Winfried Bruns, Tim Roemer, Attila Wiebe
We study initial algebras of determinantal rings, defined by minors of generic matrices, with respect to their classical generic point. This approach leads to very short proofs for the structural properties of determinantal rings. Moreover, it allows us to classify their Cohen-Macaulay and Ulrich ideals.
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This paper studies Ulrich ideals in one-dimensional Cohen-Macaulay local rings. A correspondence between Ulrich ideals and overrings is given. Using the correspondence, chains of Ulrich ideals are closely explored. The specific cases where the rings are of minimal multiplicity and \operatorname{GGL} rings are analyzed.
Let R be a Cohen-Macaulay local ring. In this paper we study the structure of Ulrich $R$-modules mainly in the case where R has minimal multiplicity. We explore generation of Ulrich R-modules, and clarify when the Ulrich R-modules are precisely the syzygies of maximal Cohen-Macaulay R-modules. We also investigate the structure of Ulrich R-modules as an exact category.
Non-existence of Ulrich modules over Cohen-Macaulay local rings
本文通过构造二维Cohen-Macaulay局部环族，否定回答了Ulrich关于是否总存在达到生成元数上界的模的问题，并给出Gorenstein正规整环与完全交整环的反例。
We provide the sufficient conditions for Rees algebras of modules to be Cohen-Macaulay, which has been proven in the case of Rees algebras of ideals by Johnson-Ulrich and Goto-Nakamura-Nishida. As it turns out the generalization from ideals to modules is not just a routine generalization, but requires a great deal of technical development. We use the technique of generic Bourbaki ideals introduced by Simis, Ulrich and Vasconcelos to obtain the Cohen-Macaulayness of Rees Algebras of modules.
We show that a smooth projective non-degenerate arithmetically Cohen-Macaulay subvariety X of P^N infinite Cohen-Macaulay type becomes of finite Cohen-Macaulay type by removing Ulrich bundles if and only if N = 5 and X is a quartic scroll or the Segre product of a line and a plane. In turn, we give a complete and explicit classification of ACM bundles over these varieties.
Ulrich modules : decomposition and relation with rings having minimal multiplicity
该研究探讨Cohen-Macaulay局部环中Ulrich模的基本性质，证明其可分解，并据此讨论具有极小重数环的若干刻画及作为模为Ulrich的理想的性质。
The structure of the complex $\operatorname{\mathbf{R}Hom}_R(R/I,R)$ is explored for an Ulrich ideal $I$ in a Cohen-Macaulay local ring $R$. As a consequence, it is proved that in a one-dimensional almost Gorenstein but non-Gorenstein local ring, the only possible Ulrich ideal is the maximal ideal. It is also studied when Ulrich ideals have the same minimal number of generators.
Full-Trace Modules
本研究在交换Noetherian局部环上引入全迹模概念，证明非正则非主理想环上剩余域的正合冲模均为全迹模，刻画了Cohen-Macaulay环极小重数与全迹Ulrich模的等价关系，并给出数值半群环上全迹Ulrich模的分解形式。
Modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings
本文研究了一维Cohen-Macaulay局部环上最小乘数模的性质，通过刻画跟踪理想和反射理想的作用，揭示了极小乘数模与Burch和Ulrich模的关系。
We study reflexive modules over one dimensional Cohen-Macaulay rings. Our key technique exploits the concept of $I$-Ulrich modules.
The Cohen-Macaulay type of idealizations of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings is explored. There are two extremal cases, one of which is closely related to the theory of Ulrich modules \cite{BHU, GOTWY1, GOTWY2, GTT2}, and the other one is closely related to the theory of residually faithful modules and the theory of closed ideals \cite{BV}.
Ulrich split rings
本文引入并研究Ulrich分裂环，建立其与剩余域合冲及上同调零化子的判据，刻画小维数情形，并给出构造方法及在检测投射/内射模等方面的应用。
We consider double determinantal varieties, a special case of Nakajima quiver varieties. Li conjectured that double determinantal varieties are normal, irreducible, Cohen-Macaulay varieties whose defining ideals have a Gröbner basis given by their natural generators. We use liaison theory to prove this conjecture in a manner that generalizes results for mixed ladder determinantal varieties. We also give a formula for the dimension of a double determinantal variety.
