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.

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.

scholarly journals New characterizations of approximately Gorenstein rings

1992 ◽  
Vol 34 (3) ◽  
pp. 361-363
Author(s):  
Johnny A. Johnson ◽  
Monty B. Taylor
Keyword(s):  
Commutative Algebra ◽  
Noetherian Ring ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Commutative Noetherian Ring ◽  
Gorenstein Rings ◽  
Inverse Systems ◽  
Metric Topology ◽  
Equivalent Conditions ◽  
Lattice Of Ideals

In [1], a condition was sought on a commutative Noetherian ring R such that whenever R satisfied the condition and R was cyclically pure in an R-algebra S, then R was necessarily pure in S. Such a condition was found and turned out not only to be a sufficient condition but also a necessary condition. Rings satisfying this condition were called approximately Gorenstein. In this paper, we give new equivalent conditions for a ring to be approximately Gorenstein. Our conditions involve a certain metric topology defined on the lattice of ideals of a commutative Noetherian ring as well as the concept of a principal system. The notion of principal systems was defined in [9] and was suggested by Macaulay's theory of inverse systems [7]. These ideas have proved to be useful in many research probems in commutative algebra.

scholarly journals Unique factorisation of normal elements in non-commutative rings

1989 ◽  
Vol 31 (1) ◽  
pp. 103-113 ◽  
Author(s):  
D. A. Jordan
Keyword(s):  
Prime Ideal ◽  
Commutative Algebra ◽  
Noetherian Ring ◽  
Commutative Rings ◽  
Noetherian Rings ◽  
Usual Sense ◽  
Commutative Noetherian Ring ◽  
Normal Element

In the literature there are several generalisations to non-commutative rings of the notion of a unique factorisation domain from commutative algebra. This paper follows in the spirit of [1, 3] and is set in the context of Noetherian rings. In [3], A. W. Chatters and the author denned a Noetherian UFR (unique factorisation ring) to be a prime Noetherian ring R in which every non-zero prime ideal contains a prime ideal generated by a non-zero normal element p, that is, by an element p such that pR = Rp. The class of Noetherian UFRs includes the Noetherian UFDs studied by Chatters in [1], while a commutative Noetherian ring is a UFR if and only if it is a UFD in the usual sense. For a Noetherian UFR R, the following are simple consequences of the definition:(i) every non-zero ideal of R contains a non-zero normal element;(ii) the set N(R) of non-zero normal elements of R is a unique factorisation monoid in the sense of [4, Chapter 3].

Cellular reductions of the Pommaret-Seiler resolution for Quasi-stable ideals

2021 ◽  
Vol 55 (3) ◽  
pp. 102-106
Author(s):  
Rodrigo Iglesias ◽  
Eduardo Sáenz de Cabezón
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Combinatorial Properties ◽  
Pommaret Bases ◽  
Involutive Bases

Involutive bases were introduced in [6] as a type of Gröbner bases with additional combinatorial properties. Pommaret bases are a particular kind of involutive bases with strong relations to commutative algebra and algebraic geometry[11, 12].

Primary decomposition of homogeneous ideal in idealization of a module

2018 ◽  
Vol 55 (3) ◽  
pp. 345-352
Author(s):  
Tran Nguyen An
Keyword(s):  
Noetherian Ring ◽  
Primary Decomposition ◽  
Associated Primes ◽  
Homogeneous Ideal ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Irreducible Decomposition

Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IM ⫅ N. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

scholarly journals On a class of dual Rickart modules

2020 ◽  
Vol 72 (7) ◽  
pp. 960-970
Author(s):  
R. Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Finite Set ◽  
Rickart Modules

UDC 512.5 Let R be a ring and let Ω R be the set of maximal right ideals of R . An R -module M is called an sd-Rickart module if for every nonzero endomorphism f of M , ℑ f is a fully invariant direct summand of M . We obtain a characterization for an arbitrary direct sum of sd-Rickart modules to be sd-Rickart. We also obtain a decomposition of an sd-Rickart R -module M , provided R is a commutative noetherian ring and A s s ( M ) ∩ Ω R is a finite set. In addition, we introduce and study ageneralization of sd-Rickart modules.

The Formalism of l-adic Sheaves

2019 ◽  
pp. 62-128
Author(s):  
Dennis Gaitsgory ◽  
Jacob Lurie
Keyword(s):  
Commutative Algebra ◽  
Mathematical Object ◽  
Product Formula ◽  
Homotopy Category ◽  
Triangulated Category

The ℓ-adic product formula discussed in Chapter 4 will need to make use of analogous structures, which are simply not visible at the level of the triangulated category Dℓ(X). This chapter attempts to remedy the situation by introducing a mathematical object Shvℓ (X), which refines the triangulated category Dℓ (X). This object is not itself a category but instead is an example of an ∞-category, which is referred to as the ∞-category of ℓ-adic sheaves on X. The triangulated category Dℓ (X) can be identified with the homotopy category of Shvℓ (X); in particular, the objects of Dℓ (X) and Shvℓ (X) are the same. However, there is a large difference between commutative algebra objects of Dℓ (X) and commutative algebra objects of the ∞-category Shvℓ (X). We can achieve (b') by viewing the complex B as a commutative algebra of the latter sort.

scholarly journals On Regular Sequences

1966 ◽  
Vol 27 (1) ◽  
pp. 355-356 ◽  
Author(s):  
J. Dieudonné
Keyword(s):  
Algebraic Geometry ◽  
Noetherian Ring ◽  
Regular Sequence ◽  
Local Rings ◽  
Regular Sequences

The concept of regular sequence of elements of a ring A (first introduced by Serre under the name of A-sequence [2]), has far-reaching uses in the theory of local rings and in algebraic geometry. It seems, however, that it loses much of its importance when A is not a noetherian ring, and in that case, it probably should be superseded by the concept of quasi-regular sequence [1].

Some Results on the Domination Number of a Zero-divisor Graph

2014 ◽  
Vol 57 (3) ◽  
pp. 573-578 ◽  
Author(s):  
Sima Kiani ◽  
Hamid Reza Maimani ◽  
Reza Nikandish
Keyword(s):  
Noetherian Ring ◽  
Zero Divisor ◽  
Domination Number ◽  
Total Domination ◽  
Ore Extension ◽  
Commutative Noetherian Ring ◽  
Zero Divisor Graph ◽  
Domination Numbers

AbstractIn this paper, we investigate the domination, total domination, and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring. Also, some relations between the domination numbers of Γ(R/I) and Γ1(R), and the domination numbers of Γ(R) and Γ(R[x, α, δ]), where R[x, α, δ] is the Ore extension of R, are studied.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

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