scholarly journals Homological Invariants of Local Rings

1963 ◽  
Vol 22 ◽  
pp. 219-227 ◽  
Author(s):  
Hiroshi Uehara
Keyword(s):  
Local Ring ◽  
Betti Numbers ◽  
Residue Field ◽  
Unit Element ◽  
Minimal System ◽  
Local Rings ◽  
Homology Algebra ◽  
Homological Invariants ◽  
Minimal System Of Generators ◽  
The Relationship

In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t1,t2,…, tn) be a minimal system of generators for M. By a complex R<T1. . ., Tp> we mean an R-algebra* obtained by the adjunction of the variables T1. . ., Tp of degree 1 which kill t1,…, tp. The main purpose of this paper is, among other things, to construct an R-algebra resolution of the field K, so that we can investigate the relationship between the homology algebra H (R < T1,…, Tn>) and the homological invariants of R such as the algebra TorR(K, K) and the Betti numbers Bp = dimk TorR(K, K) of the local ring R. The relationship was initially studied by Serre [5].

scholarly journals VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

2018 ◽  
Vol 61 (03) ◽  
pp. 705-725
Author(s):  
DIPANKAR GHOSH ◽  
TONY J. PUTHENPURAKAL
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Regular Sequence ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Syzygy Modules ◽  
Tor Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

AbstractLet R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

Reduction numbers and Hilbert polynomials of ideals in higher dimensional CohenMacaulay local rings

1992 ◽  
Vol 111 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Yinghwa Wu
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Primary Ideal ◽  
Local Rings ◽  
Hilbert Polynomials ◽  
Higher Dimensional ◽  
Reduction Number ◽  
Minimal Reduction ◽  
System Of Parameters

Throughout, (R, m) will denote a d-dimensional CohenMacaulay (CM for short) local ring having an infinite residue field and I an m-primary ideal in R. Recall that an ideal J I is said to be a reduction of I if Ir+1 = JIr for some r 0, and a reduction J of I is called a minimal reduction of I if J is generated by a system of parameters. The concepts of reduction and minimal reduction were first introduced by Northcott and Rees12. If J is a reduction of I, define the reduction number of I with respect to J, denoted by rj(I), to be min {r 0 Ir+1 = JIr}. The reduction number of I is defined as r(I) = min {rj(I)J is a minimal reduction of I}. The reduction number r(I) is said to be independent if r(I) = rj(I) for every minimal reduction J of I.

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

scholarly journals CONTRACTED, -FULL AND RELATED CLASSES OF IDEALS IN LOCAL RINGS

2013 ◽  
Vol 55 (3) ◽  
pp. 669-675
Author(s):  
DAVID E. RUSH
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Local Rings

AbstractThe class of $\mathfrak{m}$-full and four related classes of ideals in a local ring (R, $\mathfrak{m}$) are extended by replacing $\mathfrak{m}$ with other ideals and the resulting classes of ideals are compared. It is shown that contracted ideals are $\mathfrak{m}$-full in a local ring with infinite residue field.

scholarly journals On the relationship between depth and cohomological dimension

2015 ◽  
Vol 152 (4) ◽  
pp. 876-888 ◽  
Author(s):  
Hailong Dao ◽  
Shunsuke Takagi
Keyword(s):  
Finite Type ◽  
Local Ring ◽  
Characteristic Zero ◽  
Residue Field ◽  
Cohomological Dimension ◽  
Picard Group ◽  
Regular Local Ring ◽  
Sharp Bounds ◽  
The Relationship

Let $(S,\mathfrak{m})$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $\mathfrak{a}$ be an ideal of $S$. We prove that if $\text{depth}\,S/\mathfrak{a}\geqslant 3$, then the cohomological dimension $\text{cd}(S,\mathfrak{a})$ of $\mathfrak{a}$ is less than or equal to $n-3$. This settles a conjecture of Varbaro for such an $S$. We also show, under the assumption that $S$ has an algebraically closed residue field of characteristic zero, that if $\text{depth}\,S/\mathfrak{a}\geqslant 4$, then $\text{cd}(S,\mathfrak{a})\leqslant n-4$ if and only if the local Picard group of the completion $\widehat{S/\mathfrak{a}}$ is torsion. We give a number of applications, including a vanishing result on Lyubeznik’s numbers, and sharp bounds on the cohomological dimension of ideals whose quotients satisfy good depth conditions such as Serre’s conditions $(S_{i})$.

scholarly journals Some criteria for regular and Gorenstein local rings via syzygy modules

2019 ◽  
Vol 18 (05) ◽  
pp. 1950097
Author(s):  
Dipankar Ghosh
Keyword(s):  
Direct Summand ◽  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Local Rings ◽  
Canonical Module ◽  
Syzygy Module ◽  
Syzygy Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

Let [Formula: see text] be a Cohen–Macaulay local ring. We prove that the [Formula: see text]th syzygy module of a maximal Cohen–Macaulay [Formula: see text]-module cannot have a semidualizing direct summand for every [Formula: see text]. In particular, it follows that [Formula: see text] is Gorenstein if and only if some syzygy of a canonical module of [Formula: see text] has a nonzero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen–Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.

scholarly journals Connectedness and Lyubeznik Numbers

2018 ◽  
Vol 2019 (13) ◽  
pp. 4233-4259 ◽  
Author(s):  
Luis Núñez-Betancourt ◽  
Sandra Spiroff ◽  
Emily E Witt
Keyword(s):  
Projective Space ◽  
Local Ring ◽  
Projective Variety ◽  
Local Cohomology ◽  
Local Rings ◽  
Affine Cone ◽  
Numerical Invariants ◽  
Lyubeznik Numbers ◽  
The Relationship

Abstract We investigate the relationship between connectedness properties of spectra and the Lyubeznik numbers, numerical invariants defined via local cohomology. We prove that for complete equidimensional local rings, the Lyubeznik numbers characterize when connectedness dimension equals 1. More generally, these invariants determine a bound on connectedness dimension. Additionally, our methods imply that the Lyubeznik number $\lambda _{1,2}(A)$ of the local ring $A$ at the vertex of the affine cone over a projective variety is independent of the choice of its embedding into projective space.

scholarly journals Characterizations of regular local rings via syzygy modules of the residue field

2018 ◽  
Vol 10 (3) ◽  
pp. 327-337
Author(s):  
Dipankar Ghosh ◽  
Anjan Gupta ◽  
Tony J. Puthenpurakal
Keyword(s):  
Residue Field ◽  
Local Rings ◽  
Syzygy Modules ◽  
Regular Local Rings

Two-Dimensional Linear Groups Over Local Rings

1969 ◽  
Vol 21 ◽  
pp. 106-135 ◽  
Author(s):  
Norbert H. J. Lacroix
Keyword(s):  
Local Ring ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Local Rings ◽  
Two Dimensional ◽  
Normal Subgroups ◽  
Field Case

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

The Étale locus in complete local rings

2021 ◽  
pp. 49-62
Author(s):  
Raymond Heitmann
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Local Rings ◽  
Regular Local Ring ◽  
Content Type ◽  
Complete Local Ring

Let R R be a complete local ring and let Q Q be a prime ideal of R R . It is determined precisely which conditions on R R are equivalent to the existence of a complete unramified regular local ring A A and an element g ∈ A − Q g\in A-Q such that R R is a finite A A -module and A g ⟶ R g A_g\longrightarrow R_g is étale . A number of other properties of the possible embeddings A ⟶ R A\longrightarrow R are developed in the process including the determination of precisely which fields can be coefficient fields in the Cohen-Gabber Theorem.

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