scholarly journals A local-global principle for small triangulated categories

2015 ◽  
Vol 158 (3) ◽  
pp. 451-476 ◽  
Author(s):  
DAVE BENSON ◽  
SRIKANTH B. IYENGAR ◽  
HENNING KRAUSE
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Commutative Noetherian Ring ◽  
Global Principle ◽  
Compactly Generated

AbstractLocal cohomology functors are constructed for the category of cohomological functors on an essentially small triangulated category ⊺ equipped with an action of a commutative noetherian ring. This is used to establish a local-global principle and to develop a notion of stratification, for ⊺ and the cohomological functors on it, analogous to such concepts for compactly generated triangulated categories.

Remarks on the local-global principle for a subcategory consisting of extension modules

2019 ◽  
Vol 18 (12) ◽  
pp. 1950236
Author(s):  
Takeshi Yoshizawa
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Global Principles ◽  
Finitely Generated Modules ◽  
Cohomology Modules ◽  
Global Principle

Faltings presented the local-global principle for the finiteness dimension of local cohomology modules. This paper deals with the local-global principle for an extension subcategory over a commutative Noetherian ring. We prove that finitely generated modules satisfy the local-global principles for certain extension subcategories. Additionally, we provide a generalization of Faltings’ local-global principle, which also includes the local-global principles for the Artinianness and Minimaxness of local cohomology modules.

On Grothendieck's local duality theorem

1979 ◽  
Vol 85 (3) ◽  
pp. 431-437 ◽  
Author(s):  
M. H. Bijan-Zadeh ◽  
R. Y. Sharp
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Noetherian Ring ◽  
Duality Theorem ◽  
Great Emphasis ◽  
Triangulated Category ◽  
Elementary Approach ◽  
Commutative Noetherian Ring ◽  
Local Duality ◽  
Dualizing Complexes

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

On ideals preserving generalized local cohomology modules

2015 ◽  
Vol 15 (01) ◽  
pp. 1650019 ◽  
Author(s):  
Tsutomu Nakamura
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Let R be a commutative Noetherian ring, 𝔞 an ideal of R and M, N two finitely generated R-modules. Let t be a positive integer or ∞. We denote by Ωt the set of ideals 𝔠 such that [Formula: see text] for all i < t. First, we show that there exists the ideal 𝔟t which is the largest in Ωt and [Formula: see text]. Next, we prove that if 𝔡 is an ideal such that 𝔞 ⊆ 𝔡 ⊆ 𝔟t, then [Formula: see text] for all i < t.

On the Finiteness Dimension of Local Cohomology Modules

2014 ◽  
Vol 21 (03) ◽  
pp. 517-520 ◽  
Author(s):  
Hero Saremi ◽  
Amir Mafi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Negative Integer ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M a non-zero finitely generated R-module. Let t be a non-negative integer. In this paper, it is shown that [Formula: see text] for all i < t if and only if there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t. Moreover, we prove that [Formula: see text] for all i.

Cohomological Dimension of Generalized Local Cohomology Modules

2008 ◽  
Vol 15 (02) ◽  
pp. 303-308 ◽  
Author(s):  
Jafar Amjadi ◽  
Reza Naghipour
Keyword(s):  
Exact Sequence ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Cohomological Dimension ◽  
Algebraic Varieties ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

The study of the cohomological dimension of algebraic varieties has produced some interesting results and problems in local algebra. Let 𝔞 be an ideal of a commutative Noetherian ring R. For finitely generated R-modules M and N, the concept of cohomological dimension cd 𝔞(M, N) of M and N with respect to 𝔞 is introduced. If 0 → N' → N → N'' → 0 is an exact sequence of finitely generated R-modules, then it is shown that cd 𝔞(M, N) = max { cd 𝔞(M, N'), cd 𝔞(M, N'')} whenever proj dim M < ∞. Also, if L is a finitely generated R-module with Supp (N/Γ𝔞(N)) ⊆ Supp (L/Γ𝔞(L)), then it is proved that cd 𝔞(M, N) ≤ max { cd 𝔞(M,L), proj dim M}. Finally, as a consequence, a result of Brodmann is improved.

scholarly journals INJECTIVE OBJECTS IN TRIANGULATED CATEGORIES

2004 ◽  
Vol 03 (04) ◽  
pp. 367-389 ◽  
Author(s):  
GRIGORY GARKUSHA ◽  
MIKE PREST
Keyword(s):  
Compact Objects ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Graded Ring ◽  
Generating Set ◽  
Injective Modules ◽  
Compactly Generated

Given a compactly generated triangulated category [Formula: see text] and a generating set ℛ of compact objects, the class of ℛ-injective objects [Formula: see text] is introduced. If [Formula: see text] is compact and ℛ={ΣnX}n∈ℤ it is shown that there is a functor [Formula: see text] identifying the class [Formula: see text] of injective modules over the ℤ-graded ring [Formula: see text] with the class [Formula: see text]. Also, the Ziegler and Zariski spectra of [Formula: see text] and of S are discussed in the paper.

