• Communications of the Korean Mathematical Society
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• v.8 no.4
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• pp.679-687
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• 1993

• East Asian mathematical journal
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• v.9 no.1
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• pp.65-73
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• 1993

• Korean Journal of Mathematics
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• v.8 no.2
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• pp.173-180
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• 2000

Let $k$ be a real abelian field such that the conductor of every nontrivial character belonging to $k$ agrees with the conductor of $k$. Note that real quadratic fields satisfy this condition. For a prime $p$, let $k_{\infty}$ be the $\mathbb{Z}_p$-extension of $k$. The aim of this paper is to produce a set of generators of the Tate cohomology group $\hat{H}^{-1}$ of the circular units of $k_n$, the $nth$ layer of the $\mathbb{Z}_p$-extension of $k$, where $p$ is an odd prime. This result generalizes some earlier works which treated the case when $k$ is real quadratic field and used them to study ${\lambda}$-invariants of $k$.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.519-530
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• 2014

Let (R,m) be a Noetherian local ring and M a finitely generated R-module. For an integer s > -1, we say that M is Cohen-Macaulay in dimension > s if every system of parameters of M is an M-sequence in dimension > s introduced by Brodmann-Nhan [1]. In this paper, we give some characterizations for Cohen-Macaulay modules in dimension > s in terms of the Noetherian dimension of the local cohomology modules $H^i_m(M)$, the polynomial type of M introduced by Cuong [5] and the multiplicity e($\underline{x}$;M) of M with respect to a system of parameters $\underline{x}$.

• Journal of the Korean Mathematical Society
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• v.39 no.6
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• pp.963-973
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• 2002

We study torsion phenomena in the integral cohomology of finite if-spaces X through the Eilenberg-Moore spectral sequence converging to H＊($\Omega$X; Z$_{p}$). We also investigate how the difference between the Z$_{p}$-filtration length f$_{p}$(X) and the Z$_{p}$-cup length c$_{p}$(X) on a simply connected finite H-space X is reflected in the Eilenberg-Moore spectral sequence converging to H＊($\Omega$X;Z$_{p}$). Finally we get the following result: Let p be an odd prime and X an n-connected finite H-space with dim QH＊ (X;Z$_{p}$) $\leq$ m. Then H＊(X;Z) is p-torsion free if (equation omitted).tion omitted).

• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.1083-1096
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• 2012

For a given ideal I of a Noetherian ring R and an arbitrary integer ${\kappa}{\geq}-1$, we apply the concept of ${\kappa}$-regular sequences and the notion of ${\kappa}$-depth to give some results on modules called ${\kappa}$-Cohen Macaulay modules, which in local case, is exactly the ${\kappa}$-modules (as a generalization of f-modules). Meanwhile, we give an expression of local cohomology with respect to any ${\kappa}$-regular sequence in I, in a particular case. We prove that the dimension of homology modules of the Koszul complex with respect to any ${\kappa}$-regular sequence is at most ${\kappa}$. Therefore homology modules of the Koszul complex with respect to any filter regular sequence has finite length.

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.185-191
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• 2006

This paper observes that the induced homomorphisms on cohomology groups by a cyclic map are trivial. For a CW-complex X, we use the fact to obtain some conditions of X so that the n-th Gottlieb group $G_n(X)$ is trivial for an even positive integer n. As corollaries, for any positive integer m, we obtain $G_{2m}(S^{2m})\;=\;0\;and\;G_2(CP^m)\;=\;0$ which are due to D. H. Gottlieb and G. Lang respectively, where $S^{2m}$ is the 2m- dimensional sphere and $CP^m$ is the complex projective m-space. Moreover, we show that $G_4(HP^m)\;=\;0\;and\;G_8(II)\;=\;0,\;where\;HP^m$ is the quaternionic projective m-space for any positive integer m and II is the Cayley projective space.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1299-1307
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• 2020

We show that the Liechti-Strenner's example for the closed nonorientable surface in [13] minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial of the action induced on the first cohomology nonpositive. We also show that the Liechti-Strenner's example of orientation-reversing homeomorphism for the closed orientable surface in [13] minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial p(x) of the action induced on the first cohomology nonpositive or all but the first coefficient of p(x)(x ± 1)², p(x)(x² ± 1), or p(x)(x² ± x + 1) nonpositive.

• Journal of the Korean Mathematical Society
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• v.42 no.4
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• pp.621-634
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• 2005

Let$P{\rightarrow}M$ be an oriented $S^2-fiber$ bundle over a closed manifold M and let Q be its associated SO(3)-bundle, then we investigate the ring structure of the cohomology of the total space P by constructing the coupling form TA induced from an SO(3) connection A. We show that the cohomology ring of total space splits into those of the base space and the fiber space if and only if the Pontrjangin class $p_1(Q)\;{\in}\;H^4(M;\mathbb{Z})$ vanishes. We apply this result to the twistor spaces of 4-manifolds.

• Journal of the Korean Mathematical Society
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• v.50 no.3
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• pp.591-605
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• 2013

The underlying field is of characteristic $p$ > 2. In this paper, we first use the method of computing the homogeneous derivations to determine the first cohomology of the so-called odd contact Lie algebra with coefficients in the even part of the generalized Witt Lie superalgebra. In particular, we give a generating set for the Lie algebra under consideration. Finally, as an application, the derivation algebra and outer derivation algebra of the Lie algebra are completely determined.