Some model theory over hereditary noetherian domains

2005

Algebr Represent Theor

Self Cite

The Ziegler spectrum for categories enriched in closed symmetric monoidal Grothendieck categories is defined and studied in this paper. It recovers the classical Ziegler spectrum of a ring. As an application, the Ziegler spectrum as well as the category of generalised quasi-coherent sheaves of a reasonable scheme is introduced and studied. It is shown that there is a closed embedding of the injective spectrum of a coherent scheme endowed with the tensor fl-topology (respectively of a noetherian scheme endowed with the dual Zariski topology) into its Ziegler spectrum. It is also shown that quasi-coherent sheaves and generalised quasi-coherent sheaves are related to each other by a recollement.

“…Such a point, being finitely presented and the injective hull of a simple module, is also open (e.g. [PrPu,3.7]). Therefore both O and ZspΛ are clopen in ZspΛ.…”

Section: The Ziegler Spectrum Of Qf-rings and The Stable Module Categorymentioning

confidence: 99%

Triangulated Categories and the Ziegler Spectrum

Garkusha¹,

2005

Algebr Represent Theor

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“…In [17] we have shown that over a wide class of rings, so-called generalized Weyl algebras (GWA), the complete classification of indecomposable pure injective modules is hardly possible since it would include a similar classification over the free algebra k X Y -a problem which is usually considered as hopeless. For instance, this class of algebras includes the first Weyl algebra A 1 k and any primitive quotient of Usl 2 k over a field k of characteristic zero.…”

Section: Introductionmentioning

confidence: 99%

Pure Injective Envelopes of Finite Length Modules over a Generalized Weyl Algebra

Puninski

2002

Journal of Algebra

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We investigate certain pure injective modules over generalised Weyl algebras. We consider pure injective hulls of finite length modules, the elementary duals of these, torsionfree pure injective modules, and the closure in the Ziegler spectrum of the category of finite length modules supported on a nondegenerate orbit of a generalized Weyl algebra. We also show that this category is a direct sum of uniserial categories and admits almost split sequences. We find parallels to but also marked contrasts with the behaviour of pure injective modules over finitedimensional algebras and hereditary orders.  2002 Elsevier Science (USA)

“…Klingler and Levy [72] showed that the category of torsion modules over this ring is "wild" and their techniques can be used to show that there is a continuous pure-injective R-module. If M is any indecomposable R-module of finite length then the pure-injective hull,M , 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 Zg R and hence that there are no isolated points in Zg R .…”

mentioning

confidence: 99%

“…If M is any indecomposable R-module of finite length then the pure-injective hull,M , 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 Zg R and hence that there are no isolated points in Zg R . 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.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Model theory and modules

2003

Handbook of Algebra

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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 notation R-Mod denotes the category of left R-modules. By Ab we mean the category of abelian groups.In Part 1 we introduce the general concepts and in Part 2 we discuss these in more specific contexts.References within the text, as well as those in the bibliography, are neither complete nor comprehensive but are intended to lead the reader to a variety of sources. PurityPurity (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 x i r ij = a j (j = 1, ..., m) of R-linear equations with constants from A (so r ij ∈ R and a j ∈ A) and with a solution in B has a solution in A (a solution being elements b 1 , ..., b n such that i b i r ij = a j for all j). We extend this in the obvious way to define the notion of a pure embedding between modules and we also say 1 that an exact sequence 0 −→ A f − → B −→ C −→ 0 is pure-exact if f is a pure embedding.Functor categories Let D(R) = (R-mod, Ab) denote the category of additive functors (from now on we use the term "functor" to mean additive functor) from the category of finitely presented left modules to the category of abelian groups. This is a Grothendieck abelian category. It has a generating set consisting of finitely presented objects: indeed, the representable functors (L, −) for L ∈ R-mod are the finitely generated projective objects and, together, are generating. This category is locally coherent -any finitely generated subfunctor of a finitely presented functor is itself finitely presented -and of global dimension less than or equal to 2. A functor is finitely presented iff it is the cokernel of a map between two representable functors. The full subcategory C(R) = (R-mod, Ab) fp of finitely presented functors is an abelian category and the inclusion of (R-mod, Ab)fp into (R-mod, Ab) preserves exact sequences. Notice that the category (R-mod, Ab) is just the category of "modules" over the "ring with many objects" R-mod (better, over a small version of this). Concepts for modules over a ring generally make good sense in this context and largely can be understood in this way (that is, as having the same content that they have for modules). There is a full embedding of Mod-R into (R-mod, Ab) which is given on objects by sending M ∈ Mod-R to the functor M ⊗ R − : R-mod −→ Ab and which is given in the natural way on morphisms. The image o...