## Model Theory and Modules (2006)

Citations: | 73 - 20 self |

### BibTeX

@MISC{Prest06modeltheory,

author = {Mike Prest},

title = {Model Theory and Modules},

year = {2006}

}

### OpenURL

### Abstract

The model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module ” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the

### Citations

498 | Model Theory - Chang, Keisler - 1973 |

363 |
Representation Theory of Artin Algebras
- Auslander, Reiten, et al.
- 1995
(Show Context)
Citation Context ...ost split if it is not a split monomorphism but every morphism g : A −→ C which is not a split monomorphism factors through f and if every endomorphism h of B with hf = f is an automorphism of B (see =-=[5]-=-). Theorem 1.15 (a) Suppose that N ∈ ZgR. Then N is isolated by a minimal pair iff N ⊗ − is the injective hull of a finitely presented simple functor in (R-mod,Ab). (b) If N ∈ ZgR is the pure-injectiv... |

225 | Locally presentable and accessible categories - Adámek, Rosický - 1994 |

91 |
C.M.: Indecomposable representations of graphs and algebras
- Dlab, Ringel
- 1976
(Show Context)
Citation Context ... K-algebra. So, if K is algebraically closed, then R is Morita equivalent to a finite product of rings, each of which is the path algebra over K of an extended Dynkin quiver (for the general case see =-=[37]-=-). Suppose that R is indecomposable as a ring. Then the Ziegler spectrum of R is, roughly, composed of finitely many generically overlapping copies of Ziegler spectra of Dedekind domains, together wit... |

74 |
Sverre O. Smalø, Preprojective modules over Artin algebras
- Auslander
- 1980
(Show Context)
Citation Context ...inite representation type. It is a long-standing open question whether or not every right pure-semisimple ring is of finite representation type. For Artin algbras this was shown to be so by Auslander =-=[3]-=-. Herzog [60] showed that it holds for PI rings. Simson has shown (e.g. see [161]) that the general problem reduces to questions about division algebras. 3. Dedekind domains Let R be a commutative Ded... |

55 |
Ringel: Auslander-Reiten sequences with few middle terms and applications to string algebras
- Butler, M
- 1987
(Show Context)
Citation Context ...K-)dimension d lie (up to isomorphism) in the union of the images of these embeddings (but the number of representation embeddings needed might grow as d grows). Examples include string algebras (see =-=[28]-=-) and tame canonical algebras ([141]) and in these examples both KG(R) and CB(ZgR) turn out to be∞ when the algebra is not domestic. There is still, however, the hope of being able to describe the spe... |

51 |
On tame algebras and bocses
- Crawley-Boevey
- 1988
(Show Context)
Citation Context ... representation type of R. For the remainder of this subsection we assume that R is a finite-dimensional Kalgebra where K is a field. For the precise definition of domestic, tame and wild we refer to =-=[31]-=- for instance (especially since, although the situation is clearcut for finite-dimensional algebras over algebraically closed fields, it is not clear what the definitions of these terms should be in g... |

50 | Coherent functors, in - Auslander - 1965 |

46 |
Generalized Weyl algebras and their representations
- Bavula
- 1990
(Show Context)
Citation Context ... of this form is dense in ZgR and hence that there are no isolated points in ZgR. These and related results are proved in [120] for a class of rings, certain generalised Weyl algebras in the sense of =-=[11]-=-, which includes the first Weyl algebra. 5c. Pullback rings IfR, R′ are two commutative discrete valuation domains and if there is an isomorphism between their residue fields then one may form the pul... |

45 | Locally finitely presented additive categories - Crawley-Boevey - 1994 |

34 |
Algebra: Rings, Modules and Categories
- Faith
- 1976
(Show Context)
Citation Context ...ations aX = X.da (a ∈ K). By varying K and d we obtain a variety of interesting examples. For instance, suppose that (K, d) is a universal field with derivation. Then R is an example of a V-ring (see =-=[42]-=-) - a ring in which every simple module is injective. In this case, there is a unique simple R-module, S, and the Ziegler spectrum of R consists of just three points: the injective module S; the “dual... |

