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A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If Φ \Phi is a finite dimensional algebra, then each functorially finite wide subcategory of mod ⁡ ( Φ ) \operatorname {mod}( \Phi ) is of the form ϕ ∗ ( mod ⁡ ( Γ ) ) \phi _{ {\textstyle *}}\big ( \operatorname {mod}( \Gamma ) \big ) in an essentially unique way, where Γ \Gamma is a finite dimensional algebra and Φ ⟶ ϕ Γ \Phi \stackrel { \phi }{ \longrightarrow } \Gamma is an algebra epimorphism satisfying Tor 1 Φ ⁡ ( Γ , Γ ) = 0 \operatorname {Tor}^{ \Phi }_1( \Gamma ,\Gamma ) = 0 . Let F ⊆ mod ⁡ ( Φ ) \mathscr {F} \subseteq \operatorname {mod}( \Phi ) be a d d -cluster tilting subcategory as defined by Iyama. Then F \mathscr {F} is a d d -abelian category as defined by Jasso, and we call a subcategory of F \mathscr {F} wide if it is closed under sums, summands, d d -kernels, d d -cokernels, and d d -extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of F \mathscr {F} is of the form ϕ ∗ ( G ) \phi _{ {\textstyle *}}( \mathscr {G} ) in an essentially unique way, where Φ ⟶ ϕ Γ \Phi \stackrel { \phi }{ \longrightarrow } \Gamma is an algebra epimorphism satisfying Tor d Φ ⁡ ( Γ , Γ ) = 0 \operatorname {Tor}^{ \Phi }_d( \Gamma ,\Gamma ) = 0 , and G ⊆ mod ⁡ ( Γ ) \mathscr {G} \subseteq \operatorname {mod}( \Gamma ) is a d d -cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some d d -cluster tilting subcategories F ⊆ mod ⁡ ( Φ ) \mathscr {F} \subseteq \operatorname {mod}( \Phi ) over algebras of the form Φ = k A m / ( rad k A m ) ℓ \Phi = kA_m / (\operatorname {rad}\,kA_m )^{ \ell } .

Morava E-theory E V n* (-) is a much-studied theory in algebraic topology, but it is not a homology theory in the usual sense, because it fails to preserve coproducts (resp. filtered homotopy colimits). The object of this paper is to construct a spectral sequence to compute the Morava E-theory of a coproduct (resp. filtered homotopy colimit). The E 2 -term of this spectral sequence involves the derived functors of direct sum (resp. filtered colimit) in an appropriate abelian category. We show that there are at most n - 1 (resp. n) of these derived functors. When n = 1, we recover the known result that homotopy commutes with an appropriate version of direct sum in the K(1)-local stable homotopy category.

D. Happel, I. Reiten and S. Smarlo initiated an investigation of quasitilted artin K-algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring K. Here, employing a notion of *-objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.

The following are equivalent for a skeletally small abelian Hom-finite category over a field with enough injectives and each simple object being an epimorphic image of a projective object of finite length. (a) Each indecomposable injective has a simple subobject. (b) The category is equivalent to the category of socle-finitely copresented right comodules over a right semiperfect and right cocoherent coalgebra such that each simple right comodule is socle-finitely copresented. (c) The category has left almost split sequences.

Given a cotorsion pair ( A , B ) (\mathcal {A},\mathcal {B}) in an abelian category C \mathcal {C} with enough A \mathcal {A} objects and enough B \mathcal {B} objects, we define two cotorsion pairs in the category C h ( C ) \mathbf {Ch(\mathcal {C})} of unbounded chain complexes. We see that these two cotorsion pairs are related in a nice way when ( A , B ) (\mathcal {A},\mathcal {B}) is hereditary. We then show that both of these induced cotorsion pairs are complete when ( A , B ) (\mathcal {A},\mathcal {B}) is the “flat” cotorsion pair of R R -modules. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new “flat” model category structure on C h ( R ) \mathbf {Ch}(R) . In the last section we use the theory of model categories to show that we can define Ext R n ⁡ ( M , N ) \operatorname {Ext}^n_R(M,N) using a flat resolution of M M and a cotorsion coresolution of N N .

On indique comment une sous-catégorie colocalisante d’une catégorie abélienne induit une localisation dans la catégorie dérivé. Ceci permet une nouvelle construction de la tour de Taylor d’un foncteur à valeur dans une catégorie de module.

In this paper, we try to realize the unbounded derived category of an abelian category as the homotopy category of a Quillen model structure on the category of unbounded chain complexes. We construct such a model structure based on injective resolutions for an arbitrary Grothendieck category, as has apparently also been done by Morel. In particular, this works for sheaves on a ringed space, and for quasi-coherent sheaves on a quasi-compact, quasi-separated scheme. However, this injective model structure is not well suited to studying the derived tensor product, so we investigate other model structures. The most successful of these is the flat model structure on complexes of sheaves over a ringed space. This is based on flat resolutions, and is compatible with the tensor product. As a corollary, we get model categories of differential graded algebras of sheaves and differential graded modules over a given differential graded algebra of sheaves. This is the author's first attempt to understand sheaves, so comments from those more experienced with the subject are welcome.

