Reduction to multiplicity less thanp in ap-cyclic extension of a two dimensional regular local ring (p = characteristic of the residue field)

1964 ◽  
Vol 154 (1) ◽  
pp. 28-55 ◽  
Author(s):  
Shreeram Abhyankar
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Cyclic Extension ◽  
Two Dimensional ◽  
Regular Local Ring

scholarly journals The Core of a Module over a Two-Dimensional Regular Local Ring

1997 ◽  
Vol 189 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Radha Mohan
Keyword(s):  
Local Ring ◽  
Two Dimensional ◽  
Regular Local Ring ◽  
The Core

scholarly journals A formula for jumping numbers in a two-dimensional regular local ring

2018 ◽  
Vol 516 ◽  
pp. 437-470
Author(s):  
Eero Hyry ◽  
Tarmo Järvilehto
Keyword(s):  
Local Ring ◽  
Two Dimensional ◽  
Regular Local Ring

Homology, Hilbert functions, and the conservation of number

1971 ◽  
Vol 69 (1) ◽  
pp. 59-70
Author(s):  
D. G. Northcott
Keyword(s):  
Finite Number ◽  
Local Ring ◽  
Residue Field ◽  
Degree Zero ◽  
System Of Equations ◽  
Regular Local Ring ◽  
Algebraic Equations ◽  
Number Of Solutions ◽  
Homology Module ◽  
New System

1. Introduction. The principle of the Conservation of Number is concerned with the following situation. One starts with a system of algebraic equations having only a finite number of solutions and then applies a homomorphism whose domain contains the coefficients of those equations. This produces a new system. Let us suppose that the new system of equations also has only a finite number of solutions. The question then arises as to how the number of solutions before specialization compares with the number present afterwards. In a typical geometrical situation, one usually wishes to assert that the two systems have equally many solutions. However, it is easy to construct algebraic situations where the number changes,† and where the change is not to be explained away through the confluence of solutions or by their slipping off to infinity. At first sight this represents a breakdown of the conservation principle, but this principle has proved so useful in the past that one has a natural reluctance to discard it. The alternative is to attempt a reformulation and in (l) the present author gave such a reformulation for the case in which the specialization consisted in mapping a regular local ring on to its residue field. The modified theory requires that we take account of systems of equations which arise in connexion with the homology modules of a certain complex. The system associated with the homology module of degree zero is found to be the same as the one that arises in the naive theory, and usually this is the only one that makes a contribution. However, in cases where the number of solutions appears to change, the other systems become active and act in such a way that the balance is restored. For an amplification of these remarks we must refer the reader to (l). They are made here to indicate how the relevance of homological concepts first became clear in any detail. In the present paper these ideas are taken further, the principal gain being that it is no longer necessary to restrict the type of specialization to that which consists in mapping a regular local ring on to its residue field. Indeed one can use very general specializations provided that one transfers the homological requirement from the ring to the system of equations under consideration. In this way, one obtains a theory which is more general and, in some of its aspects, simpler as well.

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 $(S_{i})$.

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