scholarly journals Finite determinacy of matrices over local rings. Tangent modules to the miniversal deformation for R-linear group actions

2019 ◽  
Vol 223 (3) ◽  
pp. 1288-1321
Author(s):  
Genrich Belitskii ◽  
Dmitry Kerner
Keyword(s):  
Linear Group ◽  
Group Actions ◽  
Local Rings ◽  
Finite Determinacy

Two-Dimensional Linear Groups Over Local Rings

1969 ◽  
Vol 21 ◽  
pp. 106-135 ◽  
Author(s):  
Norbert H. J. Lacroix
Keyword(s):  
Local Ring ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Local Rings ◽  
Two Dimensional ◽  
Normal Subgroups ◽  
Field Case

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

ON INCIDENCE MODULO IDEAL RINGS

2007 ◽  
Vol 06 (04) ◽  
pp. 553-586 ◽  
Author(s):  
M. A. DOKUCHAEV ◽  
V. V. KIRICHENKO ◽  
B. V. NOVIKOV ◽  
A. P. PETRAVCHUK
Keyword(s):  
Linear Group ◽  
Partially Ordered Set ◽  
General Linear Group ◽  
Associative Ring ◽  
Ordered Set ◽  
General Linear ◽  
Local Rings ◽  
Partially Ordered ◽  
Finite Partially Ordered Set

For a given associative ring B, a two-sided ideal J ⊂ B and a finite partially ordered set P, we study the ring A = I(P, B, J) of incidence modulo J matrices determined by P. The properties of A involving its radical and quiver are investigated, and the interaction of A with serial rings is explored. The category of A-modules is studied if P is linearly ordered. Applications to the general linear group over some local rings are given.

A note on the existence of nontrivial homomorphism En(R) → En−1(R)

2017 ◽  
Vol 16 (01) ◽  
pp. 1750010
Author(s):  
Hong You
Keyword(s):  
Group Actions ◽  
Negative Answer ◽  
Group Homomorphism ◽  
Matrix Group ◽  
Local Rings ◽  
Division Rings ◽  
Nontrivial Group ◽  
Low Dimensional

We show that there is no nontrivial group homomorphism [Formula: see text] over commutative local rings and division rings for [Formula: see text], respectively. It gives a negative answer to Ye’s problem (see [S. K. Ye, Low-dimensional representations of matrix group actions on CAT(0) spaces and manifolds, J. Algebra 409 (2014) 219–243]) for the above rings.

scholarly journals The invariants of orthogonal group actions

1993 ◽  
Vol 48 (2) ◽  
pp. 313-319 ◽  
Author(s):  
Li Chiang ◽  
Yu-Ching Hung
Keyword(s):  
Quadratic Form ◽  
Finite Field ◽  
Rational Function ◽  
Linear Group ◽  
Function Field ◽  
Orthogonal Group ◽  
Group Actions ◽  
Rational Function Field ◽  
Transcendental Extension

Let Fq be the finite field of order q, an odd number, Q a non-degenerate quadratic form on , O(n, Q) the orthogonal group defined by Q. Regard O(n, Q) as a linear group of Fq -automorphisms acting linearly on the rational function field Fq(x1, …, xn). We shall prove that the invariant subfield Fq(x1,…, xn)O(n, Q) is a purely transcendental extension over Fq for n = 5 by giving a set of generators for it.

OVERGROUPS OF SYMPLECTIC GROUP IN LINEAR GROUP OVER LOCAL RINGS

2001 ◽  
Vol 29 (6) ◽  
pp. 2313-2318 ◽  
Author(s):  
Hong You ◽  
Baodong Zheng
Keyword(s):  
Linear Group ◽  
Symplectic Group ◽  
Local Rings

scholarly journals The invariants of projective linear group actions

1989 ◽  
Vol 39 (1) ◽  
pp. 107-117 ◽  
Author(s):  
Huah Chu ◽  
Ming-Chang Kang ◽  
Eng-Tjioe Tan
Keyword(s):  
Rational Function ◽  
Linear Group ◽  
Function Field ◽  
Group Actions ◽  
Direct Computation ◽  
Rational Function Field ◽  
Projective Linear Group

Let Fq be the field with q elements and let G = PGLn(Fq) or PSLn(Fq) act on Fq(x1,…,xn−1), the rational function field of n − 1 variables. Then Fq(x1,…,xn−1)G is purely transcendental over Fq. In fact, a set of n − 1 generators of Fq(x1,…xn−1)G, over Fq is exhibited. The case n = 2 is treated by direct computation.

ON INTEGRAL DOMAINS WITH CYCLIC GROUP ACTIONS

2011 ◽  
Vol 10 (03) ◽  
pp. 491-508
Author(s):  
JUNRO SATO ◽  
SUSUMU ODA ◽  
KEN-ICHI YOSHIDA
Keyword(s):  
New York ◽  
Cyclic Group ◽  
Galois Extension ◽  
Integral Domain ◽  
Group Actions ◽  
Local Rings ◽  
Quotient Field ◽  
Galois Extensions ◽  
Integral Domains ◽  
Field L

Let A be a commutative integral domain with quotient field L, and let R be a subdomain of A with quotient field K. Assuming that L is a Galois extension of K, Nagata required the condition for R to be normal when A is called a Galois extension of R (see p. 31, M. Nagata, Local Rings (Wiley, New York, 1962)). However in this paper, A is considered in the case that R is not necessarily assumed to be normal. We introduce the notion of cyclic Galois extensions of integral domains and investigate several properties of such ring extensions. In particular, we completely determine the seminormalization [Formula: see text] of A in an overdomain B such that both A ⊆B are cyclic Galois extensions of R.

Sign in / Sign up

Export Citation Format

Share Document