This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves E of arbitrary high rank on a general standard (resp. linear) determinantal scheme X\subset \PP^n of codimension c \ge 1, n-c \ge 1 and defined by the maximal minors of a t \times (t+c-1) homogeneous matrix A. The sheaves E are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme X\subset \PP^n is of wild representation type provided the degrees of the entries of the matrix A satisfy …
Complexity and rigidity of Ulrich modules, and some applications
本文研究 Ulrich 模作为测试模的可行性，证明 CM 局部环上的 Ulrich 模具有极大复杂性和曲率，给出局部环的新刻画，并探讨其 Tor 刚性性质。
In this paper we study Ulrich ideals of and Ulrich modules over Cohen--Macaulay local rings from various points of view. We determine the structure of minimal free resolutions of Ulrich modules and their associated graded modules, and classify Ulrich ideals of numerical semigroup rings and rings of finite CM-representation type.
The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen-Macaulay modules with respect to ideals, and study the relationship between those modules and Ulrich modules with respect to good ideals.
Some properties of ideals in Cohen-Macaulay local rings
本文研究了带有典范模的 Cohen-Macaulay 局部环在商去正则序列和局部化时相关不变量关系的保持条件，并给出了一维情形下理想成为 Elias 和 Burch 理想的条件，进而探讨了这三类理想之间的关系。
An Ulrich sheaf on an embedded projective variety is a normalized arithmetically Cohen-Macaulay sheaf with the maximum possible number of independent sections. Ulrich sheaves are important in the theory of Chow forms, Boij-Soderberg theory, generalized Clifford algebras, and for an approach to Lech's conjecture in commutative algebra. In this note, we give a reduction of the construction of Ulrich sheaves on a projective variety X to the construction of an Ulrich sheaf for a finite map of curves, which is in turn equivalent to a higher-rank Brill-Noether problem for any of a certain class of c…
In this article, applying the quasi-Gorenstein analogous of the Ulrich's deformation of certain Gorenstein rings we show that some homological conjectures, including the Monomial Conjecture, Big Cohen-Macaulay Algebra Conjecture as well as the Small Cohen-Macaulay Conjecture reduce to the excellent unique factorization domains. Some other reductions of the Monomial Conjecture are also proved. We, moreover, show that certain almost complete intersections adhere the Monomial Conjecture.
If $I$ is a perfect ideal in a local Cohen-Macaulay ring, the generators of ideals linked to $I$ are well understood. However, the generators of the residual intersections of $I$ have only been computed in a few special cases. In this paper, we show that the $n$-residual intersections of determinantal ideals of generic $2\times n$ matrices are sums of links.
We show that the ideal generated by maximal minors (i.e., $(k+1)$-minors) of a $(k+1) \times n$ Vandermonde matrix is radical and Cohen-Macaulay. Note that this ideal is generated by all Specht polynomials with shape $(n-k,1,...,1)$.
Cohen-Macaulay types of certain edge subrings of bipartite graphs and (generalized) Fuss-Catalan numbers
该研究给出了一类特殊凸多联骨牌的两个非同构坐标环实例，其Cohen-Macaulay型为广义Fuss-Catalan数，并推导了这类数的行列式公式。
We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.
