scholarly journals EXISTENCE OF ALMOST SPLIT SEQUENCES VIA REGULAR SEQUENCES

2013 ◽  
Vol 88 (2) ◽  
pp. 218-231 ◽  
Author(s):  
HOSSEIN ESHRAGHI
Keyword(s):  
Local Ring ◽  
Endomorphism Ring ◽  
Krull Dimension ◽  
Inductive Argument ◽  
Split Sequence ◽  
Regular Sequences ◽  
Complete Local Ring ◽  
Almost Split Sequence ◽  
Almost Split Sequences

AbstractLet $(R, \mathfrak{m})$ be a Cohen–Macaulay complete local ring. We will apply an inductive argument to show that for every nonprojective locally projective maximal Cohen–Macaulay object $ \mathcal{X} $ of the morphism category of $R$ with local endomorphism ring, there exists an almost split sequence ending in $ \mathcal{X} $. Regular sequences are exploited to reduce the Krull dimension of $R$ on which the inductive argument is established. Moreover, the Auslander–Reiten translate of certain objects is described.

scholarly journals Hilbert-Kunz Multiplicity of Three-Dimensional Local Rings

2005 ◽  
Vol 177 ◽  
pp. 47-75 ◽  
Author(s):  
Kei-ichi Watanabe ◽  
Ken-ichi Yoshida
Keyword(s):  
Lower Bound ◽  
Local Ring ◽  
Three Dimensional ◽  
Krull Dimension ◽  
Local Rings ◽  
Mild Conditions ◽  
Quadric Hypersurface ◽  
Complete Local Ring

In this paper, we investigate the lower bound sHK(p, d) of Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension d containing a field of characteristic p > 0. Especially, we focus on three-dimensional local rings. In fact, as a main result, we will prove that sHK (p, 3) = 4/3 and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degenerate quadric hypersurface k[[X, Y, Z,W]]/(X2 + Y2 + Z2 + W2) under mild conditions.Furthermore, we pose a generalization of the main theorem to the case of dim A ≥ 4 as a conjecture, and show that it is also true in case dim A = 4 using the similar method as in the proof of the main theorem.

scholarly journals On the number of terms in the middle of almost split sequences over cycle-finite artin algebras

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
Piotr Malicki ◽  
José Peña ◽  
Andrzej Skowroński
Keyword(s):  
Module Category ◽  
Artin Algebra ◽  
Split Sequence ◽  
Almost Split Sequence ◽  
Artin Algebras ◽  
Almost Split Sequences

AbstractWe prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5.

Artinian local cohomology modules of cofinie modules

2020 ◽  
pp. 2250029
Author(s):  
Kamal Bahmanpour
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Proper Ideal ◽  
Dimension Formula ◽  
Complete Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let [Formula: see text] be a commutative Noetherian complete local ring and [Formula: see text] be a proper ideal of [Formula: see text]. Suppose that [Formula: see text] is a nonzero [Formula: see text]-cofinite [Formula: see text]-module of Krull dimension [Formula: see text]. In this paper, it shown that [Formula: see text] As an application of this result, it is shown that [Formula: see text], for each [Formula: see text] Also it shown that for each [Formula: see text] the submodule [Formula: see text] and [Formula: see text] of [Formula: see text] is [Formula: see text]-cofinite, [Formula: see text] and [Formula: see text] whenever the category of all [Formula: see text]-cofinite [Formula: see text]-modules is an Abelian subcategory of the category of all [Formula: see text]-modules. Also some applications of these results will be included.

Almost Split Sequences whose Middle Term has at most Two Indecomposable Summands

1979 ◽  
Vol 31 (5) ◽  
pp. 942-960 ◽  
Author(s):  
M. Auslander ◽  
R. Bautista ◽  
M. I. Platzeck ◽  
I. Reiten ◽  
S. O. Smalø
Keyword(s):  
Exact Sequence ◽  
Representation Theory ◽  
Finitely Generated ◽  
Artin Algebra ◽  
Middle Term ◽  
Split Sequence ◽  
Almost Split Sequence ◽  
Artin Algebras ◽  
Almost Split Sequences

Let Λ be an artin algebra, and denote by mod Λ the category of finitely generated Λ-modules. All modules we consider are finitely generated.We recall from [6] that a nonsplit exact sequence in mod A is said to be almost split if A and C are indecomposable, and given a map h: X → C which is not an isomorphism and with X indecomposable, there is some t: X → B such that gt = h.Almost split sequences have turned out to be useful in the study of representation theory of artin algebras. Given a nonprojective indecomposable Λ-module C (or an indecomposable noninjective Λ-module A), we know thatthere exists a unique almost split sequence [6, Proposition 4.3], [5, Section 3].

scholarly journals Irreducible homomorphisms for lattices over orders

1977 ◽  
Vol 17 (1) ◽  
pp. 109-124
Author(s):  
Joachim W. Schmidt
Keyword(s):  
Finite Group ◽  
Direct Summand ◽  
Group Ring ◽  
Middle Term ◽  
Split Sequence ◽  
Almost Split Sequence ◽  
Almost Split Sequences ◽  
A Finite Group

Let Λ be a complete R-order in the semi-simple K-algebra A. Then it has been shown that for each indecomposable Λ-lattice M which is not projective, there exists a unique almost split sequence 0 → N → E → M → 0. Here we study the middle term E and characterize those almost split sequences where E has a projective direct summand. In the case where Λ is the group-ring RG for a finite group G, we get information about the almost split sequences for the syzygies and apply our results in an example.

