scholarly journals Generating sequences and semigroups of valuations on 2 dimensional normal local rings

2018 ◽  
Author(s):  
◽  
Arpan Dutta
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Local Rings ◽  
Closed Field ◽  
Quotient Singularity ◽  
Two Dimensional ◽  
Invariant Ring ◽  
Rational Rank ◽  
Generating Sequence ◽  
Generating Sequences

In this thesis we develop a method for constructing generating sequences for valuations dominating the ring of a two dimensional quotient singularity. Suppose that K is an algebraically closed field of characteristic zero, K[X, Y] is a polynomial ring over K and v is a rational rank 1 valuation of the field K(X, Y) which dominates K[X, Y](X,Y) . Given a finite Abelian group H acting diagonally on K[X, Y], and a generating sequence of v in K[X, Y] whose members are eigenfunctions for the action of H, we compute a generating sequence for the invariant ring K[X, Y]H. We use this to compute the semigroup SK[X,Y ]H (v) of values of elements of K[X, Y]H. We further determine when SK[X,Y ]H (v) is a finitely generated SK[X,Y ]H (v)-module.

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.

scholarly journals On the Grothendieck group of a quotient singularity defined by a finite abelian group

1992 ◽  
Vol 149 (1) ◽  
pp. 122-138 ◽  
Author(s):  
Jürgen Herzog ◽  
Eduardo Marcos ◽  
Rolf Waldi
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Grothendieck Group ◽  
Quotient Singularity

Graded Involutions on Upper-triangular Matrix Algebras

2009 ◽  
Vol 16 (01) ◽  
pp. 103-108 ◽  
Author(s):  
A. Valenti ◽  
M. V. Zaicev
Keyword(s):  
Abelian Group ◽  
Characteristic Zero ◽  
Finite Abelian Group ◽  
Triangular Matrix ◽  
Closed Field ◽  
Triangular Matrices ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Upper Triangular Matrices ◽  
Upper Triangular Matrix

Let UTn be the algebra of n × n upper-triangular matrices over an algebraically closed field of characteristic zero. We describe all G-gradings on UTn by a finite abelian group G commuting with an involution (involution gradings).

On the classification of some two-dimensional Markov shifts with group structure

1992 ◽  
Vol 12 (4) ◽  
pp. 823-833 ◽  
Author(s):  
Mark A. Shereshevsky
Keyword(s):  
Abelian Group ◽  
Measure Space ◽  
Group Structure ◽  
Finite Abelian Group ◽  
Two Dimensional ◽  
Finite Abelian Groups ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Markov Shifts

AbstractFor a finite Abelian group G define the two-dimensional Markov shift for all . Let μG be the Haar measure on the subgroup . The group ℤ2 acts on the measure space (XG, MG) by shifts. We prove that if G1, and G2 are p-groups and E(G1,) ≠ E(G2), where E(G) is the least common multiple of the orders of the elements of G, then the shift actions on and are not measure-theoretically isomorphic. For any finite Abelian groups G1 and G2 the shift actions on and are topologically conjugate if and only if G1 and G2 are isomorphic.

Group Gradings on Associative Algebras with Involution

2008 ◽  
Vol 51 (2) ◽  
pp. 182-194 ◽  
Author(s):  
Y. A. Bahturin ◽  
A. Giambruno
Keyword(s):  
Abelian Group ◽  
Matrix Algebra ◽  
Finite Abelian Group ◽  
Closed Field ◽  
Base Field ◽  
Associative Algebras ◽  
Finite Dimensional ◽  
The Matrix ◽  
Simple Algebras ◽  
Group Gradings

AbstractIn this paper we describe the group gradings by a finite abelian group G of the matrix algebra Mn(F) over an algebraically closed field F of characteristic different from 2, which respect an involution (involution gradings). We also describe, under somewhat heavier restrictions on the base field, all G-gradings on all finite-dimensional involution simple algebras.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group

2021 ◽  
Author(s):  
Weidong Gao ◽  
Siao Hong ◽  
Wanzhen Hui ◽  
Xue Li ◽  
Qiuyu Yin ◽  
...  
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum

Stable cohomotopy and cobordism of abelian groups

1981 ◽  
Vol 90 (2) ◽  
pp. 273-278 ◽  
Author(s):  
C. T. Stretch
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Formal Group ◽  
Characteristic Class ◽  
Burnside Ring ◽  
Classifying Space ◽  
Formal Group Law ◽  
Stable Cohomotopy ◽  
Group Law

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

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