scholarly journals On Fitting Ideals of Kahler Differential Module

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 413
Author(s):  
Nurbige Turan ◽  
Necati Olgun
Keyword(s):  
Characteristic Zero ◽  
Projective Dimension ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Differential Module ◽  
First Order ◽  
Fitting Ideal ◽  
Order Derivation

Let k be an algebraically closed field of characteristic zero, and R / I and S / J be algebras over k . Ω 1 ( R / I ) and Ω 1 ( S / J ) denote universal module of first order derivation over k . The main result of this paper asserts that the first nonzero Fitting ideal Ω 1 ( R / I ⊗ k S / J ) is an invertible ideal, if the first nonzero Fitting ideals Ω 1 ( R / I ) and Ω 1 ( S / J ) are invertible ideals. Then using this result, we conclude that the projective dimension of Ω 1 ( R / I ⊗ k S / J ) is less than or equal to one.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

Abelian Gradings on Upper Block Triangular Matrices

2012 ◽  
Vol 55 (1) ◽  
pp. 208-213 ◽  
Author(s):  
Angela Valenti ◽  
Mikhail Zaicev
Keyword(s):  
Abelian Group ◽  
Characteristic Zero ◽  
Finite Abelian Group ◽  
Triangular Matrix ◽  
Closed Field ◽  
Triangular Matrices ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Block Triangular Matrix

AbstractLet G be an arbitrary finite abelian group. We describe all possible G-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.

Differentially algebraic group chunks

1990 ◽  
Vol 55 (3) ◽  
pp. 1138-1142 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Algebraic Group ◽  
Characteristic Zero ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Zariski Topology ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Differential Fields ◽  
Algebraically Closed Fields

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

scholarly journals Corrections to “Seminormal rings and weakly normal varieties”

1987 ◽  
Vol 107 ◽  
pp. 147-157 ◽  
Author(s):  
Marie A. Vitulli
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Regular Function ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Regular Functions ◽  
Normal Varieties

In “Seminormal rings and weakly normal varieties” we introduced the notion of a c-regular function on an algebraic variety defined over an algebraically closed field of characteristic zero. Our intention was to describe those k-valued functions on a variety X that become regular functions when lifted to the normalization of X, but without any reference to the normalization in the definition. That is, we aspired to identify the c-regular functions on X with the regular functions on the weak normalization of X

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

Linear Operators Preserving Similarity Classes and Related Results

1994 ◽  
Vol 37 (3) ◽  
pp. 374-383 ◽  
Author(s):  
Chi-Kwong Li ◽  
Stephen Pierce
Keyword(s):  
Positive Integer ◽  
Characteristic Zero ◽  
Linear Operators ◽  
Partial Answer ◽  
Zariski Closure ◽  
Closed Field ◽  
Algebraically Closed Field

AbstractLet Mn be the algebra of n × n matrices over an algebraically closed field of characteristic zero. For A ∊ Mn, denote by the collection of all matrices in Mn that are similar to A. In this paper we characterize those invertible linear operators ϕ on Mn that satisfy , where for some given A1,..., Ak ∊ Mn and denotes the (Zariski) closure of S. Our theorem covers a result of Howard on linear operators mapping the set of matrices annihilated by a given polynomial into itself, and extends a result of Chan and Lim on linear operators commuting with the function f(x) = xk for a given positive integer k ≥ 2. The possibility of weakening the invertibility assumption in our theorem is considered, a partial answer to a conjecture of Howard is given, and some extensions of our result to arbitrary fields are discussed.

On the Existence of the Graded Exponent for Finite Dimensional ℤp-graded Algebras

2012 ◽  
Vol 55 (2) ◽  
pp. 271-284 ◽  
Author(s):  
Onofrio M. Di Vincenzo ◽  
Vincenzo Nardozza
Keyword(s):  
Characteristic Zero ◽  
Prime Order ◽  
Closed Field ◽  
Graded Algebras ◽  
Algebraically Closed Field ◽  
Finite Dimensional

AbstractLet F be an algebraically closed field of characteristic zero, and let A be an associative unitary F-algebra graded by a group of prime order. We prove that if A is finite dimensional then the graded exponent of A exists and is an integer.

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