• Journal of the Korean Mathematical Society
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• v.60 no.6
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• pp.1337-1364
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• 2023

We describe the Gorenstein derived categories of Gorenstein rings via the homotopy categories of Gorenstein injective modules. We also introduce the concept of Gorenstein cosilting complexes and study its basic properties. This concept is generalized by cosilting complexes in relative homological methods. Furthermore, we investigate the existence of the relative version of the Bongartz's theorem and construct a Bongartz's complement for a Gorenstein precosilting complex.

• Journal of the Korean Mathematical Society
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• v.61 no.3
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• pp.603-620
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• 2024

Let 𝒱, 𝒲, 𝑦, 𝒳 be four classes of left R-modules. The notion of (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein R-complexes is introduced, and it is shown that under certain mild technical assumptions on 𝒱, 𝒲, 𝑦, 𝒳, an R-complex 𝑴 is (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein if and only if the module in each degree of 𝑴 is (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein and the total Hom complexs Hom_R(𝒀, 𝑴), Hom_R(𝑴, 𝑿) are exact for any ${\mathbf{Y}}\,{\in}\,{\tilde{\mathcal{Y}}}$ and any ${\mathbf{X}}\,{\in}\,{\tilde{\mathcal{X}}}$. Many known results are recovered, and some new cases are also naturally generated.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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• pp.497-505
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• 2021

Let R → S be a ring homomorphism. The relations of Gorenstein projective dimension with respect to a semidualizing module of homologically bounded complexes between U ⊗LR X and X are considered, where X is an R-complex and U is an S-complex. Some sufficient conditions are given under which the equality ${\mathcal{GP}}_{\tilde{C}}-pd_S(S{\otimes}{L \atop R}X)={\mathcal{GP}}_C-pd_R(X)$ holds. As an application it is shown that the Auslander-Buchsbaum formula holds for GC-projective dimension.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.479-494
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• 2017

We construct various examples of simply connected minimal complex surfaces of general type with $p_g=0$ and $K^2=1,2$ using ${\mathbb{Q}}$-Gorenstein smoothing method.

• Journal of the Korean Mathematical Society
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• v.50 no.3
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• pp.493-507
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• 2013

We construct a new family of simply connected minimal complex surfaces of general type with $p_g$ = 1, $q$ = 0, and $K^2$ = 3, 4, 5, 6, 8 using a $\mathbb{Q}$-Gorenstein smoothing theory.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1269-1274
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• 2010

As the sequel to our previous work [4], we construct a minimal complex surface of general type with $p_g=0$, $K^2=3$ and $H_1$ = $\mathbb{Z}/2\mathbb{Z}$ by using a rational blow-down surgery and $\mathbb{Q}$-Gorenstein smoothing the-ory.