scholarly journals An equivalence induced by Ext and Tor applied to the finitistic weak dimension of coherent rings

1998 ◽  
Vol 40 (2) ◽  
pp. 167-176 ◽  
Author(s):  
Elwood Wilkins
Keyword(s):  
Representation Theory ◽  
Model Theory ◽  
Arbitrary Ring ◽  
Essential Component ◽  
Functor Categories ◽  
Weak Dimension ◽  
The 1960S ◽  
Almost Split Sequences ◽  
Coherent Rings

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.

Quinus ab Omni Nævo Vindicatus1

1997 ◽  
Vol 23 ◽  
pp. 25-65 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Possible Worlds ◽  
Modal Logics ◽  
Singular Terms ◽  
The 1960S

Today there appears to be a widespread impression that W. V. Quine's notorious critique of modal logic, based on certain ideas about reference, has been successfully answered. As one writer put it some years ago: “His objections have been dead for a while, even though they have not yet been completely buried.” What is supposed to have killed off the critique? Some would cite the development of a new ‘possible-worlds’ model theory for modal logics in the 1960s; others, the development of new ‘direct’ theories of reference for names in the 1970s.These developments do suggest that Quine's unfriendliness towards any formal logics but the classical and indifference towards theories of reference for any singular terms but variables were unfortunate.

scholarly journals From subcategories to the entire module categories

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Rasool Hafezi
Keyword(s):  
Representation Theory ◽  
Triangular Matrix ◽  
Projective Modules ◽  
Artin Algebra ◽  
Finite Representation ◽  
Triangular Matrix Algebra ◽  
Gorenstein Projective Modules ◽  
Category Of Modules ◽  
Almost Split Sequences ◽  
Auslander Algebra

AbstractIn this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

scholarly journals Anaphora

Linguistics ◽  
2011 ◽  
Author(s):  
Eric Reuland ◽  
Martin Everaert ◽  
Anna Volkova
Keyword(s):  
Language Acquisition ◽  
Representation Theory ◽  
Computational Linguistics ◽  
Language Processing ◽  
Discourse Representation ◽  
Linguistic Typology ◽  
Discourse Representation Theory ◽  
Subsequent Reference ◽  
The 1960S ◽  
Centering Theory

Anaphora can be generally defined as “subsequent reference to an entity already introduced in discourse” (Safir 2004a; see Representing Anaphoric Dependencies). The study of anaphora spans various fields of linguistics, from formal syntax and semantics to linguistic typology and pragmatics, and from computational linguistics to language processing and language acquisition. A major divide in this field is that between intrasentential anaphora—more specifically, binding relations—and intersentential, or discourse, anaphora. The former attracted attention in the 1960s and is one of the central topics in generative syntax and semantics, but also in current typological studies. The latter has been studied extensively since the early 1990s within computational linguistics, discourse representation theory, and functional approaches such as centering theory.

The Inductive Step of the Second Brauer-Thrall Conjecture

1980 ◽  
Vol 32 (2) ◽  
pp. 342-349 ◽  
Author(s):  
Sverre O. Smalø
Keyword(s):  
Representation Theory ◽  
Finitely Generated ◽  
Inductive Step ◽  
Artin Algebra ◽  
Indecomposable Modules ◽  
Infinite Type ◽  
Categorical Methods ◽  
Artin Algebras ◽  
Almost Split Sequences ◽  
Finitely Generated Modules

In this paper we are going to use a result of H. Harada and Y. Sai concerning composition of nonisomorphisms between indecomposable modules and the theory of almost split sequences introduced in the representation theory of Artin algebras by M. Auslander and I. Reiten to obtain the inductive step in the second Brauer-Thrall conjecture.Section 1 is devoted to giving the necessary background in the theory of almost split sequences.As an application we get the first Brauer-Thrall conjecture for Artin algebras. This conjecture says that there is no bound on the length of the finitely generated indecomposable modules over an Artin algebra of infinite type, i.e., an Artin algebra such that there are infinitely many nonisomorphic indecomposable finitely generated modules. This result was first proved by A. V. Roiter [8] and later in general for Artin rings by M. Auslander [2] using categorical methods.

Introduction

2020 ◽  
pp. 1-8
Author(s):  
Sarah Farmer
Keyword(s):  
Economic Growth ◽  
World War Ii ◽  
Social Class ◽  
World War ◽  
Modern Life ◽  
Essential Component ◽  
Post World War Ii ◽  
Economic Boom ◽  
The 1960S ◽  
Rural France

In post World War II France, the postwar drive to modernize and explosive economic growth caused the collapse of the peasantry as a social class. Peasants left the countryside en masse, villages emptied out, and fields that had been cultivated for centuries were left fallow. And yet, this book argues, rural France did not vanish in the sweeping transformations of the 1950s and 1960s. Paradoxically, postwar modernization made the French yearn for imaginative and tangible connections to the life that peasants had once lived. This, in turn, became an engine of change in its own right. Rural Inventions explores this paradox. Nostalgia for the rural is a thread that runs through the chapters of this study. Yet Rural Inventions also shows that in the postagrarian society initiated by the postwar economic boom, the rural could become a harbinger of future possibilities. Participants in France’s peasant moment of the 1960s to early 1980s reinscribed dwelling in the countryside as an essential component of contemporary modern life.

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 Duality and contravariant functors in the representation theory of artin algebras

2019 ◽  
Vol 18 (06) ◽  
pp. 1950111 ◽  
Author(s):  
Samuel Dean
Keyword(s):  
Representation Theory ◽  
Model Theory ◽  
Final Section ◽  
Artinian Ring ◽  
Inverse Limits ◽  
Artin Algebra ◽  
Artin Algebras ◽  
Natural Transformations ◽  
Equivalent Properties ◽  
Finitely Presented

We know that the model theory of modules leads to a way of obtaining definable categories of modules over a ring [Formula: see text] as the kernels of certain functors [Formula: see text] rather than of functors [Formula: see text] which are given by a pp-pair. This paper will give various algebraic characterizations of these functors in the case that [Formula: see text] is an artin algebra. Suppose that [Formula: see text] is an artin algebra. An additive functor [Formula: see text] preserves inverse limits and [Formula: see text] is finitely presented if and only if there is a sequence of natural transformations [Formula: see text] for some [Formula: see text] which is exact when evaluated at any left [Formula: see text]-module. Any additive functor [Formula: see text] with one of these equivalent properties has a definable kernel, and every definable subcategory of [Formula: see text] can be obtained as the kernel of a family of such functors. In the final section, a generalized setting is introduced, so that our results apply to more categories than those of the form [Formula: see text] for an artin algebra [Formula: see text]. That is, our results are extended to those locally finitely presented [Formula: see text]-linear categories whose finitely presented objects form a dualizing variety, where [Formula: see text] is a commutative artinian ring.

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