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On Shifted Principles for Attached Primes of the Top Local Cohomology Modules
Abstract
Let be a Noetherian local ring, let I be an ideal of R, and let M be a finitely generated R-module with . In this paper, we establish shifted principles under localization and completion for attached primes of the top local cohomology module We characterize the catenarity, the weak going-up property, and the strong Lichstenbaum-Hartshorne vanishing property of the base ring R in terms of these shifted principles of the top local cohomology modules.
Let be a Noetherian local ring. This paper deals with the annihilator of Artinian local cohomology modules in the relation with the structure of the base ring R, for non negative integers i and finitely generated R-modules M. Firstly, the catenarity and the unmixedness of local rings are characterized via the compatibility of annihilator of top local cohomology modules under localization and completion, respectively. Secondly, some necessary and sufficient conditions for a local ring being a quotient of a Cohen-Macaulay local ring are given in term of the annihilator of all local cohomology modules under localization and completion.
Let (R,m) be a Noetherian local ring, I be an ideal of R and M be a finitely generated R-module such that HIt(M) is Artinian and I-cofinite, where t=cd(I,M). In this paper, we establish shifted principles under localization and completion for attached primes of the top local cohomology module HIt(M).
We study Noetherian local rings whose all formal fibers are of dimension zero. Universal catenarity and going-up property of the canonical map to the completion are considered. We present several characterizations of these rings, including a characterization of Weierstrass preparation type. A characterization of local rings with going up property by a strong form of Lichtenbaum–Hartshorne Theorem is obtained. As an application, we give an upper bound for dimension of formal fibers of a large class of algebras over these rings.
Let [fraktur a] be an ideal of a local ring (R, [fraktur m]) and let N be a finitely generated R-module
of dimension d: It is shown that Hd[fraktur a](N) [simeq R: similar, equals]
Hd[fraktur m](N)/[sum L: summation operator]n[set membership][open face N][left angle bracket][fraktur m][right angle bracket]
(0: Hd[fraktur m](N)[fraktur a]n); where for
an Artinian R-module X we put [left angle bracket][fraktur m][right angle bracket]X
= [cap B: intersection]n[set membership][open face N][fraktur m]nX. As a consequence several
vanishing and connectedness results are deduced.
For a multiplicative set S of a commutative ring R we define the co-localization functor HomR(Rs,). It is a functor on the category of R-modules to the category of Rs-modules. It is shown to be exact on the category of Artinian R-modules. While the co-localization of an Artinian module is almost never an Artinian Rs-module it inherits many good properties of A, e.g. it has a secondary representation. The construction is applied to the dual of a result of Bourbaki, a description of asymptotic prime divisors and the co-support of an Artinian module.
Let (R,\m) be a Noetherian local ring and M a finitely generated module with This paper is concerned with the following property for the top local cohomology module H^d_\m(M): \Ann (0:_{H^d_\m(M)}\p)=\p\ \text{for all prime ideals} \p\supseteq\Ann H^d_\m(M). In this paper we will show that this property is equivalent to the catenaricity of the unmixed support \Usupp M of M which is defined by \Usupp M=\Supp M/U_M(0), where is the largest submodule of M of dimension less than d. Some characterizations of this property in terms of system of parameters as well as the relation between the unmixed supports of M and of the \m-adic completion are given.
This second edition of a successful graduate text provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, including in multi-graded situations, and provides many illustrations of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Serre's Affineness Criterion, the Lichtenbaum–Hartshorne Vanishing Theorem, Grothendieck's Finiteness Theorem and Faltings' Annihilator Theorem, local duality and canonical modules, the Fulton–Hansen Connectedness Theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. The book is designed for graduate students who have some experience of basic commutative algebra and homological algebra and also experts in commutative algebra and algebraic geometry. Over 300 exercises are interspersed among the text; these range in difficulty from routine to challenging, and hints are provided for some of the more difficult ones.
Let (R, 𝔪) be a Noetherian local ring. Denote by N-dimRA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dimRA =dim (R/AnnRA) for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.
Let (R,m) be a Noetherian local ring and M a finitely generated R-module. It is well known that the local cohomology module Hmi(M) is Artinian for all i ≥ 0. Following I.G. Macdonald [8], denote by AttRHmi(M) the set of attached primes of Hmi(M). This paper is concerned with clarifying the structure of the base ring R via a relation between AttRHmi(M) and AttR̂Hmi(M). Some characterizations for R being universally catenary with all Cohen-Macaulay formal fibers are given.
