Ideals with Trivial Conormal Bundle

1980 ◽  
Vol 32 (1) ◽  
pp. 210-218 ◽  
Author(s):  
A. V. Geramita ◽  
C. A. Weibel
Keyword(s):  
Polynomial Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Question Answering ◽  
General Theorem ◽  
Closed Field ◽  
Natural Question ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Conormal Bundle

Throughout this paper all rings considered will be commutative, noetherian with identity. If R is such a ring and M is a finitely generated R-module, we shall use v(M) to denote that non-negative integer with the property that M can be generated by v(M) elements but not by fewer.Since every ideal in a noetherian ring is finitely generated, it is a natural question to ask what v(I) is for a given ideal I. Hilbert's Nullstellensatz may be viewed as the first general theorem dealing with this question, answering it when I is a maximal ideal in a polynomial ring over an algebraically closed field.More recently, it has been noticed that the properties of an R-ideal I are intertwined with those of the R-module I/I2.

A note on the multiplicity in factor ring

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Eduard Boďa
Keyword(s):  
Polynomial Ring ◽  
Maximal Ideal ◽  
Short Note ◽  
Primary Ideal ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Factor Ring ◽  
Local Polynomial

AbstractLet (R,m) = k[x 1,..., x n](x 1,...,x n) be a local polynomial ring (k being an algebraically closed field), and Q:= (F 1,..., F r)R be a primary ideal in R with respect to a maximal ideal m ⊂ R. In this short note we give a formula for the multiplicity e 0 (QR/(F 1)R, R/(F 1)R).

scholarly journals On the length of the powers of systems of parameters in local ring

1990 ◽  
Vol 120 ◽  
pp. 77-88 ◽  
Author(s):  
Nguyen Tu Cuong
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Finitely Generated ◽  
Systems Of Parameters ◽  
System Of Parameters ◽  
The Ideal

Throughout this note, A denotes a commutative local Noetherian ring with maximal ideal m and M a finitely generated A-module with dim (M) = d. Let x1, …, xd be a system of parameters (s.o.p. for short) for M and I the ideal of A generated by x1, …, xd.

scholarly journals On the least degree of polynomials bounding above the differences between lengths and multiplicities of certain systems of parameters in local rings

1992 ◽  
Vol 125 ◽  
pp. 105-114 ◽  
Author(s):  
Nguyen Tu Cuong
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Local Rings ◽  
Finitely Generated ◽  
Parameter Ideal ◽  
Systems Of Parameters ◽  
The Difference

Let A be a commutative local Noetherian ring with the maximal ideal m and M a finitely generated A-module, d = dim M. It is well-known that the difference between the length and the multiplicity of a parameter ideal q of Mgives a lot of informations on the structure of the module M. For instance, M is a Cohen-Macaulay (CM for short) module if and only if IM(q) = 0 for some parameter ideal q or M is Buchsbaum module (see [S-V]) if and only if IM(q) is a constant for all parameter ideals q of M.

Constructing ω-stable structures: rank 2 fields

2000 ◽  
Vol 65 (1) ◽  
pp. 371-391 ◽  
Author(s):  
John T. Baldwin ◽  
Kitty Holland
Keyword(s):  
General Framework ◽  
Saturated Model ◽  
Closed Field ◽  
Morley Rank ◽  
Natural Numbers ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Fast Growing ◽  
Rank 2 ◽  
Stable Structures

AbstractWe provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one function μ from ‘primitive extensions’ to the natural numbers a theory Tμ of an expansion of an algebraically closed field which has Morley rank 2. Finally, we show that if μ is not finite-to-one the theory may not be ω-stable.

On the complement of a nef and big divisor on an algebraic variety

1996 ◽  
Vol 120 (3) ◽  
pp. 411-422 ◽  
Author(s):  
Francesco Russo
Keyword(s):  
Algebraic Variety ◽  
Cartier Divisor ◽  
Ample Divisor ◽  
Closed Field ◽  
Finitely Generated ◽  
Complete Variety ◽  
Algebraically Closed Field

Let X be an algebraic (complete) variety over a fixed algebraically closed field k. To every Cartier divisor D on X, we can associate the graded k-algebra . As is known, for a semi-ample divisor D, R(X, D) is a finitely generated k-algebra (see [21] or [9]), while this property is no longer true for arbitrary nef and big divisors (see [21]).

