• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1077-1095
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• 2015

In this paper we study equivariant crossed modules in its link with strict graded categorical groups. The resulting Schreier theory for equivariant group extensions of the type of an equivariant crossed module generalizes both the theory of group extensions of the type of a crossed module and the one of equivariant group extensions.

• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.29-43
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• 2006

The article deals with the problem of extending representations of a closed normal subgroup H to a topological group G. We show that the standard technique using group cohomology to solve the problem in the case of finite groups can be generalized in the category of topological groups if G/H is discrete.

• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.167-179
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• 1997

In the study of a manifold M, the exterior algebra $\Lambda^* M$ plays an important role. In fact, the de Rham cohomology theory gives many informations of a manifold. Another important object in the study of a manifold is its Clifford algebra (Cl(M), generated by the tangent space.

• Journal of the Korean Mathematical Society
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• v.38 no.3
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• pp.623-631
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• 2001

Let $\kappa$ be a real abelian field of conductor f and $\kappa$(sub)$\infty$ = ∪(sub)n$\geq$0$\kappa$(sub)n be its Z(sub)p-extension for an odd prime p such that płf$\phi$(f). he aim of this paper is ot compute the cohomology groups of circular units. For m>n$\geq$0, let G(sub)m,n be the Galois group Gal($\kappa$(sub)m/$\kappa$(sub)n) and C(sub)m be the group of circular units of $\kappa$(sub)m. Let l be the number of prime ideals of $\kappa$ above p. Then, for mm>n$\geq$0, we have (1) C(sub)m(sup)G(sub)m,n = C(sub)n, (2) H(sup)i(G(sub)m,n, C(sub)m) = (Z/p(sup)m-n Z)(sup)l-1 if i is even, (3) H(sup)i(G(sub)m,n, C(sub)m) = (Z/P(sup)m-n Z)(sup l) if i is odd (※Equations, See Full-text).

• Korean Journal of Mathematics
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• v.5 no.1
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• pp.61-74
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• 1997

Let $d$ be a positive integer with $d\not{\equiv}2$ mod 4, and let $K=\mathbb{Q}({\zeta}_{pd})$ for S an odd prime $p$ such that $p{\equiv}1$ mod $d$. Let $K_{\infty}={\bigcup}_{n{\geq}0}K_n$ be the cyclotomic $\mathbb{Z}_p$-extension of $K=K_0$. In this paper, explicit generators for the Tate cohomology group $\hat{H}^{-1}$($G_{m,n}$ are given when $d=qr$ is a product of two distinct primes, where $G_{m,n}$ is the Galois group Gal($K_m/K_n$) and $C_m$ is the group of cyclotomic units of $K_m$. This generalizes earlier results when $d=q$ is a prime.

• East Asian mathematical journal
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• v.14 no.2
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• pp.371-375
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• 1998

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.669-681
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• 1998

In this paper, we prove that on the complete Kahler manifold, if ${\rho}(x){\geq}-\frac{1}{2}{\lambda}_0$ and either ${\rho}(x_0)>-\frac{1}{2}{lambda}_0$ at some point $x_0$ or Vol(M)=${\infty}$, then the Clifford $L^2$ cohomology group $L^2{\mathcal H}^{\ast}(M,S)$ is trivial, where $\rho(x)$ is the least eigenvalue of ${\mathcal R}_x + \bar{{\mathcal R}}(x)\;and\;{\lambda}_0$ is the infimum of the spectrum of the Laplacian acting on $L^2$-functions on M.

• Journal of the Korean Mathematical Society
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• v.57 no.5
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• pp.1061-1078
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• 2020

We prove some results about the finiteness of co-associated primes of generalized local homology modules inspired by a conjecture of Grothendieck and a question of Huneke. We also show some equivalent properties of minimax local homology modules. By duality, we get some properties of Herzog's generalized local cohomology modules.

• Honam Mathematical Journal
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• v.1 no.1
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• pp.43-49
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• 1979

G를 유한군으로, C를 모든 복소수체로 가정하고, V를 C상에서의 유한차원 벡터공간이라 하자. V상에서의 G의 사영적 표시는, X, $y{\epsilon}G$이고 ${\alpha}:\;G{\times}G{\rightarrow}C$를 Facto set이라 할 때 $T(x)T(y)=T(xy){\alpha}(x,y)$이 되는 함수 $T=\;G{\rightarrow}GL(V)$를 말한다. 본 논문의 목적은 군에 대한 Extension theory를 사용해서, G상의 factor set들의 동치류들은 G의 Second Cohomology group과 동형이라는 것을 증명하는 것이다.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.109-114
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• 1989

This paper deals with the classifying spaces of finite groups. To any finite group G we associate a space BG with the property that .pi.$_{1}$(BG)=G, .pi.$_{i}$ (BG)=0 for i>1. BG is called the classifying space of G. Consider the problem of finding a stable splitting BG= $X_{1}$$^{V}$ $X_{1}$$^{V}$..$^{V}$ $X_{n}$ localized at pp. Ideally the $X_{i}$ 's are indecomposable, thus displaying the homotopy type of BG in the simplest terms. Such a decomposition naturally splits $H^{*}$(BG). The main purpose of this paper is to give the classification theorem in stable homotopy theory for groups with periodic cohomology i.e. cyclic Sylow p-subgroups for p an odd prime and to calculate some universal stable element. In this paper, all cohomology groups are with Z/p-coefficients and p is an odd prime.prime.