Buchsbaum Ring Research Articles

Almost Buchsbaumness of some rings arising from complexes with isolated singularities

We study properties of the Stanley–Reisner rings of simplicial complexes with isolated singularities modulo two generic linear forms. Miller, Novik, and Swartz proved that if a complex has homologically isolated singularities, then its Stanley–Reisner ring modulo one generic linear form is Buchsbaum. Here we examine the case of nonhomologically isolated singularities, providing many examples in which the Stanley–Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but not Buchsbaum ring.

$$F$$ F -injectivity and Buchsbaum singularities

Let \((R,\mathfrak {m}, K)\) be a local ring that contains a field. We show that, when \(R\) has equal characteristic \(p>0\) and when \(H_\mathfrak {m}^i(R)\) has finite length for all \(i<\dim R\), then \(R\) is \(F\)-injective if and only if every ideal generated by a system of parameters is Frobenius closed. As a corollary, we show that such an \(R\) is in fact a Buchsbaum ring. This answers positively a question of S. Takagi that \(F\)-injective singularities with isolated non-Cohen–Macaulay locus are Buchsbaum. We also study the characteristic \(0\) analogue of this question and we show that Du Bois singularities with isolated non-Cohen–Macaulay locus are Buchsbaum in the graded case.

PSEUDO BUCHSBAUMNESS FOR IDEALIZATIONS OF NOETHERIAN MODULES OVER LOCAL RINGS

Let (R, 𝔪) be a Noetherian local ring with dim R = d and M an R-module with dim M &lt; d. We prove in this paper that the idealization R ⋉ M of M over R is a pseudo Buchsbaum ring if and only if so is R.

Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

AbstractLet (A,m) be a Noetherian local ring with d = dim A ≥ 2. Then, if A is a Buchsbaum ring, the first Hilbert coefficients of A for parameter ideals Q are constant and equal to where hi(A) denotes the length of the ith local cohomology module of A with respect to the maximal ideal m. This paper studies the question of whether the converse of the assertion holds true, and proves that A is a Buchsbaum ring if A is unmixed and the values are constant, which are independent of the choice of parameter ideals Q in A. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.

Buchsbaumness in local rings possessing constant first Hilbert coefficients of parameters

AbstractLet (A,m) be a Noetherian local ring withd= dimA≥ 2. Then, ifAis a Buchsbaum ring, the first Hilbert coefficientsofAfor parameter idealsQare constant and equal towherehi(A)denotes the length of theith local cohomology moduleofAwith respect to the maximal ideal m. This paper studies the question of whether the converse of the assertion holds true, and proves thatAis a Buchsbaum ring ifAis unmixed and the valuesare constant, which are independent of the choice of parameter idealsQinA. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.

The Chern coefficients of local rings

The Chern numbers of the title are the first coefficients (after the multiplicities) of the Hilbert functions of various filtrations of ideals of a local ring (R,m). For a Noetherian (good) filtration A of m-primary ideals, the positivity and bounds for e1(A) are well-studied if R is Cohen-Macaulay, or more broadly, if R is a Buchsbaum ring or mild generalizations thereof. For arbitrary geometric local domains, we introduce techniques based on the theory of maximal CohenMacaulay modules and of extended multiplicity functions to establish the meaning of the positivity of e1(A), and to derive lower and upper bounds for e1(A). Dedicated to Professor Melvin Hochster on the occasion of his 65th birthday

