It is possible to associate a topological space to the category of modules over any ring. This space, the Ziegler spectrum, is based on the indecomposable pure-injective modules. Although the Ziegler spectrum arose within the model theory of modules and plays a central role in that subject, this book concentrates specifically on its algebraic aspects and uses. The central aim is to understand modules and the categories they form through associated structures and dimensions, which reflect the complexity of these, and similar, categories. The structures and dimensions considered arise particularly through the application of model-theoretic and functor-category ideas and methods. Purity and associated notions are central, localisation is an ever-present theme and various types of spectrum play organising roles. This book presents a unified, coherent account of material which is often presented from very different viewpoints and clarifies the relationships between these various approaches.

In recent years the interplay between model theory and other branches of mathematics has led to many deep and intriguing results. In this, the first book on the topic, the theme is the interplay between model theory and the theory of modules. The book is intended to be a self-contained introduction to the subject and introduces the requisite model theory and module theory as it is needed. Dr Prest develops the basic ideas concerning what can be said about modules using the information which may be expressed in a first-order language. Later chapters discuss stability-theoretic aspects of modules, and structure and classification theorems over various types of rings and for certain classes of modules. Both algebraists and logicians will enjoy this account of an area in which algebra and model theory interact in a significant way. The book includes numerous examples and exercises and consequently will make an ideal introduction for graduate students coming to this subject for the first time.

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We present a general construction of model category structures on the category C(Qco(X)) of unbounded chain complexes of quasi-coherent sheaves on a semi-separated scheme X. The construction is based on making compatible the filtrations of individual modules of sections at open affine subsets of X. It does not require closure under direct limits as previous methods. We apply it to describe the derived category D(Qco(X)) via various model structures on C(Qco(X)). As particular instances, we recover recent results on the flat model structure for quasi-coherent sheaves. Our approach also includes the case of (infinite-dimensional) vector bundles, and of restricted flat Mittag-Leffler quasi-coherent sheaves, as introduced by Drinfeld. Finally, we prove that the unrestricted case does not induce a model category structure as above in general. IntroductionLet X be a scheme and Qco(X) the category of all quasi-coherent sheaves on X. A convenient way of approaching the derived category D(Qco(X)) goes back to Quillen [27], and consists in introducing a model category structure on C(Qco(X)), the category of unbounded chain complexes on Qco(X). In particular, one can compute morphisms between two objects X and Y of D(Qco(X)) as the C(Qco(X))morphisms between cofibrant and fibrant replacements of X and Y , respectively, modulo chain homotopy.Recently, Hovey has shown that model category structures naturally arise from small cotorsion pairs over C(Qco(X)), [20]. Since Qco(X) is a Grothendieck category [8], there is a canonical injective model category structure on C(Qco(X)). However, this structure is not monoidal, that is, compatible with the tensor product on Qco(X), [21, pp. 111-2]. Another natural, but not monoidal, model structure on C(Qco(X)) was constructed in [22] under the assumption of X being a Noetherian separated scheme with enough locally frees.The lack of compatibility with the tensor product was partially solved in [14, 25] by using flat quasi-coherent sheaves. The main result of [14] shows that in case X is quasi-compact and semi-separated, it is possible to construct a monoidal flat model structure on C(Qco(X)). The weak equivalences of this model structure are the same as the ones for the injective model structure, hence they induce the same cohomology functors (see [25] for a different approach). However, the structure of flat quasi-coherent sheaves is rather complex, and it is difficult to compute the associated fibrant and cofibrant replacements. Moreover, the methods of [14] depend heavily on the fact that the class of all flat modules is closed under direct limits.A different approach has recently been suggested in [10] for the particular case of quasi-coherent sheaves on the projective line P 1 (k). In that paper it was shown that the class of infinite-dimensional vector bundles (i.e., those quasi-coherent sheaves whose sections in all open affine sets are projective) imposes a monoidal model category structure on C(Qco (P 1 (k)). The proofs and techniques in [10] are strongly based on the Grothendiec...

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...