The almost split sequence starting with a simple module

1994 ◽  
Vol 62 (3) ◽  
pp. 203-206 ◽  
Author(s):  
Sheila Brenner
Keyword(s):  
Simple Module ◽  
Split Sequence ◽  
Almost Split Sequence

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.

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.

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 Laurent phenomenon and simple modules of quiver Hecke algebras

2019 ◽  
Vol 155 (12) ◽  
pp. 2263-2295 ◽  
Author(s):  
Masaki Kashiwara ◽  
Myungho Kim
Keyword(s):  
Simple Module ◽  
Hecke Algebras ◽  
Duke Math ◽  
Cluster Algebras ◽  
Coordinate Ring ◽  
Simple Modules ◽  
Laurent Phenomenon ◽  
Cluster Variable ◽  
Quantum Cluster

In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$, then $[S]$ is a cluster monomial in $[\mathscr{C}]$. If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$, then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$.

scholarly journals Picard–Vessiot extensions of artinian simple module algebras

2005 ◽  
Vol 285 (2) ◽  
pp. 743-767 ◽  
Author(s):  
Katsutoshi Amano ◽  
Akira Masuoka
Keyword(s):  
Simple Module ◽  
Module Algebras

scholarly journals FAME (v1.0): a simple module to simulate the effect of planktonic foraminifer species-specific habitat on their oxygen isotopic content

2018 ◽  
Vol 11 (9) ◽  
pp. 3587-3603 ◽  
Author(s):  
Didier M. Roche ◽  
Claire Waelbroeck ◽  
Brett Metcalfe ◽  
Thibaut Caley
Keyword(s):  
Late Holocene ◽  
Simple Module ◽  
Climatic Conditions ◽  
Model Data ◽  
Planktonic Foraminifer ◽  
Planktonic Foraminifers ◽  
Oxygen Isotopic ◽  
Species Specific ◽  
Almost All ◽  
Multiproxy Approach

Abstract. The oxygen-18 to oxygen-16 ratio recorded in fossil planktonic foraminifer shells has been used for over 50 years in many geoscience applications. However, different planktonic foraminifer species generally yield distinct signals, as a consequence of their specific living habitats in the water column and along the year. This complexity is usually not taken into account in model–data integration studies. To overcome this shortcoming, we developed the Foraminifers As Modeled Entities (FAME) module. The module predicts the presence or absence of commonly used planktonic foraminifers and their oxygen-18 values. It is only forced by hydrographic data and uses a very limited number of parameters, almost all derived from culture experiments. FAME performance is evaluated using the Multiproxy Approach for the Reconstruction of the Glacial Ocean surface (MARGO) Late Holocene planktonic foraminifer calcite oxygen-18 and abundance datasets. The application of FAME to a simple cooling scenario demonstrates its utility to predict changes in planktonic foraminifer oxygen-18 to oxygen-16 ratio in response to changing climatic conditions.

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