Let (R,\m) be a Noetherian local ring and M a finitely generated
R-module. Following I. G. Macdonald \cite{Mac}, the set of all attached
primes of the Artinian local cohomology module H^i_{\m}(M) is denoted by
\Att_R(H^i_{\m}(M)). In \cite[Theorem 3.7]{Sh}, R. Y. Sharp proved that if
R is a quotient of a Gorenstein local ring then the shifted localization
principle always holds true, i.e. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \Att_{R_{\p}}\big(H^{i-\dim (R/\p)}_{\p R_{\p}}(M_{\p})\big)=\big\{\q
R_{\p}\mid \q\in\Att_RH^i_{\m}(M), \q\subseteq \p\big\} \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ (1) for any local cohomology modules H^i_{\m}(M) and any
\p\in\Spec (R). In this paper, we improve Sharp's result as follows: the
shifted localization principle always holds true if and only if R is
universally catenary and all its formal fibers are Cohen-Macaulay, if and only
if \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \displaystyle
\Att_{\R}(H^i_{\m}(M))=\bigcup_{\p\in\Att_R(H^i_{\m}(M))}\Ass_{\R}(\R/\p\R)\ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)
holds true for any finitely generated R-module M and any integer
This also improves the main result of the paper \cite{CN}.
Let
be a noetherian, local ring with completion
. We show that
satisfies the condition Going up if and only if there exists to every artinian R-module M with
a submodule
with
This is further equivalent to R being formal catenary, to α(R) = 0 and to
for all prime ideals
where
.
In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Matsumura covers the basic material, including dimension theory, depth, Cohen-Macaulay rings, Gorenstein rings, Krull rings and valuation rings. More advanced topics such as Ratliff's theorems on chains of prime ideals are also explored. The work is essentially self-contained, the only prerequisite being a sound knowledge of modern algebra, yet the reader is taken to the frontiers of the subject. Exercises are provided at the end of each section and solutions or hints to some of them are given at the end of the book.
This chapter describes the dimension of formal fibers of a local ring. If (A, m) is a noetherian local ring and A is its m-adic completion. The fibers of the morphism Spec(Â) → Spec(A) are the formal fibers of A. As  has very good properties and as  is flat over A, A has a good property if the formal fibers have related good properties. This is the philosophy of Grothendieck's theory of excellent rings. The fiber over the generic point (0) of Spec(A) is called the generic fiber.
Let (R,m) be a Noetherian local ring and I an ideal of R. Let M be a finitely generated R-module with dim. M=d. It is clear by Matlis duality that if R is complete then HId(M) satisfies the following property: AnnR(0:HId(M)p) = p for all prime ideals p ⊇ AnnRHId(M). However, HId(M) does not satisfy the property (*) in general. In this paper we characterize the property (*) of HId(M) in order to study the catenarity of the ring R/AnnRHId(M), the set of attached primes AttRHId(M), the co-support CosR(HId(M)), and the multiplicity of HId(M). We also show that if HId(M) satisfies the property (*) then HId(M)≅Hmd(M/N) for some submodule N of M.
and Eilenberg [2] formalized the techniques of cohomology into the abstract the- ory of homological algebra, including projective and injective resolutions, derived functors, and Ext and Tor. Auslander and Buchsbaum [1] introduced the notion of regular sequence and depth of a module (which they called codimension) and proved the fundamental result for a,nitely generated module over a regular lo- cal ring, that its homological dimension plus its depth (the maximum length of a regular sequence) is equal to the dimension of the local ring. On the other hand, the work of Kodaira and Spencer showed the usefulness of cohomology in the study of complex manifolds, and Serre in his famous paper FAC [6] showed that these same techniques could be extended to a theory of cohomology of coherent sheaves in abstract algebraic geometry. Grothendieck’s extension of this theory to arbitrary schemes further strengthened the ties between algebra and geometry by making any ring A a subject for study in algebraic geometry via its
Let (R, m) be a Noetherian local ring and M a finitely generated R-module. Let i ≥ 0 be an integer. Consider the following property for the Artinian local cohomology module Hmi (M)(*)AnnR (0 :Hmi (M) p) = p for all p ∈ Var (AnnR (Hmi (M))) . In this paper, we study the property (*) of Hmi (M) in order to investigate the universal catenaricity of the ring R / AnnR M and the unmixedness of the ring R / p for certain p in SuppM. We also characterize the property (*) for Hmi (M) and obtain the associativity formulae for multiplicity of Hmi (M) in case where Hmi (M) satisfies the property (*).
Let T be a complete local ring and C a finite set of incomparable prime ideals of T. We find necessary and sufficient conditions for T to be the completion of an integral domain whose generic formal fiber is semilocal with maximal ideals the elements of C. In addition, if the characteristic of T is zero, we give necessary and sufficient conditions for T to be the completion of an excellent integral domain whose generic formal fiber is semilocal with maximal ideals the elements of C.
We propose a definition of co-support for modules over commutative, Noetherian rings that we hope will relate to the local homology functors of Greenlees and May in the same way ordinary support relates to local cohomology. By expressing local homology in terms of the Koszul complex, we prove some vanishing theorems involving this co-support. We also investigate the co-localization functor which gives rise to our definition of co-support. This functor treats Artiniantype constructions the same way the ordinary localization functor treats Noetherian-type constructions, and this duality extends to one between co-support and ordinary support.
On the dimension and multiplicity of local cohomology modules
- M Brodmannm
- R Y Sharp
Test ideals in local rings
- K E Smith
- KE Smith
Local Rings. Interscience
- M Nagata