scholarly journals Polynomial rings over rings integral over their centres

1986 ◽  
Vol 34 (1) ◽  
pp. 11-23
Author(s):  
C. L. Wangneo
Keyword(s):  
Polynomial Ring ◽  
Noetherian Ring ◽  
Polynomial Rings ◽  
Finitely Generated

We prove in this paper that every finitely generated critical module over A[X] is compressible where A is a Noetherian ring integral over a subring of its centre. Here A[X] denotes the polynomial ring over A in a commuting indeterminate X.

scholarly journals GK DIMENSION AND LOCALLY NILPOTENT SKEW DERIVATIONS

2014 ◽  
Vol 57 (3) ◽  
pp. 555-567
Author(s):  
JEFFREY BERGEN ◽  
PIOTR GRZESZCZUK
Keyword(s):  
Closed Field ◽  
Finitely Generated ◽  
Natural Conditions ◽  
Skew Derivation ◽  
Algebraically Closed Field ◽  
Locally Nilpotent ◽  
Gk Dimension ◽  
Kirillov Dimension

AbstractLet A be a domain over an algebraically closed field with Gelfand–Kirillov dimension in the interval [2,3). We prove that if A has two locally nilpotent skew derivations satisfying some natural conditions, then A must be one of five algebras. All five algebras are Noetherian, finitely generated, and have Gelfand–Kirillov dimension equal to 2. We also obtain some results comparing the Gelfand–Kirillov dimension of an algebra to its subring of invariants under a locally nilpotent skew derivation.

scholarly journals Rings of differential operators on toric varieties

1994 ◽  
Vol 37 (1) ◽  
pp. 143-160 ◽  
Author(s):  
A. G. Jones
Keyword(s):  
Line Bundle ◽  
Toric Variety ◽  
Characteristic Zero ◽  
Differential Operators ◽  
Toric Varieties ◽  
Closed Field ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Rings Of Differential Operators

Let be a finite dimensional toric variety over an algebraically closed field of characteristic zero, k. Let be the sheaf of differential operators on . We show that the ring of global sections, is a finitely generated Noetherian k-algebra and that its generators can be explicitly found. We prove a similar result for the sheaf of differential operators with coefficients in a line bundle.

scholarly journals Unit Equations on Quaternions

2020 ◽  
Vol 71 (4) ◽  
pp. 1521-1534
Author(s):  
Yifeng Huang
Keyword(s):  
Classical Result ◽  
Quaternion Algebra ◽  
Multiplicative Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Forward Orbit ◽  
Unit Equations

Abstract A classical result about unit equations says that if Γ1 and Γ2 are finitely generated subgroups of ${\mathbb C}^\times$, then the equation x + y = 1 has only finitely many solutions with x ∈ Γ1 and y ∈ Γ2. We study a non-commutative analogue of the result, where $\Gamma_1,\Gamma_2$ are finitely generated subsemigroups of the multiplicative group of a quaternion algebra. We prove an analogous conclusion when both semigroups are generated by algebraic quaternions with norms greater than 1 and one of the semigroups is commutative. As an application in dynamics, we prove that if f and g are endomorphisms of a curve C of genus 1 over an algebraically closed field k, and deg( f), deg(g)≥ 2, then f and g have a common iterate if and only if some forward orbit of f on C(k) has infinite intersection with an orbit of g.

scholarly journals Bi-artinian noetherian rings

2001 ◽  
Vol 43 (1) ◽  
pp. 9-21
Author(s):  
E. A. Whelan
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Minimal Ideal ◽  
Chain Condition ◽  
Noetherian Rings ◽  
Composition Series ◽  
Finitely Generated ◽  
Analogous Statement ◽  
Descending Chain Condition ◽  
Unique Maximal Ideal

A noetherian ring R satisfies the descending chain condition on two-sided ideals (“is bi-artinian”) if and only if, for each prime P ∈ spec(R), R/P has a unique minimal ideal (necessarily idempotent and left-right essential in R/P). The analogous statement for merely right noetherian rings is false, although our proof does not use the full noetherian condition on both sides, requiring only that two-sided ideals be finitely generated on both sides and that R/Q be right Goldie for each Q ∈ spec(R). Examples exist, for each n∈ℕ and in all characteristics, of bi-artinian noetherian domains Dn with composition series of length 2n and with a unique maximal ideal of height n. Noetherian rings which satisfy the related E-restricted bi-d.c.c. do not, in general, satisfy the second layer condition (on either side), but do satisfy the Jacobson conjecture.

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