𝐾-groups of rings of integers and 𝐾-groups of group rings

AbstractBott periodicity for the unitary and symplectic groups is fundamental to topologicalK-theory. Analogous to unitary topologicalK-theory, for algebraicK-groups with finite coefficients, similar results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraicK-groups for any ring implies periodicity for the hermitianK-groups, analogous to orthogonal and symplectic topologicalK-theory.The proofs use in an essential way higherKSC-theories, extending those of Anderson and Green. They also provide an upper bound for the higher hermitianK-groups in terms of higher algebraicK-groups.We also relate periodicity to étale hermitianK-groups by proving a hermitian version of Thomason's étale descent theorem. The results are illustrated in detail for local fields, rings of integers in number fields, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings.

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Commutative nil-clean and $\pi$-Regular Group Rings

Uzbek Mathematical Journal ◽

10.29229/uzmj.2019-3-4 ◽

2019 ◽

Vol 2019 (3) ◽

pp. 33-39

Author(s):

P.V. Danchev

Keyword(s):

Group Rings ◽

Regular Group

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A note on Lie nilpotent group rings

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700030409 ◽

1992 ◽

Vol 45 (3) ◽

pp. 503-506 ◽

Cited By ~ 16

Author(s):

R.K. Sharma ◽

Vikas Bist

Keyword(s):

Nilpotent Group ◽

Group Algebra ◽

Group Rings ◽

Group Of Units

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.

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On free products inside the unit group of integral group rings

Communications in Algebra ◽

10.1080/00927872.2021.1894167 ◽

2021 ◽

pp. 1-9

Author(s):

Feliks Rączka

Keyword(s):

Unit Group ◽

Group Rings ◽

Integral Group ◽

Free Products ◽

Integral Group Rings

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Subgroups induced by certain ideals of free group rings

Communications in Algebra ◽

10.1080/00927878308822978 ◽

1983 ◽

Vol 11 (22) ◽

pp. 2519-2525 ◽

Cited By ~ 5

Author(s):

Chander Kanta Gupta

Keyword(s):

Free Group ◽

Group Rings

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On *-clean non-commutative group rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498816501504 ◽

2016 ◽

Vol 15 (08) ◽

pp. 1650150 ◽

Cited By ~ 4

Author(s):

Hongdi Huang ◽

Yuanlin Li ◽

Gaohua Tang

Keyword(s):

Finite Field ◽

Local Ring ◽

Group Algebra ◽

Rational Numbers ◽

Semisimple Group ◽

Group Algebras ◽

Group Rings ◽

Dihedral Groups ◽

Commutative Local Ring ◽

Generalized Quaternion

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

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