Every k-algebra a which is essentially of finite type over k and simple is a regular local ring

1982 ◽  
pp. 35-44
Author(s):  
Richard Sot
Keyword(s):  
Finite Type ◽  
Local Ring ◽  
Regular Local Ring

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 $(S_{i})$.

On Generalized Regular Local Ring

2015 ◽  
Vol 3 (1) ◽  
pp. 145-152
Author(s):  
Zubayda Ibraheem ◽  
Naeema Shereef
Keyword(s):  
Local Ring ◽  
Regular Local Ring

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

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.

Lyubeznik tables of linked ideals

2019 ◽  
Vol 19 (07) ◽  
pp. 2050138
Author(s):  
Parvaneh Nadi ◽  
Farhad Rahmati ◽  
Majid Eghbali
Keyword(s):  
Local Ring ◽  
Regular Local Ring ◽  
Dimension 2

In this paper, we examine the Lyubeznik tables of two linked ideals [Formula: see text] and [Formula: see text] of a complete regular local ring [Formula: see text] containing a field. More precisely, we prove that the Lyubeznik tables of two evenly linked ideals [Formula: see text] and [Formula: see text] are the same when [Formula: see text] and [Formula: see text] both satisfy one of the following properties: (1) canonically Cohen–Macaulay, (2) generalized Cohen–Macaulay and (3) Buchsbaum. Furthermore, we give some conditions for equality of Lyubeznik tables of two linked ideals of dimension 2.

scholarly journals Almost Gorenstein rings arising from fiber products

2020 ◽  
pp. 1-18
Author(s):  
Naoki Endo ◽  
Shiro Goto ◽  
Ryotaro Isobe
Keyword(s):  
Local Ring ◽  
The Other ◽  
Gorenstein Ring ◽  
Local Rings ◽  
Regular Local Ring ◽  
Gorenstein Rings ◽  
Fiber Product

Abstract The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times _T S$ of Cohen–Macaulay local rings R, S of the same dimension $d>0$ over a regular local ring T with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are R and S. In addition, the other generalizations of Gorenstein properties are also explored.

scholarly journals A Note on Gorenstein Rings of Embedding Codimension Three

1973 ◽  
Vol 50 ◽  
pp. 227-232 ◽  
Author(s):  
Junzo Watanabe
Keyword(s):  
Local Ring ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Regular Sequence ◽  
Complete Intersections ◽  
Gorenstein Ring ◽  
Regular Local Ring ◽  
Gorenstein Rings ◽  
Number Of Generators ◽  
Five Elements

Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.

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