Buchsbaum and Monomial Conjecture Dimension#

We define two new homological invariants for a finitely generated module M over a commutative Noetherian local ring R, its Buchsbaum dimension B-dim R M, and its Monomial conjecture dimension MC-dim R M. It will be shown that these new invariants have certain nice properties we have come to expect from homological dimensions. Over a Buchsbaum ring R, every finite module M has B-dim R M < ∞; conversely, if the residue field has finite B-dimension, then the ring R is Buchsbaum. Similarly R satisfies the Hochster Monomial Conjecture if only if MC-dim R k is finite, where k is the residue field of R. MC-dimension fits between the B-dimension and restricted flat dimension Rfd of Christensen et al. [Christensen, L. W., Foxby, H.-B., Frankild, A. (2002). Restricted homological dimensions and Cohen–Macaulayness. J. Algebra 251(1):479–502]. B-dimension itself is finer than CM-dimension of Gerko [Gerko, A. A. (2001). On homological dimensions. Sb. Math. 192(7–8):1165–1179] and we have equality if CM-dimension is finite. It also satisfies an analog of the Auslander–Buchsbaum formula.

Stanley-Reisner rings, sheaves, and Poincaré-Verdier duality

Recently, I defined a squarefree module over a polynomial ring $S = k[x_1, >..., x_n]$ generalizing the Stanley-Reisner ring $k[\Delta] = S/I_\Delta$ of a simplicial complex $\Delta \subset 2^{1, ..., n}$. In this paper, from a squarefree module $M$, we construct the $k$-sheaf $M^+$ on an $(n-1)$ simplex $B$ which is the geometric realization of $2^{1, ..., n}$. For example, $k[\Delta]^+$ is (the direct image to $B$ of) the constant sheaf on the geometric realization $|\Delta| \subset B$. We have $H^i(B, M^+) = [H^{i+1}_m(M)]_0$ for all $i > 0$. The Poincare-Verdier duality for sheaves $M^+$ on $B$ corresponds to the local duality for squarefree modules over $S$. For example, if $|\Delta|$ is a manifold, then $k[\Delta]$ is a Buchsbaum ring whose canonical module is a squarefree module giving the orientation sheaf of $|\Delta|$ with the coefficients in $k$.

Descent from the form ring and buchsbaum rings

There is a spectral sequence technique in order to estimate the local cohomology of a ring by the local cohomology of a certain form ring. As applications there are information on the descent of homological properties (Cohen-Macaulay, Buchsbaum etc.) from the form ring to the ring itself. In the case of Buchsbaum ring there is a discussion of the descent of the surjectivity of a natural map into the local cohomology.

Type of blowing-up over a Buchsbaum ring

Throughout this paper (A, , k) denotes a Noetherian local ring of dimension d and stands for the ith local cohomology functor with respect to . We refer the reader to [16] for any unexplained terminology.

On the Cohen-Macaulay and Buchsbaum property for unions of planes in affine space

The coordinate ring A of a union of planes in affine space is studied and it is asked when A is a Cohen-Macaulay or Buchsbaum ring. These properties are related to the position of the planes via the notion of seminormality. It is shown that A is Cohen-Macaulay iff A is connected in codimension 2 and seminormal in an appropriate sense. Consideration of the Cohen-Macaulification then yields a simple criterion for A to be Buchsbaum. Methods with several examples are illustrated.

