. We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pure-injective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. Introduction Smashing subcategories naturally arise in the stable homotopy category S from localization functors l : S ! S which induce for every spectrum X a natural isomorphism l(X) ' X l(S) between the localization of X and the smash product of X with the localization of the sphere spectrum S. In fact, a localization functor has this property if and only if it preserv...

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this paper is to develop further the analysis of existence and properties of generic modules. Our approach depends to a large extent on the embedding of a module category into a bigger functor category. These general concepts are explained in the first two sections. We continue in Section 3 with a new characterization of the pure-injective modules which occur as the source of a minimal left almost split morphism. This is of interest in our context because generic modules are pure-injective. Next we consider indecomposable endofinite modules. Recall that a module is endofinite if it is of finite length when regarded in the natural way as a module over its endomorphism ring. Changing slightly the original definition, we say that a module is generic if it is indecomposable endofinite but not finitely presented. Section 4 is devoted to several characterizations of generic modules in order to justify the choice of the non-finitely presented modules as the generic objects. We prove them for dualizing rings, i.e. a class of rings which includes noetherian algebras and artinian PIrings. Existence results for generic modules over dualizing rings follow in Section 5. Several results in this paper depend on the fact that a functor f : Mod(\Gamma) ! Mod() which commutes with direct limits and products, preserves certain finiteness conditions. For example, if a \Gamma-module M is endofinite then f(M) is endofinite. If in addition End \Gamma (M) is a PI-ring, then End (N) is a PI-ring for every indecomposable direct summand N of f(M ). This material is collected in Section 6 and 7. In Section 8 we introduce an effective method to construct generic modules over artin algebras from so-called generalized tubes. The special case of a tube in the Auslander-Reiten quiver is discussed in t...

We investigate categorical versions of algebraically closed (= pure) embeddings, existentially closed embeddings, and the like, in the context of locally presentable categories. The definitions of S. Fakir [Fa, 75], as well as some of his results, are revisited and extended. Related preservation theorems are obtained, and a new proof of the main result of Rosicky, Adamek and Borceux ([RAB, 02]), characterizing #-injectivity classes in locally #-presentable categories, is given.

this paper we show that mod determines the representation type of . Recall that the algebra is either tame, i.e. all finite dimensional indecomposable-modules belong to oneparameter families, or is wild, i.e. there are two-parameter families of finite dimensional indecomposable-modules [8]. Of course, one feels that this dichotomy should not depend on the deletion of finitely many objects in the category mod , and this is precisely one of the main results of this paper. More precisely, given another algebra \Gamma and an equivalence mod ! mod \Gamma, then \Gamma is tame if is tame. Moreover, we show that the equivalence sends the one-parameter families in mod to one-parameter families in mod \Gamma. The fact that mod determines the representation type of also follows, for some classes of symmetric algebras, from recent work of Assem, de la Pe~na and Erdmann [2, 9]; however their methods are completely different. Equivalences between stable module categories have been studied by many authors. They naturally occur for instance in representation theory of finite groups. Another source of examples, which includes every algebra of Loewy length 2, is the class of algebras stably equivalent to a hereditary algebra. Usually the analysis concentrates on homological properties of the category mod which are preserved by an equivalence mod ! mod \Gamma. In this paper we follow a different approach. We investigate pure-injective modules which are not necessarily finitely presented. Among them the endofinite modules are of particular interest. Recall that a module is endofinite if it is of finite length when regarded in the natural way as a module over its endomorphism ring. In order to study the non-finitely presented -modules we introduce a new category

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this paper. The representation type of a finite dimensional algebra is traditionally defined using the concept of a continuous one-parameter family in the category mod of finite dimensional-modules. For instance, is of tame representation type if for every n 2 N only finitely many such families are needed to parametrize all indecomposable -modules of dimension n. An alternative approach uses so-called generic modules. This was suggested by Crawley-Boevey and he established for tame algebras a correspondence between continuous one-parameter families and generic modules [3]. In [6], we used generic modules to show that a stable equivalence mod ! mod \Gamma induces a bijection between continuous one-parameter families in mod and mod \Gamma. However, without any extra assumption on the stable equivalence it remains an open question how the dimensions of the modules in mod and mod \Gamma are related. In this paper we settle the problem for stable equivalences which are induced by an appropriate functor mod ! mod \Gamma. Stable equivalences of Morita type are of this form, and therefore we can prove that two finite dimesional algebras have the same representation type if they are stably equivalent of Morita type. I would like to thank Thorsten Holm for drawing my attention to the representation type problem for derived equivalent algebras. We now begin with some notation and recall briefly the definitions which are needed in this paper. Let be an associative ring with identity. Denote by Mod the category of (right) -modules and let mod be the full subcategory of finitely presented-modules. A sequence of morphisms 0 ! L ! M ! N ! 0 in Mod is pure-exact if the induced sequence 0 ! Hom(X;L) ! Hom(X;M) !

: Given a two-sided artinian ring , it is shown that the Ziegler spectrum of forms a test class for certain homological properties of . We discuss the finitistic dimension of , Nunke's condition, and also the relation between the big and the little finitistic dimension. Let be a two-sided artinian ring. Denote by Mod the category of (right) -modules and by mod the full subcategory of all finitely presented-modules. Given a-module M , we denote by pdM its projective dimension, and the finitistic dimension Fd of is the supremum of the projective dimensions of the-modules with finite projective dimension. The Ziegler spectrum Zsp of is by definition the set of isomorphism classes of indecomposable pure-injective-modules. The aim of this note is to show that the Ziegler spectrum forms a test class for certain homological properties of . Furthermore, the Ziegler spectrum carries a topology and we shall use its compactness to obtain an equivalent formulation of Nunke's condition. In the fin...

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This is essentially the talk I gave on definable additive categories; I define these categories, say where they came from, describe some of what is around them and then point out the 2-category which they form.