• Title, Summary, Keyword: local homology modules

Search Result 6, Processing Time 0.036 seconds

SOME FINITENESS RESULTS FOR CO-ASSOCIATED PRIMES OF GENERALIZED LOCAL HOMOLOGY MODULES AND APPLICATIONS

  • Do, Yen Ngoc;Nguyen, Tri Minh;Tran, Nam Tuan
    • Journal of the Korean Mathematical Society
    • /
    • v.57 no.5
    • /
    • pp.1061-1078
    • /
    • 2020
  • We prove some results about the finiteness of co-associated primes of generalized local homology modules inspired by a conjecture of Grothendieck and a question of Huneke. We also show some equivalent properties of minimax local homology modules. By duality, we get some properties of Herzog's generalized local cohomology modules.

ON THE κ-REGULAR SEQUENCES AND THE GENERALIZATION OF F-MODULES

  • Ahmadi-Amoli, Khadijeh;Sanaei, Navid
    • Journal of the Korean Mathematical Society
    • /
    • v.49 no.5
    • /
    • pp.1083-1096
    • /
    • 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.

COLOCALIZATION OF LOCAL HOMOLOGY MODULES

  • Rezaei, Shahram
    • Bulletin of the Korean Mathematical Society
    • /
    • v.57 no.1
    • /
    • pp.167-177
    • /
    • 2020
  • Let I be an ideal of Noetherian local ring (R, m) and M an artinian R-module. In this paper, we study colocalization of local homology modules. In fact we give Colocal-global Principle for the artinianness and minimaxness of local homology modules, which is a dual case of Local-global Principle for the finiteness of local cohomology modules. We define the representation dimension rI (M) of M and the artinianness dimension aI (M) of M relative to I by rI (M) = inf{i ∈ ℕ0 : HIi (M) is not representable}, and aI (M) = inf{i ∈ ℕ0 : HIi (M) is not artinian} and we will prove that i) aI (M) = rI (M) = inf{rIR𝖕 (𝖕M) : 𝖕 ∈ Spec(R)} ≥ inf{aIR𝖕 (𝖕M) : 𝖕 ∈ Spec(R)}, ii) inf{i ∈ ℕ0 : HIi (M) is not minimax} = inf{rIR𝖕 (𝖕M) : 𝖕 ∈ Spec(R) ∖ {𝔪}}. Also, we define the upper representation dimension RI (M) of M relative to I by RI (M) = sup{i ∈ ℕ0 : HIi (M) is not representable}, and we will show that i) sup{i ∈ ℕ0 : HIi (M) ≠ 0} = sup{i ∈ ℕ0 : HIi (M) is not artinian} = sup{RIR𝖕 (𝖕M) : 𝖕 ∈ Spec(R)}, ii) sup{i ∈ ℕ0 : HIi (M) is not finitely generated} = sup{i ∈ ℕ0 : HIi (M) is not minimax} = sup{RIR𝖕 (𝖕M) : 𝖕 ∈ Spec(R) ∖ {𝔪}}.

A NOTE ON THE LOCAL HOMOLOGY

  • Rasoulyar, S.
    • Bulletin of the Korean Mathematical Society
    • /
    • v.41 no.2
    • /
    • pp.387-391
    • /
    • 2004
  • Let A be Noetherian ring, a= (${\tau}_1..., \tau_n$ an ideal of A and $C_{A}$ be category of A-modules and A-homomorphisms. We show that the connected left sequences of covariant functors ${limH_i(K.(t^t,-))}_{i\geq0}$ and ${lim{{Tor^A}_i}(\frac{A}{a^f}-)}_{i\geq0}$ are isomorphic from $C_A$ to itself, where $\tau^t\;=\;{{\tau_^t}_1$, ㆍㆍㆍ${\tau^t}_n$.

TORSION THEORY, CO-COHEN-MACAULAY AND LOCAL HOMOLOGY

  • Bujan-Zadeh, Mohamad Hosin;Rasoulyar, S.
    • Bulletin of the Korean Mathematical Society
    • /
    • v.39 no.4
    • /
    • pp.577-587
    • /
    • 2002
  • Let A be a commutative ring and M an Artinian .A-module. Let $\sigma$ be a torsion radical functor and (T, F) it's corresponding partition of Spec(A) In [1] the concept of Cohen-Macauly modules was generalized . In this paper we shall define $\sigma$-co-Cohen-Macaulay (abbr. $\sigma$-co-CM). Indeed this is one of the aims of this paper, we obtain some satisfactory properties of such modules. An-other aim of this paper is to generalize the concept of cograde by using the left derived functor $U^{\alpha}$$_{I}$(-) of the $\alpha$-adic completion functor, where a is contained in Jacobson radical of A.A.