Higher syzygy bundles and the Eisenbud-Huneke-Ulrich conjecture
研究具有部分线性（虚拟）分解的多项式环中的理想，证明了Eisenbud-Huneke-Ulrich猜想的一个稍弱版本，并推广了Macaulay界和Eisenbud-Huneke-Ulrich定理。
Generic Bourbaki ideals were introduced by Simis, Ulrich and Vasconcelos to study the Cohen-Macaulay property of Rees algebras of modules. In this article we prove that the same technique can sometimes be used to investigate the Cohen-Macaulay property of fiber cones of modules and to study the defining ideal of Rees algebras. This is possible as long as the Rees algebra of a given module $E$ is a deformation of the Rees algebra of a generic Bourbaki ideal $I$ of $E$. Our main technical result provides a deformation condition that in fact extends the applicability of generic Bourbaki ideals to…
The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In a Cohen-Macaulay complete local ring $R$ with a parameter ideal $Q$, the Auslander-Reiten conjecture holds for $R$ if and only if it holds for the residue ring $R/Q$. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring $R/Q^\ell$ in connection with that for $R$, and prove the equivalence of them for the case where $R$ is Gorenstein and $\ell\le \dim R$. In the latter part, we generalize the result of the minimal multiplicity by J. Sally. Due to these two of our…
In the present paper we investigate a question stemming from a long-standing conjecture of Vasconcelos: given a generically a complete intersection perfect ideal I in a regular local ring R, is it true that if I/I^2 (or R/I^2) is Cohen-Macaulay then R/I is Gorenstein? Huneke and Ulrich, Minh and Trung, Trung and Tuan and - very recently - Rinaldo Terai and Yoshida, already considered this question and gave a positive answer for special classes of ideals. We give a positive answer for some classes of ideals, however, we also exhibit prime ideals in regular local rings and homogeneous level idea…
We compute the canonical trace of generic determinantal rings and provide a sufficient condition for the trace to specialize. As an application we determine the canonical trace $\mbox{tr}(ω_R)$ of a Cohen-Macaulay ring $R$ of codimension two, which is generically Gorenstein. It is shown that if the defining ideal $I$ of $R$ is generated by $n$ elements, then $\mbox{tr}(ω_R)$ is generated by the $(n-2)$-minors of the Hilbert-Burch matrix of $I$.
We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a $2\times n$ sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic.
In this paper, we study the family of determinantal ideals of "close" cuts of Hankel matrices, say $ \mathcal{f} $. We show that the multi-Rees algebra of ideals in $ \mathcal{f} $ is defined by a quadratic Gröbner basis, it is Koszul, normal Cohen-Macaulay domain and it has a nice Sagbi basis. As a consequence of Koszulness, we prove that every product $ I_1\ldots I_l $ of ideals of $ \mathcal{f} $ has linear resolution. Moreover, we show that natural generators of every product $ I_1\ldots I_l $ form a Gröbner basis.
Linkage and $F$-Regularity of Determinantal Rings
本文证明了由极大子式定义的generic determinantal ring的generic link具有强$F$-正则性，强化了Chardin和Ulrich在分次情形下的结果，并为determinantal rings的强$F$-正则性提供了新的简单证明。
Rees Algebras of Unit Interval Determinantal Facet Ideals
本文利用 SAGBI 基技术，为单位区间行列式面理想的 Rees 代数与特殊纤维环构造 Gröbner 基，证明其纤维型、Koszul 性及正规 Cohen-Macaulay 与有理奇点性质。
Blowup algebras of determinantal modules
本文研究由阶梯矩阵极大子式或单位区间行列式理想生成的模的爆破代数，确定其多重Rees代数及特殊纤维环的Gröbner基，并证明所得代数为Koszul Cohen–Macaulay正规整环，且在特征零时具有有理奇点、正特征时为F-有理。
Let $R=k[x_1, ..., x_n]$ be a polynomial ring and let $I\subset R$ be a graded ideal. In \cite{R}, Römer asked whether under the Cohen-Macaulay assumption the $i$-th Betti number $β_{i}(R/I)$ can be bounded above by a function of the maximal shifts in the minimal graded free $R$-resolution of $R/I$ as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen-Macaulay algebras $k[x_1, ..., x_n]/I$ when $I$ is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds ar…
In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong and T. Kobayashi concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified …
A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $n\times m$ matrix where $n\leq m$ indexed by the facets of a simplicial complex $Δ$. We consider the more general notion of an $r$-DFI, which is generated by a subset of $r$-minors of a generic matrix indexed by the facets of $Δ$ for some $1\leq r\leq n$. We define and study so-called lcm-closed and unit interval $r$-DFIs, and show that the minors parametrized by the facets of $Δ$ form a reduced Gröbner basis with respect to \emph{any} term order for an lcm-closed $r$-DFI. We also see that being lcm-…
Let $R$ be a Noetherian local ring and $m$ a positive integer. Let $I$ be the ideal of $R$ generated by the maximal minors of an $m \times (m + 1)$ matrix $M$ with entries in $R$. Assuming that the grade of the ideal generated by the $k$-minors of $M$ is at least $m - k + 2$ for $1 \leq \forall k \leq m$, we will study the associated primes of $I^n$ for $\forall n > 0$. Moreover, we compute the saturation of $I^n$ for $1 \leq \forall n \leq m$ in the case where $R$ is a Cohen-Macaulay ring and the entries of $M$ are powers of elements that form an sop for $R$.