31 |
Tame algebras and generic modules
- Crawley-Boevey
- 1991
(Show Context)
Citation Context ...ed is said to be generic. At least in the context of finite-dimensional algebras over algebraically closed fields, such modules correspond to one-parameter families of finite-dimensional modules, see =-=[32]-=- and, for more general contexts, [34]. The next result indicates the model-theoretic relevance of pure-injectivity and its second part points to the special role played by the indecomposable pure-inje... |

26 |
Isolated singularities and existence of almost split sequences, Representation theory
- Auslander
- 1984
(Show Context)
Citation Context ...tween right and left modules. The basic duality, which is valid for all rings R, is between the categories, C(R) = (R-mod,Ab)fp and C(Rop) = (mod-R,Ab)fp of finitely presented functors. Theorem 1.26 (=-=[4]-=-, [55]) For any ring R we have C(Rop) ' C(R)op via the contravariant functor which is defined on objects by taking F ∈ C(Rop) to the functor, DF , in C(R) which is given on objects by taking L ∈ R-mod... |

26 |
Modules of finite length over their endomorphism ring, in: Representations of algebras and related
- Crawley-Boevey
(Show Context)
Citation Context ... the context of finite-dimensional algebras over algebraically closed fields, such modules correspond to one-parameter families of finite-dimensional modules, see [32] and, for more general contexts, =-=[34]-=-. The next result indicates the model-theoretic relevance of pure-injectivity and its second part points to the special role played by the indecomposable pure-injectives. Theorem 1.13 ([43], [151]) Ev... |

24 | Relative homological algebra and purity in triangulated categories
- Beligiannis
(Show Context)
Citation Context ...tion may be made for functors. (Indeed, everything that we do here for modules can be done in the setting of any locally finitely presented Grothendieck category and, to some extent, beyond (e.g. see =-=[14]-=-, [82]).) An object E of an abelian category is injective if every inclusion of the form E −→ F in the category is split. Every object of a Grothendieck category has an injective hull (a “smallest” in... |

17 |
On the free product of associative rings
- Cohn
- 1959
(Show Context)
Citation Context ...sources. 1 Purity Purity (pure embeddings, pure-injective modules) undoubtedly plays the central role so we will start with that. The notion of a pure embedding between modules was introduced by Cohn =-=[30]-=-. We say that the module A is a pure submodule of the module B if every finite system ∑ i xirij = aj (j = 1, ...,m) of R-linear equations with constants from A (so rij ∈ R and aj ∈ A) and with a solut... |

17 | Infinite-dimensional modules in the representation theory of finite-dimensional algebras
- Crawley-Boevey
- 1998
(Show Context)
Citation Context ... not onto}. These are, indeed, equivalent: any set of one of the above forms is of the other 7 forms (and the transformation from one to the other form is quite explicit and can be found variously in =-=[36]-=-, [77], [115]). Since every functor in (mod-R,Ab)fp is isomorphic to Fφ/Fψ for some pp formulas φ ≥ ψ a fourth way of giving the open sets is as {N ∈ ZgR : −→ GN 6= 0} as G ranges over (mod-R,Ab)fp an... |

15 |
Some model-theoretic properties of functor categories for modules
- Burke
- 1994
(Show Context)
Citation Context ...ctive R-module which cogenerates a torsion theory, τ , of finite type on Mod-R. Then the ring of definable scalars of E is precisely the 13 corresponding localisation R −→ Rτ . A finer topology Burke =-=[21]-=- introduced another topology on the underlying set of ZgR which he (re-)named, in [22], the full support topology (in his thesis he called it the “types-over-formulas” topology because the basic open ... |

14 | Phantom maps and purity in modular representation theory, I, Fundamenta Mathematicae 161 - Gnacadja - 1999 |

14 |
Regular modules for tame hereditary algebras
- Crawley-Boevey
- 1991
(Show Context)
Citation Context ...e ring of quotients, a matrix ring over a division ring which is finite-dimensional over its centre. Equivalently, R is a hereditary noetherian prime ring which satisfies some polynomial identity. In =-=[33]-=- CrawleyBoevey draws a parallel between and, indeed, links the categories of finite-length modules over a tame hereditary artin algebra and over a hereditary order, with 18 the maximal orders correspo... |

