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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 } .

The main goal of this paper is to prove the following: for a triangulated category $ \underline{C}$ and $E\subset \operatorname{Obj} \underline{C}$ there exists a cohomological functor $F$ (with values in some abelian category) such that $E$ is its set of zeros if (and only if) $E$ is closed with respect to retracts and extensions (so, we obtain a certain Nullstellensatz for functors of this type). Moreover, for $ \underline{C}$ being an $R$-linear category (where $R$ is a commutative ring) this is also equivalent to the existence of an $R$-linear $F: \underline{C}^{op}\to R-\operatorname{mod}$ satisfying this property. As a corollary, we prove that an object $Y$ belongs to the corresponding "envelope" of some $D\subset \operatorname{Obj} \underline{C}$ whenever the same is true for the images of $Y$ and $D$ in all the categories $ \underline{C}_p$ obtained from $ \underline{C}$ by means of "localizing the coefficients" at maximal ideals $p\triangleleft R$. Moreover, to prove our theorem we develop certain new methods for relating triangulated categories to their (non-full) countable triangulated subcategories. The results of this paper can be applied to the study of weight structures and of triangulated categories of motives.

We give a short proof of the tilting theorem by Happel, Reiten and Smalo: two abelian categories A and B are derived equivalent, provided that B is tilted from A.

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.

To any left system of diagram categories or to any left pointed derivateur (in the sense of Grothendieck) a K-theory space is associated. This K-theory space is shown to be canonically an infinite loop space and to have a lot of common properties with Waldhausen's K-theory. A weaker version of additivity is shown. Also, Quillen's K-theory of a large class of exact categories including the abelian categories is proved to be a retract of the K-theory of the associated derivateur., 50 pages

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.

It is shown that there exists an ordered abelian group that has no smallest positive element and that has no sequence of nonzero elements converging to zero. Some formulae for the rank of ordered abelian groups have been derived and a necessary condition for an order type to be rank of an ordered abelian group has been discussed. These facts have been translated to the spectrum of a valuation ring using some well-known results in valuation theory.

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 .

This paper studies twisting functors on the main block of the Bernstein-Gelfand-Gelfand category O \mathcal {O} and describes what happens to (dual) Verma modules. We consider properties of the right adjoint functors and show that they induce an auto-equivalence of derived categories. This allows us to give a very precise description of twisted simple objects. We explain how these results give a reformulation of the Kazhdan-Lusztig conjectures in terms of twisting functors.

The category of finite dimensional (type 1) representations of a quantum affine algebra U q ( g ^ ) U_q(\widehat {{\mathfrak g}}) is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition in a direct sum of indecomposable subcategories (blocks). We define the elliptic central character of a finite dimensional (type 1) representation of U q ( g ^ ) U_q(\widehat {{\mathfrak g}}) and show that the block decomposition of this category is parametrized by these elliptic central characters.

The category of local torsion-free abelian groups of finite rank is known to have the cancellation and n n -th root properties but not the Krull-Schmidt property. It is shown that 10 is the least rank of a local torsion-free abelian group with two non-equivalent direct sum decompositions into indecomposable summands. This answers a question posed by M.C.R. Butler in the 1960’s.

Let H be a connected hereditary abelian category over an algebraically closed field k, with finite dimensional homomorphism and extension spaces. There are two main known types of such categories: those derived equivalent to mod λ for some finite dimensional hereditary k-algebra λ and those derived equivalent to some category cohX of coherent sheaves on a weighted projective line X in the sense of Geigle and Lenzing (1987). The aim of this paper is to give a characterization of the second class in terms of some properties known to hold for these hereditary categories.

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.