Applications of dualizing complexes to Buchsbaum rings

Let (A, m, k) denote a local noetherian ring with d = dim A admitting a normalized dualizing complex D’. By well-known theorems on local duality A is Cohen-Macaulay ring if and only if the (-d)th place truncated dualizing complex r-,D’ is isomorphic to the zero complex in the derived category D(A). (See Section 1 for more precise statements.) In the following we consider a class of local rings for which the truncated dualizing complex t-,D’ is isomorphic to a complex of vector spaces over A/m = k. There have been several efforts to get results about local rings which generalize the Cohen-Macaulay property, see, for instance [7,13,14]. A connection to our consideration is given in the following sense: In 2.3 we prove one of our main, theorems: the complex t-,D’ of A is isomorphic to a complex of vector spaces over k if and only if A is Buchsbaum ring in the sense of [ 13, 141. They are those rings for which the difference L(A/d) e,(x; A) between length and multiplicity of a system of parameters x does ndt depend on x. See also 2.1 for a brief summary of essential facts about Buchsbaum rings and relation to a conjecture of D. A. Buchsbaum. In [7] M. Hochster and J. L. Roberts have studied purity of the Frobenius map and related things to show that the local cohomology modules of certain rings of such type behave nicely. Applying our criterion 3.1 to this situation, we can show in 4.1 that such rings have a truncated dualizing complex t-,D’ which is isomorphic to a complex of vector spaces over k. Following the examples given in [7] we get a number of Buchsbaum rings. For instance: Let Vc IP~ be a projective variety defined by an ideal which is generated by square-free monomials, then V is arithmetically Buchsbaum if and only if V is a Cohen-Macaulay variety. Our Theorem 2.3 is related to a result of R. Kiehl in [8]. Using the “surjective map” criterion of [ 141 he showed that the canonical module of A is Buchsbaum if t-,D’ is isomorphic to a complex of vector spaces over k. This yields in [8] that the local rings at the cusps of Hilbert modular groups are factorial Buchsbaum rings which are not Cohen-Macaulay. 61 OOOl-8708/82/040061-17$05.00/O

On rings with linear resolutions

Let R denote a positively graded noetherian ring over a field k, R = @EoRi with R, = k and m the irrelevant maximal ideal generated by forms of degree 1, or let R be the m-adic completion of such a ring. If we consider X as a homomorphic image of a polynomial ring, resp. of a power series ring over k with a homogeneous defining ideal, one may ask whether R has a (minimal) linear resolution as an S-module. Cohen-Macaulay rings R of maximal embedding dimension, e.g., have a linear resolution as Sally showed in [9]. She also proved formulas for the Hilbert series HR and the Poincari: series Pi, Pz. Schenzel [lo] later characterized CM-rings R with a linear resolution and called these rings “extremal.” He also gave formulas for HR and Pi. S. Goto (See (3.1) below) proved, besides other facts, the linearity of the resolution for Buchsbaum rings of maximal embedding dimension (i.e., edim R = dim R + e(R) + I(R) 1; I(R) denotes the invariant of the Buchsbaum ring R) and a formula for HH. In Section 1 we generalize these results and characterize rings with a linear resolution, which have a parameter system that forms a d-sequence. In the Buchsbaum case we then show that the H,, Pi and Pf of such rings are determined by the embedding dimension, the Krull dimension, the dimensions of the local cohomology modules as k-vector spaces and what we call the type of R. In Section 2 we determine all Buchsbaum rings R = k[ [R,]] of maximal embedding dimension and multiplicity 2. Sally proved in [8] that, for a local CM-ring (R, m) of maximal embedding dimension, the associated graded ring is also CM of maximal embedding dimension. Goto showed a similar result for Buchsbaum rings of

On the associated graded ring of a Buchsbaum ring

Mathematische NachrichtenVolume 107, Issue 1 p. 209-220 Article On the associated graded ring of a Buchsbaum ring Ngo Viet Trung, Ngo Viet Trung Viên Toán hoc–Viên Khoa hoc Nghia Dǒ, Tù Liém Hanoi/VietnamSearch for more papers by this author Ngo Viet Trung, Ngo Viet Trung Viên Toán hoc–Viên Khoa hoc Nghia Dǒ, Tù Liém Hanoi/VietnamSearch for more papers by this author First published: 1982 https://doi.org/10.1002/mana.19821070116Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume107, Issue11982Pages 209-220 RelatedInformation

On Buchsbaum rings obtained by gluing

Let A be a Noetherian local ring with maximal ideal m. In 1973 J. Barshay [1] showed that, if A is a Cohen-Macaulay ring, then so is the Rees algebra R(q) = ⊕n≧0qn for every parameter ideal q of A (cf. p. 93, Corollary). Recently the author and Y. Shimoda [5] have proved that the Rees algebra R(q) is a Cohen-Macaulay ring for every parameter ideal q of A if and only if(#) A is a Buchsbaum ring and for i ≠ 1, dim A.