With any subset of a countable discrete abelian we associate with it three Banach space properties. These properties are Radon-Nikodym type properties. The relationship between these properties is investigated. The results are applied to give new characterizations of Riesz subsets and Rosenthal subsets of countable discrete abelian groups.

Given a generic Mordell-Weil group over a function field, we can specialize it down to a number field. It has been known for some time that the resulting homomorphism of groups is injective "infinitely often". We prove that this is in fact true "almost always", in a sense that is quantitatively nearly best possible.

Let Γ \Gamma be a discrete abelian group, and E ⊂ Γ E \subset \Gamma . For F ⊂ E F \subset E , we say that F ∈ P ( E ) F \in \mathcal {P}(E) , if for all Λ \Lambda , finite subsets of Γ , 0 ∉ Λ , Λ + F ∩ F \Gamma ,0 \notin \Lambda ,\Lambda + F \cap F is finite. Having defined the Banach algebra, A ~ ( E ) = c ( E ) ∩ B ( E ) \tilde A(E) = c(E) \cap B(E) , we prove the following: (i) E ⊂ Γ E \subset \Gamma is a Sidon set if and only if every F ∈ P ( E ) F \in \mathcal {P}(E) is a Sidon set; (ii) E ∈ P ( Γ ) E \in \mathcal {P}(\Gamma ) is a Sidon set if and only if A ~ ( E ) = A ( E ) \tilde A(E) = A(E) .

Quillen has constructed a K K -theory K ∗ C {K_{\ast }}C for nice categories, one of which is the category of projective R R -modules. We construct a theory K V ∗ C K{V_{\ast }}C for the nice categories parametrized by rings. When applied to projective modules we recover the Karoubi-Villamayor K K -theory K V ∗ ( R ) K{V_{\ast }}(R) . As an application, we show that the Cartan map from K ∗ ( R ) {K_{\ast }}(R) to G ∗ ( R ) {G_{\ast }}(R) factors through the groups K V ∗ ( R ) K{V_{\ast }}(R) . We also compute K V ∗ K{V_{\ast }} for the categories of faithful projectives and Azumaya algebras, generalizing results of Bass.

Both the classical approach to decomposition theories and Fisher’s technique of constructing decomposition theories from radical functions are extended to and exploited in the context of abelian categories. These two different approaches to decomposition theories for abelian categories intertwine in one theorem from which flows necessary and sufficient conditions for the existence of the tertiary, primary, and Bourbaki’s P \mathcal {P} -primary decomposition theories.

Let p: $ -+ £ be a sheaf (espace etale) of abelian groups. Applying singular functor S one obtains a simplicial map tt: A-> X with A=5(«f), X=S(X)and tt — S(p). The fibers 77_1(x), x e X, form a "local system of groups" over X which will be called a costack of abelian groups over the simplicial set X. In general, a costack is defined as a functor on X, regarded as a category. This is a generalized dual of the notion of a stack defined by Spanier [5]. The main objects of this note are (1) to develop a general theory of stacks and costacks over simplicial sets, (2) to construct a semisimplicial homology theory with "variable" coefficients which is unique in the sense of Eilenberg-Steenrod. The coefficients of the homology are a costack in an abelian category. In particular, when the coefficient costack is a locally constant costack the homology becomes the usual homology with local coefficients. There are three chapters in this note. Chapter I is devoted to a study of stacks and costacks. It is partially a preparatory chapter. In Chapter II we define homology of costacks via usual chain complexes and prove that the homology so defined can be computed by projective resolutions by introducing a generalized torsion product functor. Under the equivalence of costacks and modules, this generalized functor is essentially the genuine torsion product functor of modules. The rest of Chapter II is a preparation for Chapter III, in which a homology theory of pairs of simplicial sets over a fixed simplicial set Ais defined. Results of Chapter II ensure the existence of such a theory. Chapter III concludes with a proof of the uniqueness of this homology theory. This is a generalization of Eilenberg-Steenrod uniqueness theorem [1]. The results presented in this note are a part of the author's Ph.D. thesis at the City University of New York written under the direction of Professor Alex Heller.

As a consequence of this theorem, every object A of (i has "elements"namely, the elements of the image A' of A under the imbedding-and all the usual propositions and constructions performed by means of "diagram chasing" may be carried out in an arbitrary abelian category precisely as in the category of abelian groups. In fact, if we identify (a with its image Wt' under the imbedding, then a sequence is exact in (t if and only if it is an exact sequence of abelian groups. The kernel, cokernel, image, and coimage of a map f of et are the kernel, cokernel, image, and coimage of f in the category of abelian groups; the map f is an epimorphism, monomorphism, or isomorphism if and only if it has the corresponding property considered as a map of abelian groups. The direct product of finitely many objects of et is their direct product as abelian groups. If A (c, then every subobject [4] of A is a subgroup of A, and the intersection (or sum) of finitely many subobjects of A is their ordinary intersection (or sum); the direct (or inverse) image of a subobject of A by a map of Q is the usual set-theoretic direct (or inverse) image. Moreover, if

