scholarly journals Multiplicity and t-isomultiple ideals

1988 ◽  
Vol 110 ◽  
pp. 81-111 ◽  
Author(s):  
M.E. Rossi ◽  
G. Valla
Keyword(s):  
Local Ring ◽  
Tangent Cone ◽  
Minimal Degree ◽  
Local Version ◽  
Regular Local Ring ◽  
Geometric Result ◽  
Del Pezzo ◽  
Intersection Of Quadrics ◽  
Minimal Multiplicity

Let V be an irreducible non degenerate variety in Pn; a classical geometric result says that degree (V) ≥ codim V + 1 and, if equality holds, V is said to be of minimal degree. Varieties of minimal degree has been classified by Del Pezzo and Bertini and they all are intersections of quadrics. The local version of this result is due to J. Sally who proved that if is a regular local ring and is a Cohen-Macaulay local ring of minimal multiplicity, according to the bound e(R) ≥ height (I) + 1 given by Abhyankar, then the tangent cone of R is intersection of quadrics and it is Cohen-Macaulay.

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.

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.

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.

Ordinary Singularities of Algebraic Curves

1981 ◽  
Vol 24 (4) ◽  
pp. 423-431 ◽  
Author(s):  
Ferruccio Orecchia
Keyword(s):  
Singular Point ◽  
Vector Space ◽  
Local Ring ◽  
Maximal Ideal ◽  
Tangent Cone ◽  
Algebraic Curves ◽  
Vector Spaces ◽  
Generic Position

AbstractLet A be the local ring at a singular point p of an algebraic reduced curve. Let M (resp. Ml,..., Mh) be the maximal ideal of A (resp. of Ā). In this paper we want to classify ordinary singularities p with reduced tangent cone: Spec(G(A)). We prove that G(A) is reduced if and only if: p is an ordinary singularity, and the vector spaces span the vector space . If the points of the projectivized tangent cone Proj(G(A)) are in generic position then p is an ordinary singularity if and only if G(A) is reduced. We give an example which shows that the preceding equivalence is not true in general.

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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