# Purity, Spectra and Localisation (Encyclopedia of Mathematics and its Applications, Volume 21)

## Mike Priest

Language: English

Pages: 799

ISBN: 2:00277872

Format: PDF / Kindle (mobi) / ePub

The central aim of this book 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, localization is an ever-present theme and various types of spectrum play organizing 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.

Topologie générale: Chapitres 1 à 4

Proofs without Words: Exercises in Visual Thinking (Classroom Resource Materials, Volume 1)

Chromatic Graph Theory (Discrete Mathematics and Its Applications)

finitely generated, by s 1 , . . . , s m say, then g 15:41 smartpdf CUUK636-Prest 18 February 24, 2010 Pp conditions ML = {a ∈ M n : ∃b1 , . . . , bm such that a = m 1 bj s j } and this is clearly a ppn m n definable subgroup of M . Indeed, if φ(x) is ∃y1 , . . . , ym i=1 xi = j =1 yj sij , where s j = (sij )i , then L = φ(RR ) and φ(M) = ML = Mφ(R) (cf. 1.1.10). Note that for right modules we have annihilation by right ideals, divisibility by left ideals. From given pp conditions we can

and natural way to prove Ziegler’s results, and many subsequent ones, is to move to the appropriate functor category. Indeed it was already appreciated that work, particularly of Gruson and Jensen, ran, in places, parallel to pre-Ziegler results in the model theory of modules and some of the translation between the two languages (model-theoretic and functorial) was already known. Furthermore, many applications have been to the representation theory of finitedimensional algebras, where functorial

spectrum which corresponds in the sense of 5.1.6 to the definable category X . In this example we can say a little more. Since, 4.3.22 (which is a continuation of this example), the pure-injective hull of M is the direct product, H (M) = p Zp , the modules in X are precisely those of the form M ⊕ Q(κ) , where κ is any cardinal number and where M purely embeds in a product of copies of the simple abelian groups Zp . The same analysis applies to the definable subcategory generated by all the simple

is such a subcategory of mod-R and is also closed under direct sums, then its lim-closure is the definable subcategory − → g 15:41 smartpdf CUUK636-Prest 122 February 24, 2010 Pp-pairs and definable subcategories of Mod-R that it generates. Examples include, over a tame hereditary algebra, the closure under finite direct sums of the modules in the preprojective component together with any collection of tubes (Section 8.1.2), and, over a tubular canonical algebra, the closure under direct

Corollary 4.3.38. There is just a set of indecomposable pure-injective R-modules up to isomorphism, indeed there are at most 2card(R)+ℵ0 . Proof. There is just a set of pp-types (without parameters) and, by 4.3.37, every indecomposable pure-injective is isomorphic to the hull of a pp-type. More precisely there are card(R) + ℵ0 pp conditions therefore no more than 2card(R)+ℵ0 pp-types for R-modules. This approach gives an alternative route to showing that pp-types are always realised, cf. 3.3.6.