A functor T : A → B T:\mathcal {A} \to \mathcal {B} acts faithfully on the right of a class of objects A ′ \mathcal {A}’ of A \mathcal {A} if it distinguishes morphisms out of objects of A ′ \mathcal {A}’ (that is, A ′ ∈ A ′ , X ∈ A , f , g ∈ A ( A ′ , X ) A’ \in \mathcal {A}’,X \in \mathcal {A},f,g \in \mathcal {A}(A’,X) and f ≠ g f \ne g implies T ( f ) ≠ T ( g ) ) T(f) \ne T(g)) . We define a full subcategory R F ( T ) \mathcal {R}\mathcal {F}(T) such that T T acts faithfully on the right of the objects of R F ( T ) \mathcal {R}\mathcal {F}(T) . An object U ∈ A U \in \mathcal {A} is a generator if H U : A → E n s {H^U}:\mathcal {A} \to \mathcal {E}ns is faithful, and if H U {H^U} is not faithful, we may still consider R F ( H U ) \mathcal {R}\mathcal {F}({H^U}) . This gives rise to the notion of a generating structure. Cogenerating structures are defined dually, and various canonical generating and cogenerating structures are defined for the category of R R -modules. Relationships between these can be used in the homological classification of rings.

This note constitutes a sequel to [AC], the terminology and notation of which are used throughout. Its purpose is to contribute some technical devices, viz. the notion of an ideal, and that of the loop-space functor, to the study of homological algebra. The concept of an ideal of an additive category is introduced (?1) in analogy with the familiar notion of the theory of rings. In particular the quotient of a category by a two-sided ideal is defined. Of especial interest are the ideals generated by identity maps, which are studied in ?2. For an abelian category with enough projectives a functor Q, defined on the quotient of the category by its projectives (?3), bears a close analogy to the loop-space functor of topology. This functor, as an operation on modules, has also been considered by P. J. Hilton and B. Eckmann [H-E]. If 3e is an abelian category with enough projectives then for 'C8, the category of proper s.e.s. in 3X, a functor r is introduced (?4). Pursuing the topological analogy, r plays the role of the operation which turns inclusion maps into fibre maps. This is related to the loop-space functor by the operation (?5) that its threefold iterate is just Q for the category 3C' (a fact which is known for the topological analogue). Using this result it is possible to associate to a proper s.e.s. exact sequences of homomorphism groups in the original category modulo its projectives. Finally the Ext groups of the original category are related to the homomorphism groups in the quotient by projectives (?6) by a functor which exhibits the latter as quotient groups of the former. The kernel measures the extent to which projectives fail to be injective; if they are all injective the study of Ext is reduced to that of the quotient category and the functor U. As in [AC] the theory is subject to straightforward dualization which however is barely indicated. The general reference is [AC]. To this should be added [G] which had not yet appeared when [AC] was written. It should be noted that "additive category" in [G] is used in a sense different from the present one, and that "abelian category" in [G] is equivalent to the "exact category" of Buchsbaum and [AC]. 1. Ideals in an additive category. A left ideal i of an additive category 3C is a subset of the maps of 3C with the properties

For any additive functor from a category of modules into an abelian category there is a largest Giraud subcategory for which the functor acts faithfully on homomorphisms into the subcategory. It is the largest Giraud subcategory into which the functor reflects exact sequences, and under certain conditions it is just the largest Giraud subcategory on which the functor acts faithfully. If the functor is exact and has a right adjoint, then the subcategory is equivalent to the quotient category determined by the kernel of the functor. In certain cases, the construction can be applied to a Morita context in order to obtain a recent theorem of Mueller. Similarly, the functor defines a certain reflective subcategory and an associated radical, which is a torsion radical in case the functor preserves monomorphisms. Certain results concerning this radical, when defined by an adjoint functor, can be applied to obtain two theorems of Morita on balanced modules. This paper is a continuation of the author's work [21 on T-faithful subcategories of module categories. For the category RX of unital left R-modules over a ring R with identity, and for any additive functor T: RX -2 into an abelian category $, a subcategory (i of RN will be called T-faithful if T is faithful when restricted to (i. (Similarly, (i is left T-faithful if T(f) 7 0 for every nonzero R-homomorphism /: M --A, with A e d.) If T is exact and has an adjoint, one can form the quotient category R)R/Ker (T), which is then equivalent to a certain Giraud subcategory of RX This Giraud subcategory can be characterized as the largest left T-faithful Giraud subcategory, and if R is commutative and Noetherian, it is the largest T-faithful Giraud subcategory of RI1 (Corollary 3.10). For any functor T: RX $ S it is possible to construct the largest Giraud subcategory of RX which is left T-faithful, thus generalizing in a certain sense the construction of the quotient category R)R/Ker (T). If T is exact, this subcategory is determined by the smallest hereditary torsion class of RV which contains Ker (T). The construction is useful even when T is not exact. Mueller Received by the editors April 2, 1973. AMS (MOS) subject classifications (1970). Primary 16AO8; Secondary 18E35, 18E40.