Lower bounds on Loewy lengths of modules of finite projective dimension
本文研究局部环上有限长度模的Loewy长度下界，证明严格Cohen-Macaulay环上该长度不小于正则度，证实多个猜想并改进已知下界。
We provide two examples of smooth projective surfaces of tame CM type, by showing that any parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in projective 5-space is either a single point or a projective line. For surfaces of minimal degree and wild CM type, we classify rigid Ulrich bundles as Fibonacci extensions. For the rational normal scrolls S(2,3) and S(3,3), a complete classification of rigid ACM bundles is given in terms of the action of the braid group in three strands.
We show in this paper that the principal component of the first order jet scheme over the classical determinantal variety of m x n matrices of rank at most 1 is arithmetically Cohen-Macaulay, by showing that an associated Stanley-Reisner simplicial complex is shellable.
Given two determinantal rings over a field k. We consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety. When the Rees algebra and the Symmetric algebra coincide, we show that the Rees algebra is Cohen-Macaulay.
Hankel determinantal rings, i.e., determinantal rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves; they may also be viewed as linear specializations of generic determinantal rings. We prove that, over fields of characteristic zero, Hankel determinantal rings have rational singularities; in the case of positive prime characteristic, we prove that they are F-pure. Independent of the characteristic, we give a complete description of the divisor class groups of these rings, and show that ea…
Herzog, Huneke, and Srinivasan have conjectured that for any homogeneous $k$-algebra, the multiplicity is bounded above by a function of the maximal degrees of the syzygies and below by a function of the minimal degrees of the syzygies. The goal of this paper is to establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field $k$ for $k$-algebras $k[x_1, ..., x_n]/I$ being $I$ a determinantal ideal of arbitrary codimension.
Schemes defined by residual intersections have been extensively studied in the case when they are Cohen-Macaulay, but this is a very restrictive condition. In this paper we make the first study of a class of natural examples far from satisfying this condition, the rank 1 loci of generic $2\times n$ matrices. Here we compute their depths and many other properties. These computations require a number of novel tools.
We prove new results on the connections between reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension (and allowing…
Let R be a Cohen-Macaulay local ring with a canonical module. We consider Auslander's (higher) delta invariants of powers of certain ideals of R. Firstly, we shall provide some conditions for an ideal to be a parameter ideal in terms of delta invarints. As an application of this result, we give upper bounds for orders of Ulrich ideals of R when R has Gorenstein punctured spectrum. Secondly, we extend the definition of indices to the ideal case, and generalize the result of Avramov-Buchweitz-Iyengar-Miller on the relationship between the index and regularity.
