Higher Class Groups of Locally Triangular Orders over Number Fields

Guo, Xuejun; Kuku, Aderemi O.

doi:10.1142/s1005386709000091

Cited by 2 publications

(13 citation statements)

References 8 publications

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“…The key idea behind these computation is that derived equivalences of rings preserve the

K

‐theory and

G

‐theory (see Theorem (5) or ). In the literature, there are many papers dealing with

K

‐groups

K_{n}

by exploiting excision, Mayer–Vietoris exact sequences or other related sequences (for example, see ), but there are few works using derived equivalences to calculate algebraic

K

‐groups. In the following, we will survey some results in this direction.…”

Section: Applicationsmentioning

confidence: 99%

“…In , the authors furthered the above result and considered the following matrix ring

S

: Let

I

be an ideal of a

Z_{p}

‐algebra

R

with identity, where

Z_{p}

is the

p

‐adic integers (or, equivalently,

0true Z_{p} = \underset{n}{\underset{\leftarrow}{falselim}} double-struckZ / p^{n} double-struckZ

), and define

S = (\begin{matrix} R & I^{t_{12}} & \dots & I^{t_{1 n}} \\ R & R & ⋱ & ⋮ \\ ⋮ & ⋮ & ⋱ & I^{t_{n - 1 n}} \\ R & R & \dots & R \end{matrix}),

where

t_{i j}

are positive integers. Assume that

S

is a ring and

R / I^{n}

is a finite ring for all

n ⩾ 1

.…”

Section: Applicationsmentioning

confidence: 99%

“…Assume that

S

is a ring and

R / I^{n}

is a finite ring for all

n ⩾ 1

. If both

_{R} I

and

I_{R}

are projective, it was proved in that the isomorphism of algebraic

K

‐groups holds:

K_{*} false(S false) false(1 / s false) ≃ K_{*} false(R false) false(1 / s false) \oplus false(n - 1 false) K_{*} false(R / I false) false(1 / s false),

where

s

is any rational integer such that

p

divides

s

, and where

G (1 / s)

denotes the group

G \otimes_{double-struckZ} Z false[\frac{1}{s} false]

for an abelian group

G

.…”

Section: Applicationsmentioning

confidence: 99%

“…In [46], the authors furthered the above result and considered the following matrix ring S: Let I be an ideal of a Z p -algebra R with identity, where Z p is the p-adic integers (or, equivalently, Z p = lim ← − n Z/p n Z), and define…”

Section: Algebraic K-theory Of Matrix Subringsmentioning

confidence: 99%

“…Assume that S is a ring and R/I n is a finite ring for all n 1. If both R I and I R are projective, it was proved in [46] that the isomorphism of algebraic K-groups holds:…”

Section: Algebraic K-theory Of Matrix Subringsmentioning

confidence: 99%

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Derived equivalences of algebras

2018

Bull. London Math. Soc.

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Derived categories and equivalences between them are the pièce de résistance of modern homological algebra. They are widely used in many branches of mathematics, especially in algebraic geometry and representation theory. In this note, we shall survey some recently developed construction methods of derived equivalences for algebras and rings, with applications to homological conjectures, such as Broué's abelian defect group conjecture and the finitistic dimension conjecture, and to computation of higher algebraic K‐groups of algebras and rings.

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“…The key idea behind these computation is that derived equivalences of rings preserve the

K

‐theory and

G

‐theory (see Theorem (5) or ). In the literature, there are many papers dealing with

K

‐groups

K_{n}

by exploiting excision, Mayer–Vietoris exact sequences or other related sequences (for example, see ), but there are few works using derived equivalences to calculate algebraic

K

‐groups. In the following, we will survey some results in this direction.…”

Section: Applicationsmentioning

confidence: 99%

“…In , the authors furthered the above result and considered the following matrix ring

S

: Let

I

be an ideal of a

Z_{p}

‐algebra

R

with identity, where

Z_{p}

is the

p

‐adic integers (or, equivalently,

0true Z_{p} = \underset{n}{\underset{\leftarrow}{falselim}} double-struckZ / p^{n} double-struckZ

), and define

S = (\begin{matrix} R & I^{t_{12}} & \dots & I^{t_{1 n}} \\ R & R & ⋱ & ⋮ \\ ⋮ & ⋮ & ⋱ & I^{t_{n - 1 n}} \\ R & R & \dots & R \end{matrix}),

where

t_{i j}

are positive integers. Assume that

S

is a ring and

R / I^{n}

is a finite ring for all

n ⩾ 1

.…”

Section: Applicationsmentioning

confidence: 99%

“…Assume that

S

is a ring and

R / I^{n}

is a finite ring for all

n ⩾ 1

. If both

_{R} I

and

I_{R}

are projective, it was proved in that the isomorphism of algebraic

K

‐groups holds:

K_{*} false(S false) false(1 / s false) ≃ K_{*} false(R false) false(1 / s false) \oplus false(n - 1 false) K_{*} false(R / I false) false(1 / s false),

where

s

is any rational integer such that

p

divides

s

, and where

G (1 / s)

denotes the group

G \otimes_{double-struckZ} Z false[\frac{1}{s} false]

for an abelian group

G

.…”

Section: Applicationsmentioning

confidence: 99%

Section: Algebraic K-theory Of Matrix Subringsmentioning

confidence: 99%

“…Assume that S is a ring and R/I n is a finite ring for all n 1. If both R I and I R are projective, it was proved in [46] that the isomorphism of algebraic K-groups holds:…”

Section: Algebraic K-theory Of Matrix Subringsmentioning

confidence: 99%

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Derived equivalences of algebras

2018

Bull. London Math. Soc.

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Higher algebraic K-groups and $${\mathcal D}$$ -split sequences

2012

Math. Z.

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In this paper, we use D-split sequences and derived equivalences to provide formulas for calculation of higher algebraic K-groups (or mod-p K-groups) of certain matrix subrings which cover tiled orders, rings related to chains of Glaz-Vasconcelos ideals, and some other classes of rings. In our results, we do not assume any homological requirements on rings and ideals under investigation, and therefore extend sharply many existing results of this type in the algebraic K-theory literature to a more general context.

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Higher Class Groups of Locally Triangular Orders over Number Fields

Abstract: In this paper, we study the K-theory of triangular rings. As an application, we show that for a locally triangular order Λ, the p-torsion in the higher class group Cl2n(Λ) can only occur for primes p which lie under the prime ideals ℘ of [Formula: see text], at which Λ is not maximal.

Cited by 2 publications

References 8 publications

Derived equivalences of algebras

Derived equivalences of algebras

Higher algebraic K-groups and $${\mathcal D}$$ -split sequences

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