• Honam Mathematical Journal
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• v.32 no.2
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• pp.227-237
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• 2010

For a discrete group G, the kernel of a homomorphism from bounded cohomology $\hat{H}^*(G)$ of G to the ordinary cohomology $H^*(G)$ of G is called the singular part of $\hat{H}^*(G)$. We give some results on the space of the singular part of the second bounded cohomology of G. Also some results on the second bounded cohomology of a uniformly perfect group are given.

• Korean Journal of Mathematics
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• v.21 no.3
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• pp.331-344
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• 2013

In the present article, we describe specific elements in a motivic cohomology group $H^1_{\mathcal{M}}(Spec\mathbb{Q}({\zeta}_l),\;\mathbb{Z}(2))$ of cyclotomic fields, which generate a subgroup of finite index for an odd prime $l$. As $H^1_{\mathcal{M}}(Spec\mathbb{Q}({\zeta}_l),\;\mathbb{Z}(1))$ is identified with the group of units in the ring of integers in $\mathbb{Q}({\zeta}_l)$ and cyclotomic units generate a subgroup of finite index, these elements play similar roles in the motivic cohomology group.

• Honam Mathematical Journal
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• v.41 no.3
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• pp.631-650
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• 2019

In this paper, for discrete groups G and K, we show that the bounded cohomology group of $G{\times}K$ is isomorphic to the cohomology group of the complex of the projective tensor product $B^*(G){\hat{\otimes}}B^*(K)$, where $B^*(G)$ and $B^*(G)$ are the complexes of bounded cochains with real coefficients ${\mathbb{R}}$ of G and K, respectively.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.151-167
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• 2007

If k is a subfield of $\mathbb{Q}(\varepsilon_m)$ then the cohomology group $H^2(k(\varepsilon_n)/k)$ is isomorphic to $H^2(k(\varepsilon_{n'})/k)$ with gcd(m, n') = 1. This enables us to reduce a cyclotomic k-algebra over $k(\varepsilon_n)$ to the one over $k(\varepsilon_{n'})$. A radical extension in projective Schur algebra theory is regarded as an analog of cyclotomic extension in Schur algebra theory. We will study a reduction of cohomology group of radical extension and show that a Galois cohomology group of a radical extension is isomorphic to that of a certain subextension of radical extension. We then draw a cohomological characterization of radical group.

• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.391-405
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• 1996

In this article, we calculate the cohomology groups of flat vector bundles on some manifolds.

• Honam Mathematical Journal
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• v.42 no.1
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• pp.105-121
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• 2020

The main result of this article is to factorize the first (σ, τ)-cohomology group of triangular Banach algebra 𝓣 of order three with coefficients in 𝓣 -bimodule 𝓧 to the first (σ, τ)-cohomology groups of Banach algbras 𝓐, 𝓑 and 𝓒, where σ, τ are continuous homomorphisms on 𝓣. As a direct consequence, we find necessary and sufficient conditions for 𝓣 to be (σ, τ)-weakly amenable.

• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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• pp.1725-1741
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• 2017

For a complex smooth projective surface M with an action of a finite cyclic group G we give a uniform proof of the isomorphism between the invariant $H^1(G,\;H^2(M,\;{\mathbb{Z}}))$ and the first cohomology of the divisors fixed by the action, using G-equivariant cohomology. This generalizes the main result of Bogomolov and Prokhorov [4].

• East Asian mathematical journal
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• v.17 no.2
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• pp.339-348
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• 2001

• Journal of the Chungcheong Mathematical Society
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• v.30 no.1
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• pp.77-102
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• 2017

As an analogy of $Poincar{\acute{e}}$ series in the space of modular forms, T. Ono associated a module $M_c/P_c$ for ${\gamma}=[c]{\in}H^1(G,R^{\times})$ where finite group G is acting on a ring R. $M_c/P_c$ is regarded as the 0-dimensional twisted Tate cohomology ${\hat{H}}^0(G,R^+)_{\gamma}$. In the case that G is the Galois group of a Galois extension K of a number field k and R is the ring of integers of K, the vanishing properties of $M_c/P_c$ are related to the ramification of K/k. We generalize this to arbitrary n-dimensional twisted cohomology of the ring of integers and obtain the extended version of theorems. Moreover, some explicit examples on quadratic and biquadratic number fields are given.

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.299-309
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• 2010

The Weyl group of a compact connected Lie group is a reflection group. If such Lie groups are locally isomorphic, the representations of the Weyl groups are rationally equivalent. They need not however be equivalent as integral representations. Turning to the invariant theory, the rational cohomology of a classifying space is a ring of invariants, which is a polynomial ring. In the modular case, we will ask if rings of invariants are polynomial algebras, and if each of them can be realized as the mod p cohomology of a space, particularly for dihedral groups.