For spaces with a group action, we introduce Bredon cohomology with local (or twisted) coefficients and show that it is invariant under weak equivariant homotopy equivalence. We use this new cohomology to construct a Serre spectral sequence for equivariant fibrations. Bredon (1) introduced what is now called Bredon cohomology, with the pur- pose of developing obstruction theory in the context of spaces equipped with an action of a fixed group G. The kind of coefficients needed in this theory are not abelian groups but rather contravariant functors from the orbit category tf (tr) into abelian groups. The purpose of this paper is to prove the existence of a spectral sequence for a Serre fibration of (7-spaces. As in the nonequivariant case, if no further restrictions are made, it is necessary to use cohomology with twisted coefficients. Such twisted coefficients are not functors on the orbit category cf(G), but on some augmentation of cf (G) depending on the base space of the fibration. Our approach is to give a new definition of Bredon cohomology in terms of cohomology of categories. Indeed, with a G-space X we shall associate a category AG(X) of "equivariant singular simplices in X." Any abelian group- valued functor Af from the orbit category cf(G) can be viewed as a system of coefficients on the category AG(X). A key result is that for any such Af, the cohomology groups H*(AGX, M) of this category are naturally isomorphic to the Bredon cohomology groups HG(X, M); see Theorem 2.2. The category AG(X) allows us to define cohomology groups of a G-space with more general coefficients. In particular, we shall construct a category HG(X) of "equivariant homotopy classes of paths in X," which sits in between AG(X) and cf (G) by functors AG(X) -► UG(X) -► tf(G). A twisted or local system of coefficients Af on X is then defined as an abelian group-valued functor on HG(X), and the Bredon cohomology of X with twisted coefficients can now be constructed as the cohomology H*(AG(X), M) of the category AG(X). This is invariant under weak G-homotopy equivalence, see Theorem 2.3. (The converse is also true: a map of G-spaces is a weak G-homotopy equivalence whenever it induces an equivalence of fundamental groupoids as well as an

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.

We give an appropriate extension of the concept of “tower of surjections” to arbitrary inverse systems. We introduce a natural closed model structure (in the sense of D. Quillen) on the category of pro-(Simplicial Abelian Groups) and interpret our condition as the definition of fibrant object.

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) .

By choosing the mappings for a category of l l -groups carefully, some significant categorical results are obtainable. The intent of this paper is to investigate the reasons why epimorphisms in the category of l l -homomorphisms between abelian l l -groups are not “onto” functions.

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.

The projective dimension of an abelian group with torsion reduced part, as a module over its endomorphism ring, is determined. In particular, a group of projective dimension 2 2 is exhibited.

In this note we present a complete set of quasi-isomorphism invariants for strongly indecomposable abelian groups of the form G = G ( A 1 , … , A n ) G = G({A_1}, \ldots ,{A_n}) . Here A 1 , … , A n {A_1}, \ldots ,{A_n} are subgroups of the rationals Q Q and G G is the kernel of f : A 1 ⊕ ⋯ ⊕ A n → Q f:{A_1} \oplus \cdots \oplus {A_n} \to Q , where f ( a 1 , … , a n ) = Σ a i f({a_1}, \ldots ,{a_n}) = \Sigma {a_i} . The invariants are the collection of numbers rank ∩ { G [ σ ] | σ ∈ M } {\text {rank}} \cap \{ G[\sigma ]|\sigma \in M\} , where M M ranges over all subsets of the type lattice generated by { type ( A i ) } \left \{ {{\text {type}}({A_i})} \right \} . Our results generalize the classical result of Baer for finite rank completely decomposable groups, as well as a result of F. Richman on a subset of the groups of the form G ( A 1 , … , A n ) G({A_1}, \ldots ,{A_n}) .

We deal with objects and morphisms in an abelian category, e.g., with modules and module-homomorphisms. Any morphism a: A -+B has a standard factorization a=aamae, where am is a monomorphism and a,e an epimorphism. DEFINITION. A morphism a: A-+B is said to be quasi-invertible if it satisfies any one of the following equivalent conditions2 (i) There is a morphism a': B-+A such that aa'a = a. (ii) There is a morphism a: B--A such that

In his paper On the groups J ( X ) J(X) . IV, Adams suggested that one might try to continue his d d and e e invariants to a sequence of higher homotopy invariants, each defined upon the vanishing of its predecessors and each taking its value in a certain Ext group. Recently he pointed out the efficacy of relocating his d d and e e invariants in Ext groups formed over a certain abelian category of comodules. It is the purpose of this note to carry out the program suggested above in a setting of the sort just mentioned. More specifically, for each homology theory which is representable by a comutative ring spectrum and whose ring of cooperations is flat over the coefficient ring, a sequence of higher homotopy invariants is constructed whose first term is Adams’ e e invariant for this theory.

