• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.701-718
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• 2019

In this paper we compute the U-projective resolution of kQ-modules where Q is quiver of type $A_n$ and ${\tilde{A}}_n$. The behavior of the sequence can be seen through its geometric representation.

• Honam Mathematical Journal
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• v.40 no.2
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• pp.355-366
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• 2018

Let R be a commutative ring with identity and let M be a finitely generated module over a Noetherian local ring R. Then it is well-known that M has a minimal projective resolution, which is unique up to isomorphisms of exact sequences. We provide a new proof of its uniqueness. Moreover, we deal with the cohomologies of (M, R/m).

• Journal of the Korean Mathematical Society
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• v.59 no.4
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• pp.733-756
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• 2022

In this paper, we introduce the notion of Gorenstein quasiresolving subcategories (denoted by 𝒢𝒬𝓡_𝒳 (𝓐)) in term of quasi-resolving subcategory 𝒳. We define a resolution dimension relative to the Gorenstein quasi-resolving categories 𝒢𝒬𝓡_𝒳 (𝓐). In addition, we study the stability of 𝒢𝒬𝓡_𝒳 (𝓐) and apply the obtained properties to special subcategories and in particular to modules categories. Finally, we use the restricted flat dimension of right B-module M to characterize the finitistic dimension of the endomorphism algebra B of a 𝒢𝒬_𝒳-projective A-module M.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.779-791
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• 2016

Transfer of homological properties under base change is a classical field of study. Let $R{\rightarrow}S$ be a ring homomorphism. The relations of Gorenstein projective (or Gorenstein injective) dimensions of unbounded complexes between $U{\otimes}^L_RX$(or $RHom_R(X,U)$) and X are considered, where X is an R-complex and U is an S-complex. In addition, some sufficient conditions are given under which the equalities $G-dim_S(U{\otimes}^L_RX)=G-dim_RX+pd_SU$ and $Gid_S(RHom_R(X,U))=G-dim_RX+id_SU$ hold.

• East Asian mathematical journal
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• v.40 no.3
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• pp.287-293
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• 2024

For a nondegenerate irreducible projective variety, it is a classical problem to describe its defining equations and the syzygies among them. In this paper, we precisely determine a minimal generating set and the minimal free resolution of defining ideals of some rational curves of maximal genus in ℙ³.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.35-54
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• 2007

Let $M_c$ = M(2, 0, c) be the moduli space of O(l)-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0\;and\;c_2=c$ on a K3 surface X, where O(1) is a generic ample line bundle on X. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a holomorphic symplectic structure on the smooth locus. In particular, $M_c$ has trivial canonical divisor. In [22], O'Grady asks if there is any symplectic desingularization of $M_{2n}$ for $n\geq3$. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq3$. This obviously implies that there is no symplectic desingularization.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1707-1714
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• 2016

Let $C{\subset}{\mathbb{P}}^r$ be a nondegenerate projective curve of degree d > r + 1 and of maximal regularity. Such curves are always contained in the threefold scroll S(0, 0, r - 2). Also some of such curves are even contained in a rational normal surface scroll. In this paper we study the minimal free resolution of the homogeneous coordinate ring of C in the case where $d{\leq}2r-2$ and C is contained in a rational normal surface scroll. Our main result provides all the graded Betti numbers of C explicitly.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.873-890
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• 2020

In this paper, for discrete groups G and K, we show that the cohomology of the complex of projective tensor product B*(G)⨶B*(K) is isomorphic to the bounded cohomology Ĥ*(G × K) of G × K, which is the cohomology of B*(G × K) as topological vector spaces, where B*(G) is a complex of bounded cochains of G with real coefficients ℝ. In fact, we construct an isomorphism between these two cohomology groups that carries the canonical seminorm in Ĥ*(G × K) to the seminorm in the cohomology of B*(G)⨶B*(K).

• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.977-986
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• 2015

Let G be a graph on the vertex set $V(G)=\{x_1,{\cdots},x_n\}$ with the edge set E(G), and let $R=K[x_1,{\cdots},x_n]$ be the polynomial ring over a field K. Two monomial ideals are associated to G, the edge ideal I(G) generated by all monomials $x_i,x_j$ with $\{x_i,x_j\}{\in}E(G)$, and the vertex cover ideal $I_G$ generated by monomials ${\prod}_{x_i{\in}C}{^{x_i}}$ for all minimal vertex covers C of G. A minimal vertex cover of G is a subset $C{\subset}V(G)$ such that each edge has at least one vertex in C and no proper subset of C has the same property. Indeed, the vertex cover ideal of G is the Alexander dual of the edge ideal of G. In this paper, for an unmixed bipartite graph G we consider the lattice of vertex covers $L_G$ and we explicitly describe the minimal free resolution of the ideal associated to $L_G$ which is exactly the vertex cover ideal of G. Then we compute depth, projective dimension, regularity and extremal Betti numbers of R/I(G) in terms of the associated lattice.

• Kyungpook Mathematical Journal
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• v.53 no.2
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• pp.149-174
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• 2013

In a previous paper, we described some Siegel modular threefolds which admit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties. Basic for this method is a paper of van Geemen and Nygaard. They study a variety $\mathcal{X}$ that is the complete intersection of four quadrics in $\mathbb{P}^7(\mathbb{C})$. This is biholomorphic equivalent to the Satake compactification of $\mathcal{H}_2/{\Gamma}^{\prime}$ for a certain subgroup ${\Gamma}^{\prime}{\subset}Sp(2,\mathbb{Z})$ and it will be the starting point of our investigation. It has been pointed out that a (projective) small resolution of this variety is a rigid Calabi-Yau manifold $\tilde{\mathcal{X}}$. Then we will consider the action of quotients of modular groups on $\mathcal{X}$ and study possible resolutions that admits a Calabi-Yau model in the category of complex spaces.