The Étale locus in complete local rings

2021 ◽  
pp. 49-62
Author(s):  
Raymond Heitmann
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Local Rings ◽  
Regular Local Ring ◽  
Content Type ◽  
Complete Local Ring

Let R R be a complete local ring and let Q Q be a prime ideal of R R . It is determined precisely which conditions on R R are equivalent to the existence of a complete unramified regular local ring A A and an element g ∈ A − Q g\in A-Q such that R R is a finite A A -module and A g ⟶ R g A_g\longrightarrow R_g is étale . A number of other properties of the possible embeddings A ⟶ R A\longrightarrow R are developed in the process including the determination of precisely which fields can be coefficient fields in the Cohen-Gabber Theorem.

scholarly journals Rank Element of a Projective Module

1965 ◽  
Vol 25 ◽  
pp. 113-120 ◽  
Author(s):  
Akira Hattori
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Local Ring ◽  
Projective Module ◽  
Projective Modules ◽  
Local Theorem ◽  
Complete Local Ring ◽  
A Finite Group

In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we shall illustrate by several examples. For a projective module P over the groupalgebra of a finite group G, the rank element of P is essentially the character of G in P. In § 2 we prove that under certain assumption two projective modules Pi and P2 over an algebra over a complete local ring o are isomorphic if and only if their rank elements are identical. This is a type of proposition asserting that two representations are equivalent if and only if their characters are identical, and in fact, when A is the groupalgebra, the above theorem may be considered as another formulation of Swan’s local theorem [9]).

scholarly journals Quasi-coefficient rings of a local ring

1977 ◽  
Vol 68 ◽  
pp. 123-130 ◽  
Author(s):  
Hideyuki Matsumura
Keyword(s):  
Local Ring ◽  
Structure Theorem ◽  
Local Rings ◽  
Complete Case ◽  
Complete Local Ring

In this note we will make a few observations on the structure of fields and local rings. The main point is to show that a weaker version of Cohen structure theorem for complete local rings holds for any (not necessarily complete) local ring. The consideration of non-complete case makes the meaning of Cohen’s theorem itself clearer. Moreover, quasi-coefficient fields (or rings) are handy when we consider derivations of a local ring.

A Complex of Modules and Its Applications to Local Cohomology and Extension Functors

2015 ◽  
Vol 117 (1) ◽  
pp. 150 ◽  
Author(s):  
Kamal Bahmanpour
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Residue Field ◽  
Regular Sequence ◽  
Injective Hull ◽  
Finitely Generated ◽  
Complete Local Ring ◽  
Dual Functor ◽  
Matlis Dual Functor

Let $(R,m)$ be a commutative Noetherian complete local ring and let $M$ be a non-zero Cohen-Macaulay $R$-module of dimension $n$. It is shown that, if $\operatorname{projdim}_R(M)<\infty$, then $\operatorname{injdim}_R(D(H^n_{\mathfrak{m}}(M)))<\infty$, and if $\operatorname{injdim}_R(M)<\infty$, then $\operatorname{projdim}_R(D(H^n_{\mathfrak{m}}(M)))<\infty$, where $D(-):= \operatorname{Hom}_{R}(-,E)$ denotes the Matlis dual functor and $E := E_R(R/\mathfrak{m})$ is the injective hull of the residue field $R/\mathfrak{m}$. Also, it is shown that if $(R,\mathfrak{m})$ is a Noetherian complete local ring, $M$ is a non-zero finitely generated $R$-module and $x_1,\ldots,x_k$, $(k\geq 1)$, is an $M$-regular sequence, then \[ D(H^k_{(x_1,\ldots,x_k)}(D(H^k_{(x_1,\ldots,x_k)}(M))))\simeq M. \] In particular, $\operatorname{Ann} H^k_{(x_1,\ldots,x_k)}(M)=\operatorname{Ann} M$. Moreover, it is shown that if $R$ is a Noetherian ring, $M$ is a finitely generated $R$-module and $x_1,\ldots,x_k$ is an $M$-regular sequence, then \[ \operatorname{Ext}^{k+1}_R(R/(x_1,\ldots,x_k),M)=0. \]

On Weakly Laskerian and Weakly Cofinite Modules

2012 ◽  
Vol 19 (04) ◽  
pp. 693-698
Author(s):  
Kazem Khashyarmanesh ◽  
M. Tamer Koşan ◽  
Serap Şahinkaya
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Cofinite Modules ◽  
Gorenstein Local Ring

Let R be a commutative Noetherian ring with non-zero identity, 𝔞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module [Formula: see text] is weakly Laskerian for all i < r. Then we prove that [Formula: see text] is also weakly Laskerian and so [Formula: see text] is finite. Moreover, we show that if s is a non-negative integer such that [Formula: see text] is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then [Formula: see text] is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of [Formula: see text] is weakly Laskerian?

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