $Z$ dropping rank in the expected codimension, where $H^0_*(Z,\cP)$ is a free R-module. The main purpose of this paper is to study properties of schemes associated to the degeneracy locus $S$ of $ψ$. It turns out that $S…
In this paper we first give a lower bound on multiplicities for Buchsbaum homogeneous $k$-algebras $A$ in terms of the dimension $d$, the codimension $c$, the initial degree $q$, and the length of the local cohomology modules of $A$. Next, we introduce the notion of Buchsbaum $k$-algebras with minimal multiplicity of degree $q$, and give several characterizations for those rings. In particular, we will show that those algebras have linear free resolutions. Further, we will give many examples of those algebras.
Let $ϕ$ be a generically surjective morphism between direct sums of line bundles on $\proj{n}$ and assume that the degeneracy locus, $X$, of $ϕ$ has the expected codimension. We call $B_ϕ = \ker ϕ$ a (first) Buchsbaum-Rim sheaf and we call $X$ a standard determinantal scheme. Viewing $ϕ$ as a matrix (after choosing bases), we say that $X$ is good if one can delete a generalized row from $ϕ$ and have the maximal minors of the resulting submatrix define a scheme of the expected codimension. In this paper we give several characterizations of good determinantal schemes. In particular, it is shown …
In this paper, we study non-Cohen--Macaulay Buchsbaum Stanley--Reisner rings with linear free resolution. In particular, for given integers $c$, $d$, $q$ with $c \ge 1$, $2 \le q \le d$, we give an upper bound $h_{c,d,q}$ on the dimension of the unique non-vanishing homology $\widetilde{H}_{q-2}(Δ;k)$ of a $d$-dimensional Buchsbaum ring $k[Δ]$ with $q$-linear resolution and codimension $c$. Also, we discuss about existence for such Buchsbaum rings with $\dim_k \widetilde{H}_{q-2}(Δ;k) = h$ for any $h$ with $0 \le h \le h_{c,d,q}$, and prove an existence theorem in the case of $q=d=3$ using the…
Multiplicity Versus Buchsbaumness of the special fiber cone
研究特殊纤维锥的Buchsbaum性与广义Cohen-Macaulay性质，基于局部环的性质和滤波器条件，证明了在特定条件下，特殊纤维环的性质与原环一致。
Note on length and multiplicity of modules over two-dimensional regular local rings
本文研究了二维正则局部环上有限生成无挠模的Buchsbaum-Rim重数，给出了其上下界及达到条件，并推导出整闭模重数的公式。
A Buchsbaum theory for Frobenius closure
本文研究了在素特征局部环中，Frobenius闭包与Hilbert-Samuel乘数差值的无关性，并结合导出范畴和Buchsbaum性质提出特征化方法。
Let A be a Noetherian local ring with the maximal ideal m and d=dimA. Let Q be a parameter ideal in A. Let I=Q:m. The problem of when the equality I^2=QI holds true is explored. When A is a Cohen-Macaulay ring, this problem was completely solved by A. Corso, C. Huneke, C. Polini, and W. Vasconcelos, while nothing is known when A is not a Cohen-Macaulay ring. The present purpose is to show that within a huge class of Buchsbaum local rings A the equality I^2=QI holds true for all parameter ideals Q. The result will supply theorems of K. Yamagishi, S. Goto and K. Nishida with ample examples of id…
Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ with infinite residue field. Let $M$ be a finitely generated proper $R$-submodule of a free $R$-module $F$ with $\ell (F/M) < \infty$ and having rank $r$. In this article, we study the fiber multiplicity $f_0(M)$ of the module $M$. We prove that if $(R,\mathfrak{m})$ is a two dimensional Cohen-Macaulay local ring, then $f_0(M)\le br_1(M)-br_0(M)+\ell (F/M)+μ(M)-r$, where $br_i(M)$ denotes the $i^{th}$ Buchsbaum-Rim coefficient of $M$.
Mixed multiplicity and Converse of Rees' theorem for modules
本文研究模的 Rees 混合重数定理逆命题，在高度、根及局部 Buchsbaum-Rim 重数相等等条件下证明元素组构成联合约化，并建立相关性质。
Let $(R, \mathfrak m)$ be a $d$-dimensional Noetherian local ring and $E$ a finitely generated $R$-submodule of a free module $R^p.$ In this work we introduce a multiplicity sequence $c_k(E), k=0,..., d+p-1$ for $E$ that generalize the Buchsbaum-Rim multiplicity defined when $E$ has finite colength in $R^p$ as well as the Achilles-Manaresi multiplicity sequence that applies when $E\subseteq R$ is an ideal. Our main result is that the new multiplicity sequence can indeed be used to detect integral dependence of modules. Our proof is self-contained and implies known numerical criteria for integr…
We study the relationship between the reduction number of a primary ideal of a local ring relative to one of its minimal reductions and the multiplicity of the corresponding Sally module. This paper is focused on three goals: (i) To develop a change of rings technique for the Sally module of an ideal to allow extension of results from Cohen-Macaulay rings to more general rings. (ii) To use the fiber of the Sally modules of almost complete intersection ideals to connect its structure to the Cohen-Macaulayness of the special fiber ring. (iii) To extend some of the results of (i) to two-dimension…
We prove a projection formula, expressing a relative Buchsbaum--Rim multiplicity in terms of corresponding ones over a module-finite algebra of pure degree, generalizing an old formula for the ordinary (Samuel) multiplicity. Our proof is simple in spirit: after the multiplicities are expressed as sums of intersection numbers, the desired formula results from two projection formulas, one for cycles and another for Chern classes. Similarly, but without using any projection formula, we prove an expansion formula, generalizing the additivity formula for the ordinary multiplicity, a case of the ass…
Given a local Noetherian ring $(R, {\mathfrak m})$ of dimension $d>0$ and infinite residue field, we study the invariants $($dimension and multiplicity$)$ of the Sally module $S_J(I)$ of any ${\mathfrak m}$-primary ideal $I$ with respect to a minimal reduction $J$. As a by-product we obtain an estimate for the Hilbert coefficients of ${\mathfrak m}$ that generalizes a bound established by J. Elias and G. Valla in a local Cohen-Macaulay setting. We also find sharp estimates for the multiplicity of the special fiber ring ${\mathcal F}(I)$, which recover previous bounds established by C. Polini, …
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