12 | Pure injectives and the spectrum of the cohomology ring of a finite group
- Benson, Krause
(Show Context)
Citation Context ...oup are pure-injective. These results have been extended by Benson 21 and Krause [18]. Krause [81] has shown how to define the Ziegler spectrum of any compactly generated triangulated category and in =-=[17]-=- Benson and Krause find the Zariski spectrum (in the classical sense) of the Tate cohomology ring of a finite group as a part of the Zariski spectrum (in the sense used in this paper) of the group rin... |

12 |
Rings of definable scalars and biendomorphism rings
- Burke, Prest
- 1997
(Show Context)
Citation Context ...e localisation, (RR,−)τ , of the forgetful functor at the torsion theory τ = τsupp(M). Then the endomorphism ring, in the localised category, D(R)τ , of (RR,−)τ is isomorphic to RX . Theorem 1.25 (a) =-=[26]-=- Let M be a Σ-pure-injective module which is finitely generated over its endomorphism ring (e.g. let M be a module of finite endolength). Then RM ' Biend(M). (b) [108] Suppose that R f−→ S is an epimo... |

12 | The Ziegler and Zariski spectra of some domestic string algebras, Algebras and Representation Theory
- Burke, Prest
(Show Context)
Citation Context ...he name. Despite the name, however, the space ZarR is only “algebraic-geometric” in parts. For example it is seldom compact and it may have infinitely many clopen points. In some examples (see, e.g., =-=[27]-=-) it seems to be a partial amalgamation of “geometric” and “combinatorial” pieces. Nonetheless, there is a natural sheaf of rings over it which directly generalises the structure sheaf of a commutativ... |

9 |
Phantom maps and purity in modular representation theory
- Benson, Gnacadja
- 1999
(Show Context)
Citation Context ...e (continuum many) points of C as its simple torsion representations. He also extends some of the theory of weights to these representations. 9. Stable and triangulated categories Benson and Gnacadja =-=[15]-=- show that certain of the idempotent modules of Rickard [139] in the stable module category for a finite group are pure-injective. These results have been extended by Benson 21 and Krause [18]. Krause... |

8 | Oystaeyen, The simple modules of certain generalised crossed products - Bavula, van - 1997 |

5 | On universal Horn classes categorical in some infinite power - Baldwin, Lachlan - 1973 |

4 | Pure injective modules over the dihedral algebras - Baratella, Prest - 1997 |

4 |
Undecidability of the theory of Abelian groups with a subgroup
- Baur
- 1976
(Show Context)
Citation Context ...nput with any word w and words w1, ..., wn will decide whether or not w represents the identity element in the free group factored by the normal subgroup generated by the words w1, ..., wn. Baur [9], =-=[10]-=- and others (for references see [102, Chapter 17]) showed that this unsolvable word problem for groups can be encoded in the theory of modules over various rings. For example the theory of K〈X,Y 〉-mod... |

4 |
Some connections between finitely generated functors and pptypes
- Burke
- 1997
(Show Context)
Citation Context ...he ring of definable scalars of E is precisely the 13 corresponding localisation R −→ Rτ . A finer topology Burke [21] introduced another topology on the underlying set of ZgR which he (re-)named, in =-=[22]-=-, the full support topology (in his thesis he called it the “types-over-formulas” topology because the basic open sets are of the form (p/ψ) where p is a pp-type and ψ a pp formula). The closed sets f... |

4 | Co-existence of Krull filtrations - Burke |

4 |
Abelian Structures, Yale University, 1974/5, partly published as Abelian Structures I
- Fisher
- 1977
(Show Context)
Citation Context ...ules Nλ, together with their multiplicities, as well as the module Nc, are determined up to isomorphism by N . The next result has been extensively used in the model theory of modules. 5 Theorem 1.9 (=-=[44]-=-, also see [47]) Let N be a pure-injective module and let A be a submodule of N . Then there is a direct summand of N , denoted H(A), which is determined up to isomorphism over A and which is minimal ... |

