Model Theory and Modules

Prest, Mike

doi:10.1017/cbo9780511600562

Cited by 180 publications

(105 citation statements)

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“…A detailed exposition can be found in [13]. Instead of working with formulas all the time, we fix a structure and work with the definable subsets of finite cartesian powers of that structure.…”

Section: Model Theory Of Modulesmentioning

confidence: 99%

“…If a subgroup of M n is pp-definable, then its cosets are also pp-definable. The following lemma is well known and a proof can be found in [13 Let L n denote the meet-semilattice of all pp-subsets of M n ordered by the inclusion relation ⊆. We will use the notation L n (M R ), specifying the module, when we work with more than one module at a time.…”

Section: Model Theory Of Modulesmentioning

confidence: 99%

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Grothendieck rings of theories of modules

Kuber

2015

Annals of Pure and Applied Logic

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Abstract. The model-theoretic Grothendieck ring of a first order structure, as defined by Krajicěk and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the Grothendieck ring, K 0 (M R ), of a right R-module M , where R is any unital ring. As a corollary we prove a conjecture of Prest that K 0 (M ) is non-trivial, whenever M is non-zero. The main proof uses various techniques from the homology theory of simplicial complexes.

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Section: Model Theory Of Modulesmentioning

confidence: 99%

Section: Model Theory Of Modulesmentioning

confidence: 99%

Grothendieck rings of theories of modules

Kuber

2015

Annals of Pure and Applied Logic

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“…We recall just a little about m-dimension from [15], based on [31] (which in turn refined Garavaglia's elementary Krull dimension [6]). We define (see [15, [31, 8.7], see [15, 10.23…”

Section: Krull-gabriel Dimension M-dimension and Cb-rankmentioning

confidence: 99%

“…This dimension also turns out [14,Lemma B.9] to be the m-dimension of the lattice of pp formulas for R, equivalently of the lattice of pointed finite-dimensional modules; that dimension is a modification of the elementary Krull dimension, introduced by Garavaglia [6], and developed further in [31, §8] and [15,Chpt. 10].…”

Section: Introductionmentioning

confidence: 99%

Krull–Gabriel Dimension of 1-domestic String Algebras

Prest

Puninski

2006

Algebr Represent Theor

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“…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-/?, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-/?, Ab) to the study of Artinian algebras. The category ((/?-mod) op , Ab) has been less fully exploited.…”

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confidence: 99%

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

Wilkins

1998

Glasgow Math. J.

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Let R be a ring, see below for other notation. The functor categories (mod-/?,Ab) and ((/?-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-/?, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-/?, Ab) to the study of Artinian algebras. The category ((/?-mod) op , Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between 7?-mod and mod-/?. 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-/?, Ab) and ((/?-mod) op , Ab) for an arbitrary ring /?.The first two sections are concerned with detailing an equivalence between ((/?-mod ) op , Ab) and (mod-/?, Ab). The main result, Theorem 2.5, states that the equivalence between these categories may be built from the functors Tor(-,-) and Ext(-,-). Parts of this theorem are well known. For instance the functor Tor(-,-) has been extensively studied in this setting when R is an Artinian algebra, see [3]. In the third section this equivalence is applied to the Ziegler spectra of /?. In Theorem 3.3, it is shown that the sets {M | fp-inj.dimM < n] and {jV|w.dinr/V < n\,n a natural number, are closed in their respective spectra when /? is left coherent. Furthermore, it is also demonstrated that these sets are mapped onto each other under Herzog's correspondence between the closed sets of the Ziegler spectra, [7, Theorem 5.5]. The main result of Section 3, Corollary 3.4, states that when R is left coherent both spectra are test spaces for right finititistic weak dimension of /?. Two special cases are noted: when R is left Noetherian and when /? is left coherent and right perfect.Throughout R denotes an associative ring with unity. Mod-/? is the category of right /?-modules, mod-/? the category of finitely presented right /?-modules, mod-/? is the quotient category of mod-/? modulo the ideal of those maps that factor through a projective. The left handed analogues of these categories are written as /?-Mod, /?-mod and /?-mod, respectively. If C is a small additive category then (C, Ab) (respectively (C op , Ab)) is the category of covariant (respectively contravariant) additive functors from C to the category of abelian groups; (C, Ab) fp is the category of finitely presented objects of (C, Ab). If

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Model Theory and Modules

Cited by 180 publications

References 0 publications

Grothendieck rings of theories of modules

Grothendieck rings of theories of modules

Krull–Gabriel Dimension of 1-domestic String Algebras

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

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