• Title, Summary, Keyword: homological algebra

Search Result 8, Processing Time 0.065 seconds

On Depth Formula and Tor Game (깊이의 식과 토르 게임에 대하여)

  • Choi Sangki
    • Journal for History of Mathematics
    • /
    • v.17 no.4
    • /
    • pp.37-44
    • /
    • 2004
  • Homological algebra has emerged and developed since 1950s. However, in 1890's Hilbert investigated the resolutions in his Syzygy Theorem which is a vital ingredient in homological algebra. In 1956 Serre has proved the finite global dimension of regular local rings. His result give a basic tool in homological algebra. This paper also deals with the depth formula that was raised by Auslander in 1961.

  • PDF

A GENERALIZATION OF HOMOLOGICAL ALGEBRA

  • Davvaz, B.;Shabani-Solt, H.
    • Journal of the Korean Mathematical Society
    • /
    • v.39 no.6
    • /
    • pp.881-898
    • /
    • 2002
  • Our aim in this paper is to introduce a generalization of some notions in homological algebra. We define the concepts of chain U-complex, U-homology, chain (U, U')-map, chain (U, U')-homotopy and $\mu$-functor. We also obtain some interesting results. We use these results to find a generalization of Lambek Lemma, Snake Lemma, Connecting Homomorphism and Exact Triangle.

THE KÜNNETH SPECTRAL SEQUENCE FOR COMPLEXES OF BANACH SPACES

  • Park, HeeSook
    • Journal of the Korean Mathematical Society
    • /
    • v.55 no.4
    • /
    • pp.809-832
    • /
    • 2018
  • In this paper, we form the basis of the abstract theory for constructing the $K{\ddot{u}}nneth$ spectral sequence for a complex of Banach spaces. As the category of Banach spaces is not abelian, several difficulties occur and hinder us from applying the usual method of homological algebra directly. The most notable facts are the image of a morphism of Banach spaces is not necessarily a Banach space, and also the closed summand of a Banach space need not be a topological direct summand. So, we consider some conditions and categorical terms that fit the category of Banach spaces to modify the familiar method of homological algebra.

SOME REMARKS FOR KÜNNETH FORMULA ON BOUNDED COHOMOLOGY

  • Park, HeeSook
    • Honam Mathematical Journal
    • /
    • v.37 no.1
    • /
    • pp.7-27
    • /
    • 2015
  • Kuneth formula is to compute (co)-homology of $A{\otimes}B$ for known (co)-homology of the complexes A and B. In the ordinary case, this is done by using elementary homological methods in an abelian category. However, when we consider the bounded cochain complex with values in $\mathbb{R}$ and its structure as a real Banach space, the techniques of homological algebra for constructing K$\ddot{u}$nneth type formulas on it are not effective. The most notable facts are the image of a morphism of Banach spaces is not necessarily closed, and also the closed summand of a Banach space need not be a topological direct summand. The main goal of this paper is to construct the theory of K$\ddot{u}$nneth type formula on bounded cohomology with real coefficients in the suitable category of Banach spaces with some restricted conditions.

A Note on the Fuzzy Linear Maps

  • Kim, Chang-Bum
    • Journal of Korean Institute of Intelligent Systems
    • /
    • v.21 no.4
    • /
    • pp.506-511
    • /
    • 2011
  • In this paper we investigate some situations in connection with two exact sequences of fuzzy linear maps. Also we obtain a generalization of the work [Theorem4] of Pan [5], and we study the analogies of The Four Lemma and The Five Lemma of homological algebra. Finally we obtain a special exact sequence.

COMMUTING POWERS AND EXTERIOR DEGREE OF FINITE GROUPS

  • Niroomand, Peyman;Rezaei, Rashid;Russo, Francesco G.
    • Journal of the Korean Mathematical Society
    • /
    • v.49 no.4
    • /
    • pp.855-865
    • /
    • 2012
  • Recently, we have introduced a group invariant, which is related to the number of elements $x$ and $y$ of a finite group $G$ such that $x{\wedge}y=1_{G{\wedge}G}$ in the exterior square $G{\wedge}G$ of $G$. This number gives restrictions on the Schur multiplier of $G$ and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form $h^m{\wedge}k$ of $H{\wedge}K$ such that $h^m{\wedge}k=1_{H{\wedge}K}$, where $m{\geq}1$ and $H$ and $K$ are arbitrary subgroups of $G$.