• 대한수학회논문집
• /
• 제33권1호
• /
• pp.53-71
• /
• 2018

In this paper, we provide two different descriptions for a relative derived category with respect to a subcategory ${\mathcal{X}}$ of an abelian category ${\mathcal{A}}$. First, we construct an exact model structure on certain exact category which has as its homotopy category the relative derived category of ${\mathcal{A}}$. We also show that a relative derived category is equivalent to homotopy category of certain complexes. Moreover, we investigate the existence of certain recollements in such categories.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제28권2호
• /
• pp.187-198
• /
• 2021

This work presents an exposition of both the internal structure of derived category of an abelian category D^*(𝓐) and its contribution in solving problems, particularly in algebraic geometry. Calculation of some morphisms will be presented between objects in D^*(𝓐) as elements in appropriate cohomology groups along with their compositions with the help of Yoneda construction under the assumption that the homological dimension of D^*(𝓐) is greater than or equal to 2. These computational settings will then be considered under sheaf cohomological context with a particular case from projective geometry.

• 대한수학회보
• /
• 제54권5호
• /
• pp.1803-1825
• /
• 2017

Let X be a smooth scheme with an action of an algebraic group G. We establish an equivalence of two categories related to the corresponding moment map ${\mu}:T^{\ast}X{\rightarrow}g^{\ast}$ - the derived category of G-equivariant coherent sheaves on the derived fiber ${\mu}^{-1}(0)$ and the derived category of G-equivariant matrix factorizations on $T^{\ast}X{\times}g$ with potential given by ${\mu}$.

• Korean Journal of Mathematics
• /
• 제26권3호
• /
• pp.439-458
• /
• 2018

In this study, we interpret the notion of homotopy of morphisms in the category of crossed modules in a category C of groups with operations using the categorical equivalence between the categories of crossed modules and of internal categories in C. Further, we characterize the derivations of crossed modules in a category C and obtain new crossed modules using regular derivations of old one.

• 한국멀티미디어학회논문지
• /
• 제13권6호
• /
• pp.841-848
• /
• 2010

Conventional filters using email header and body information equally judge whether an incoming email is spam or not. However this is unrealistic in everyday life because each person has different criteria to judge what is spam or not. To resolve this problem, we consider user preference information as well as email category information derived from the email content. In this paper, we have developed a personalized anti-spam system using ontologies constructed from rules derived in a data mining process. The reason why traditional content-based filters are not applicable to the proposed experimental situation is described. In also, several experiments constructing classifiers to decide email category and comparing classification rule learners are performed. Especially, an ID3 decision tree algorithm improved the overall accuracy around 17% compared to a conventional SVM text miner on the decision of email category. Some discussions about the axioms generated from the experimental dataset are given too.

• 대한수학회지
• /
• 제57권5호
• /
• pp.1135-1165
• /
• 2020

Polishchuk-Zaslow explained the homological mirror symmetry between Fukaya category of symplectic torus and the derived category of coherent sheaves of elliptic curves via Lagrangian torus fibration. Recently, Cho-Hong-Lau found another proof of homological mirror symmetry using localized mirror functor, whose target category is given by graded matrix factorizations. We find an explicit relation between these two approaches.

• Korean Journal of Mathematics
• /
• 제27권1호
• /
• pp.175-192
• /
• 2019

(Multiplicative) Hom-Lie-Yamaguti superalgebras are defined as a ${\mathbb{Z}}_2$-graded generalization of Hom-Lie Yamaguti algebras and also as a twisted generalization of Lie-Yamaguti superalgebras. Hom-Lie-Yamaguti superalgebras generalize also Hom-Lie supertriple systems (and subsequently ternary multiplicative Hom-Nambu superalgebras) and Hom-Lie superalgebras in the same way as Lie-Yamaguti superalgebras generalize Lie supertriple systems and Lie superalgebras. Hom-Lie-Yamaguti superalgebras are obtained from Lie-Yamaguti superalgebras by twisting along superalgebra even endomorphisms. We show that the category of (multiplicative) Hom-Lie-Yamaguti superalgebras is closed under twisting by self-morphisms. Constructions of some examples of Hom-Lie-Yamaguti superalgebras are given. The notion of an nth derived (binary) Hom-superalgebras is extended to the one of an nth derived binary-ternary Hom-superalgebras and it is shown that the category of Hom-Lie-Yamaguti superalgebras is closed under the process of taking nth derived Hom-superalgebras.

• 대한수학회지
• /
• 제59권2호
• /
• pp.379-405
• /
• 2022

We introduce the notions of Koszul N-complex, Čech N-complex and telescope N-complex, explicit derived torsion and derived completion functors in the derived category D_N (R) of N-complexes using the Čech N-complex and the telescope N-complex. Moreover, we give an equivalence between the categories of cohomologically 𝖆-torsion N-complexes and cohomologically 𝖆-adic complete N-complexes, and prove that over a commutative Noetherian ring, via Koszul cohomology, via RHom cohomology (resp. ⊗ cohomology) and via local cohomology (resp. derived completion), all yield the same invariant.

• 대한수학회보
• /
• 제40권1호
• /
• pp.1-8
• /
• 2003

In this paper, we further develop the weak polygroup theory, we define quotient weak polygroup and then the fundamental homomorphism theorem of group theory is derived in the context of weak polygroups. Also, we consider the fundamental relation $\beta$$^{*}$ defined on a weak polygroup and define a functor from the category of all weak polygroups into the category of all fundamental groups.s.

• 대한수학회지
• /
• 제58권1호
• /
• pp.219-236
• /
• 2021

Let �� be a weighted projective line defined over the algebraic closure $k={\bar{\mathbb{F}}}_q$ of the finite field ��q and σ be a weight permutation of ��. By folding the category coh-�� of coherent sheaves on �� in terms of the Frobenius twist functor induced by σ, we obtain an ��q-category, denoted by coh-(��, σ; q). We then prove that coh-(��, σ; q) is derived equivalent to the valued canonical algebra associated with (��, σ).