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.

We construct a category of quotients over a category satisfying a condition similar to the Ore condition. Addition of quotients is briefly discussed. In a previous paper [1 ] a theory of relations in categories of a general type was introduced. In trying to adapt the elegant approach of Hilton [2] to the nonabelian case, a difficulty is encountered, but Hilton's method works in the associative case, in which by [1] an Ore condition holds. In this case a generalization is given which seems to be of interest; it describes a construction of a category which is algebraically a sort of category of quotients. Finally, addition is briefly discussed. Notations and definitions of [1 ] are freely used. Let C be a finitely complete [3] bicategory [4] with classes of monics 4 and epics S. In [1 ] we have constructed a near-category (Re with class of objects I Re I = I eC| and with relations as morphisms. We employ the method of Hilton [2] in the nonabelian case, but we have to use pairs of coinitial morphisms. With fixed objects A, B in C consider the collection (P(A, B) of pairs of morphisms (a, $, AAX 2B, and declare (a, f) -(a', f') in 6P(A, B) if and only if there are a-, a-' in 8 that satisfy ao-=a'o-', AS=3'a'. In [1 ] we have proved that the extension G1e of C is a category if and only if a condition denoted (A) holds, and that condition reads as follows: For every $, -q in C with common codomain and with tCS there is a common right multiple Qu =?7v with vES. I. If (A) holds in e then -is an equivalence in P(A, B). For, if (ao-, Oa) = (uo', Af'o') and (a'r, O ',r) = (a"r', f r') with o&'s and r's in 8 then, by (A) and the pullback condition, there are X, {ES such that o-'c =4; hence r'q, o-kES and (aoc0, fo4) = (a"r'/, flruO. If (A) holds we denote Z(A, B) =6(A, B)/-. Representatives for elements of Z(A, B) can be chosen according to: (a, 3)--(a', f3') in V(A, B) if and only if the s-parts in i-s-factorizations of {a, f } and a', f'}, namely {a, f}i, {a', 3'}i are equivalent monics. (For, if a, NI = {I, t } and {oa', A3'} = {I, q }I' with ~, D'G then there are Received by the editors March 31, 1970. AMS 1969 subject classifications. Primary 1810.

A pseudocompact ring A is a complete Hausdorff topological ring which admits a system of open neighborhoods of 0 consisting of two sided ideals I for which A/I is an Artin ring. A complete HausdorR topological A-module M is said to be pseudocompact if it has a system of open neighborhoods of 0 consisting of submodules N for which M/N has finite length. The category G?,, of pseudocompact (I-modules is an Abelian category with exact inverse limits and enough projectives. Such generalities, which are more or less well known, are gathered in the first section for the convenience of the reader. To get more interesting results, we must introduce some commutativity by assuming that, in addition, (1 is a pseudocompact algebra over a commutative pseudocompact ring Q (see definition in Section 2). We may then define a tensor product on %A and introduce its derived functor 554. The category %+,, of discrete cl-modules, which is dual to VA0 by Proposition 2.3, also plays an important role through the bifunctor Horn : V, x 5BA + go and its derived functor &zt. We then have the proper setting for doing homological algebra. This is done in Section 3 which generalizes the elementary results on homological dimension in complete Noetherian semilocal rings. As an immediate application we find that if s2({xi}} is the algebra of noncommutative formal power series in {xi} over Q, then gl dim J2{{xi}} = gl dim Sa + 1 (Theorem 3.9). We recall that a profinite group is a compact totally disconnected topological group, i.e., an inverse limit of finite groups. We define the complete group

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