Let $X$ be a set of $4$ generic points in $\mathbb{P}^2$ with homogeneous coordinate ring $R$. We classify indecomposable graded MCM modules over $R$ by reducing the classification to the Four Subspace problem solved by Nazarova and Gel$'$fand-Ponomarev, or equivalently to the representation theory of the $\widetilde{D}_4$ quiver. In particular, the $\mathbb{P}^1$ tubular family of regular representations corresponds to matrix factorizations of the pencil of conics going through $X$, with smooth conics $Q_{t}$ corresponding to rank one tubes and the singular conics $Q_0, Q_1, Q_{\infty}$ givin…
We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Groebner bases with respect to a skew-diagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties are arithmetically Cohen-Macaulay. We characterize the arithmetically Gorenstein ones, among those that satisfy a technical condition. Our main result consists in proving that mixed ladder determinanta…
Segre-Determinantal Loci and the Image Variety for Three Flatland Cameras
受代数几何在计算机视觉重建问题中的应用启发，研究Segre簇上线性相关点的退化轨迹方程，证明其理想是素理想、Cohen-Macaulay且由自然极大子式生成，这些子式构成通用Gröbner基。
On generalized Gorenstein local rings
本文提出广义戈林斯坦局部环的概念，并探讨其与Cohen-Macaulay环相关代数结构的联系，同时提供了多种实例。
Let k be a field and R a pure subring of the infinite-dimensional polynomial ring k[X1;...]. If R is generated by monomials, then we show that the equality of height and grade holds for all ideals of R. Also, we show R satisfies the weak Bourbaki unmixed property. As an application, we give the Cohen-Macaulay property of the invariant ring of the action of a linearly reductive group acting by k-automorphism on k[X1;...]. This provides several examples of non-Noetherian Cohen-Macaulay rings (e.g. Veronese, determinantal and Grassmanian rings).
A level initial ideal of the 2-minors determinantal ideal
本文针对域上$m\times n$变量矩阵的2-子式生成的行列式理想$I_2$，构造合适单项式序得到其水平初始理想，比较二者Betti表，并证明$m<n$时对应单纯复形可壳。
Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero
本文在特征零下构造了无算术 Cohen-Macaulay 丛的极化曲面，其截面环无分次极大 Cohen-Macaulay 模，通过数值判据和 Harder-Narasimhan 滤过与 Horrocks 分裂的比较，反驳了分次细化猜想。
Goto rings
本文引入并发展 Goto 环理论，推广几乎 Gorenstein 环，通过扩展典范理想约化数至多 2 定义，并用 Sally 模刻画其结构、确定 Hilbert 函数，且给出多类例子。
On Rees algebras of 2-determinantal ideals
本文证明了2×n线性形式矩阵子式理想的Rees代数及其特殊纤维环具有Cohen-Macaulay和Koszul性质，通过分析Hilbert概形分层并结合Groebner退化揭示了其二次代数特征。
In a Cohen-Macaulay local ring $(A, \mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequality $\ell_A(A/I) \ge \mathrm{e}_0(I) - \mathrm{e}_1(I) + \dfrac{\mathrm{e}_2(I) + \ell_A(I^2/QI)}{2}$, where $\mathrm{e}_i(I)$ denotes the $i$th Hilbert coefficients and $Q$ denotes a minimal reduction of $I$. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore our inequality becomes an equality if and only if the depth of the associated graded ring of …
Let $R$ be a Cohen-Macaulay local ring of dimension $d$ with infinite residue field. Let $I$ be an $R$-ideal that has analytic spread $\ell(I)=d$, $G_d$ condition and the Artin-Nagata property $AN^-_{d-2}$. We provide a formula relating the length $λ(I^{n+1}/JI^{n})$ to the difference $P_I(n)-H_I(n)$, where $J$ is a general minimal reduction of $I$, $P_I(n)$ and $H_I(n)$ are the generalized Hilbert-Samuel polynomial and the generalized Hilbert-Samuel function in the sense of C. Polini and Y. Xie. We then use it to establish formulas to compute the higher generalized Hilbert coefficients of $I$…
Let $R$ be a Noetherian local ring and let $I$ be an ideal in $R$. The ideal $I$ is called balanced if the colon ideal $J:I$ is independent of the choice of the minimal reduction $J$ of $I$. Under suitable assumptions, Ulrich showed that $I$ is balanced if and only if the reduction number, $r(I)$, of $I$ is at most the `expected' one, namely $\ell(I)- \height I+1$, where $\ell(I)$ is the analytic spread of $I$. In this article we propose a generalization of balanced. We prove under suitable assumptions that if either $R$ is one-dimensional or the associated graded ring of $I$ is Cohen-Macaulay…
Special fibers of coordinate sections of Hankel Matrices
本文研究Hankel矩阵坐标截面的特殊纤维，揭示其定义方程的结构特征，并证明其Cohen-Macaulay性质及极小生成元的次数增长规律。
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