3 |
The extension group of the simple modules over the first Weyl algebra
- Bavula
(Show Context)
Citation Context ...at there is a continuous pure-injective R-module. If M is any indecomposable R-module of finite length then the pure-injective hull, M̄ , of M is indecomposable and it follows from a result of Bavula =-=[12]-=- that no such point is isolated (see [120]). In [120] it is shown that the set of points of this form is dense in ZgR and hence that there are no isolated points in ZgR. These and related results are ... |

3 |
Pure-injective hulls of expanding string modules
- Burke
- 1997
(Show Context)
Citation Context ...g algebras which turn out to 17 give all integer values 3 ≤ n < ω. See the references for details, including the explicit description of the points and the topology, which relies heavily on [143] and =-=[25]-=-. It has been shown by Krause [79] (for finite-dimensional algebras over an algebraically closed field) and Herzog [63] (for artin algebras in general) that there is no artin algebra R with KG(R) = 1.... |

3 |
Model theory of modules over a serial ring
- Eklof, Herzog
- 1995
(Show Context)
Citation Context ...tains further results about the relationship between maxspec(R) and ZgR in the general regular case. 7. Serial rings The model theory of modules over serial rings was investigated by Eklof and Herzog =-=[39]-=- and by Puninski [128]. In both these papers a particularly nice basis of the Ziegler topology was found and general characterisations of indecomposable pure-injectives in terms of the ideals of R wer... |

3 | Σ-pure-injective modules over a serial ring, Abelian groups and modules, Kluwee Acad - Facchini, Puninski - 1995 |

2 |
Decidability and undecidability for theories of abelian groups with predicates for subgroups
- Baur
- 1975
(Show Context)
Citation Context ...hen input with any word w and words w1, ..., wn will decide whether or not w represents the identity element in the free group factored by the normal subgroup generated by the words w1, ..., wn. Baur =-=[9]-=-, [10] and others (for references see [102, Chapter 17]) showed that this unsolvable word problem for groups can be encoded in the theory of modules over various rings. For example the theory of K〈X,Y... |

1 |
The theory of ZC(2)2-lattices is decidable
- Baratella, Toffalori
- 1998
(Show Context)
Citation Context ... Toffalori, especially with a view to showing that the tame/wild dichotomy corresponds to the split between (a ring having) decidable/undecidable theory of modules. See [165], [166] and, for example, =-=[8]-=-, [86], [88], [167]. These papers also provide a great variety of examples of interpretations of classes of additive structures in other such classes. 11. Decidability/undecidability The word problem ... |

1 |
Generic idempotent modules for a finite group, Algebras and Representation Theory 3
- Benson, Krause
- 2000
(Show Context)
Citation Context ...nacadja [15] show that certain of the idempotent modules of Rickard [139] in the stable module category for a finite group are pure-injective. These results have been extended by Benson 21 and Krause =-=[18]-=-. Krause [81] has shown how to define the Ziegler spectrum of any compactly generated triangulated category and in [17] Benson and Krause find the Zariski spectrum (in the classical sense) of the Tate... |

1 | Logical aspects of ring and module theory, Fundamentalnaya i Prikladnaya - Beidar, Mikhalev, et al. - 1995 |

1 | A general character theory for modules - Burke |

1 |
On pure-injective modules over pullback rings
- Ehrahimi-Atani
(Show Context)
Citation Context ... which are, for m,n ≥ 2, m+n ≥ 5, tame non-domestic string algebras. For such pullback rings Toffalori [168], [169] classified the indecomposable pure-injective “separated” modules and Ebrahimi-Atani =-=[38]-=- classified all the indecomposable pure-injectives, N , such that N/N.J(R) is of finite length. Note that the complete description of the Ziegler spectrum for such rings would include that for the tam... |

1 |
Powers of saturated modules
- Fisher
- 1972
(Show Context)
Citation Context ... contexts, [34]. The next result indicates the model-theoretic relevance of pure-injectivity and its second part points to the special role played by the indecomposable pure-injectives. Theorem 1.13 (=-=[43]-=-, [151]) Every module is elementarily equivalent to a pureinjective module (in fact, is an elementary substructure of its pure-injective hull). ([178]) Every module is elementarily equivalent to a dir... |