On the Endomorphism Rings of Local Cohomology Modules

2010 ◽  
Vol 53 (4) ◽  
pp. 667-673 ◽  
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Nonnegative Integer ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Ring Homomorphism ◽  
Endomorphism Rings ◽  
Commutative Noetherian Ring ◽  
Ring Of Fractions ◽  
Local Cohomology Modules ◽  
Canonical Ring ◽  
Cohomology Modules

AbstractLet R be a commutative Noetherian ring and a a proper ideal of R. We show that if n := gradeRa, then . We also prove that, for a nonnegative integer n such that = 0 for every i ≠ n, if for all i > 0 and z ∈ a, then is a homomorphic image of R, where Rz is the ring of fractions of R with respect to a multiplicatively closed subset ﹛z j | j ⩾ 0﹜ of R. Moreover, if HomR(Rz , R) = 0 for all z ∈ a, then is an isomorphism, where is the canonical ring homomorphism R → .

scholarly journals On the abelianization of derived categories and a negative solution to Rosický’s problem

2012 ◽  
Vol 149 (1) ◽  
pp. 125-147 ◽  
Author(s):  
Silvana Bazzoni ◽  
Jan Šťovíček
Keyword(s):  
Regular Cardinal ◽  
Large Family ◽  
Global Dimension ◽  
Derived Categories ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Negative Solution ◽  
Compactly Generated

AbstractWe prove for a large family of rings R that their λ-pure global dimension is greater than one for each infinite regular cardinal λ. This answers in the negative a problem posed by Rosický. The derived categories of such rings then do not satisfy, for any λ, the Adams λ-representability for morphisms. Equivalently, they are examples of well-generated triangulated categories whose λ-abelianization in the sense of Neeman is not a full functor for any λ. In particular, we show that given a compactly generated triangulated category, one may not be able to find a Rosický functor among the λ-abelianization functors.

scholarly journals Stable under specialization sets and cofiniteness

2019 ◽  
Vol 18 (01) ◽  
pp. 1950015 ◽  
Author(s):  
K. Divaani-Aazar ◽  
H. Faridian ◽  
M. Tousi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring

Let [Formula: see text] be a commutative noetherian ring, and [Formula: see text] a stable under specialization subset of [Formula: see text]. We introduce a notion of [Formula: see text]-cofiniteness and study its main properties. In the case [Formula: see text], or [Formula: see text], or [Formula: see text] is semilocal with [Formula: see text], we show that the category of [Formula: see text]-cofinite [Formula: see text]-modules is abelian. Also, in each of these cases, we prove that the local cohomology module [Formula: see text] is [Formula: see text]-cofinite for every homologically left-bounded [Formula: see text]-complex [Formula: see text] whose homology modules are finitely generated and every [Formula: see text].

Cofiniteness of local cohomology modules for a pair of ideals for small dimensions

2018 ◽  
Vol 17 (02) ◽  
pp. 1850020 ◽  
Author(s):  
Moharram Aghapournahr
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] and [Formula: see text] be two ideals of [Formula: see text] and [Formula: see text] be an [Formula: see text]-module (not necessary [Formula: see text]-torsion). In this paper among other things, it is shown that if dim [Formula: see text], then the [Formula: see text]-module [Formula: see text] is finitely generated, for all [Formula: see text], if and only if the [Formula: see text]-module [Formula: see text] is finitely generated, for [Formula: see text]. As a consequence, we prove that if [Formula: see text] is finitely generated and [Formula: see text] such that the [Formula: see text]-module [Formula: see text] is [Formula: see text] (or weakly Laskerian) for all [Formula: see text], then [Formula: see text] is [Formula: see text]-cofinite for all [Formula: see text] and for any [Formula: see text] (or minimax) submodule [Formula: see text] of [Formula: see text], the [Formula: see text]-modules [Formula: see text] and [Formula: see text] are finitely generated. Also it is shown that if dim [Formula: see text] (e.g. dim [Formula: see text]) for all [Formula: see text], then the local cohomology module [Formula: see text] is [Formula: see text]-cofinite for all [Formula: see text].

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