scholarly journals Cohomology of groups and splendid equivalences of derived categories

2001 ◽  
Vol 131 (3) ◽  
pp. 459-472 ◽  
Author(s):  
ALEXANDER ZIMMERMANN
Keyword(s):  
Group Ring ◽  
Local Structure ◽  
Cohomology Ring ◽  
Derived Categories ◽  
Derived Category ◽  
Restriction Map ◽  
Group Rings ◽  
Cohomology Rings ◽  
The Impact ◽  
Cohomology Of Groups

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences TrPic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

scholarly journals The number of idempotents in abelian group rings

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 719-723
Author(s):  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Group Rings ◽  
Arbitrary Characteristic

Suppose that R is a commutative unitary ring of arbitrary characteristic and G is a multiplicative abelian group. Our main theorem completely determines the cardinality of the set id(RG), consisting of all idempotent elements in the group ring RG. It is explicitly calculated only in terms associated with R, G and their divisions. This result strengthens previous estimates obtained in the literature recently.

TRIVIAL TORSION UNITS IN G-ADAPTED GROUP RINGS

2006 ◽  
Vol 05 (06) ◽  
pp. 781-791
Author(s):  
ALLEN HERMAN ◽  
YUANLIN LI
Keyword(s):  
Group Ring ◽  
Torsion Group ◽  
Unit Group ◽  
Group Rings ◽  
Quaternion Group ◽  
Torsion Units ◽  
Equivalent Conditions

Let G be a torsion group and let R be a G-adapted ring. In this note we study the question of when the group ring RG has only trivial torsion units. It turns out that the above question is closely related to the question of when the quaternion group ring RQ8 has only trivial torsion units. We first give a ring-theoretic condition on R which determines exactly when the quaternion group ring has only trivial torsion units. Then several equivalent conditions for RG to have only trivial torsion units are provided. We also investigate the hypercenter of the unit group of a G-adapted group ring RG, and show that when R satisfies the torsion trivial involution condition, this hypercenter is not equal to the center if and only if G is a Q*-group.

Some Results on Semi-Perfect Group Rings

1974 ◽  
Vol 26 (1) ◽  
pp. 121-129 ◽  
Author(s):  
S. M. Woods
Keyword(s):  
Group Ring ◽  
Polynomial Ring ◽  
Strong Interaction ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Commutative Case ◽  
Perfect Group ◽  
Case Examples ◽  
Complete Answer ◽  
Necessary And Sufficient

The aim of this paper is to find necessary and sufficient conditions on a group G and a ring A for the group ring AG to be semi-perfect. A complete answer is given in the commutative case, in terms of the polynomial ring A[X] (Theorem 5.8). In the general case examples are given which indicate a very strong interaction between the properties of A and those of G. Partial answers to the question are given in Theorem 3.2, Proposition 4.2 and Corollary 4.3.

On Automorphisms of Complete Algebras And The Isomorphism Problem for Modular Group Rings

1990 ◽  
Vol 42 (3) ◽  
pp. 383-394 ◽  
Author(s):  
Frank Röhl
Keyword(s):  
Group Ring ◽  
Modular Group ◽  
Isomorphism Problem ◽  
Nilpotency Class ◽  
Affirmative Answer ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Analogous Question

In [5], Roggenkamp and Scott gave an affirmative answer to the isomorphism problem for integral group rings of finite p-groups G and H, i.e. to the question whether ZG ⥲ ZH implies G ⥲ H (in this case, G is said to be characterized by its integral group ring). Progress on the analogous question with Z replaced by the field Fp of p elements has been very little during the last couple of years; and the most far reaching result in this area in a certain sense - due to Passi and Sehgal, see [8] - may be compared to the integral case, where the group G is of nilpotency class 2.

Outer Partial Actions and Partial Skew Group Rings

2015 ◽  
Vol 67 (5) ◽  
pp. 1144-1160 ◽  
Author(s):  
Patrik Nystedt ◽  
Johan Öinert
Keyword(s):  
Abelian Group ◽  
Dynamical Systems ◽  
Group Ring ◽  
Group Rings ◽  
Partial Action ◽  
Unital Ring ◽  
Different Types ◽  
Outer Action ◽  
Skew Group Rings ◽  
Skew Group Ring

AbstractWe extend the classical notion of an outer action α of a group G on a unital ring A to the case when α is a partial action on ideals, all of which have local units. We show that if α is an outer partial action of an abelian group G, then its associated partial skew group ring A *α G is simple if and only if A is G-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

Functors on Finite Vector Spaces and Units in Abelian Group Rings

1986 ◽  
Vol 29 (1) ◽  
pp. 79-83 ◽  
Author(s):  
Klaus Hoechsmann
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Elementary Abelian Group ◽  
Vector Spaces ◽  
Group Rings ◽  
Cyclic Subgroups ◽  
Group Of Units ◽  
Finite Vector ◽  
Finite Vector Spaces

AbstractIf A is an elementary abelian group, let (A) denote the group of units, modulo torsion, of the group ring Z[A]. We study (A) by means of the compositewhere C and B run over all cyclic subgroups and factor-groups, respectively. This map, γ, is known to be injective; its index, too, is known. In this paper, we determine the rank of γ tensored (over Z);with various fields. Our main result depends only on the functoriality of

Units in Group Rings of the Infinite Dihedral Group

1991 ◽  
Vol 34 (1) ◽  
pp. 83-89 ◽  
Author(s):  
Maciej Mirowicz
Keyword(s):  
Group Ring ◽  
Integral Domain ◽  
Dihedral Group ◽  
Group Rings ◽  
Finitely Generated ◽  
Group Of Units ◽  
Group D

AbstractThis paper studies the group of units U(RD∞) of the group ring of the infinite dihedral group D∞ over a commutative integral domain R. The structures of U(Z2D∞) and U(Z3D∞) are determined, and it is shown that U(ZD∞) is not finitely generated.

Trivial Units in Group Rings

2000 ◽  
Vol 43 (1) ◽  
pp. 60-62 ◽  
Author(s):  
Daniel R. Farkas ◽  
Peter A. Linnell
Keyword(s):  
Recent Result ◽  
Group Ring ◽  
Finite Index ◽  
Integral Group Ring ◽  
Alternative Proof ◽  
Crystallographic Group ◽  
Group Rings ◽  
Integral Group ◽  
Arbitrary Group ◽  
Normalized Units

AbstractLet G be an arbitrary group and let U be a subgroup of the normalized units in ℤG. We show that if U contains G as a subgroup of finite index, then U = G. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

scholarly journals A note on thick subcategories of stable derived categories

2013 ◽  
Vol 212 ◽  
pp. 87-96
Author(s):  
Henning Krause ◽  
Greg Stevenson
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Complete Intersections ◽  
Line Bundles ◽  
Finitely Generated ◽  
Exact Category ◽  
Coherent Sheaves ◽  
Noetherian Scheme ◽  
Finitely Generated Modules ◽  
Local Complete Intersections

AbstractFor an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection, we classify thick subcategories of finitely generated modules over strict local complete intersections and produce generators for the category of coherent sheaves on a separated Noetherian scheme with an ample family of line bundles.

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