Split Monomorphism Research Articles

On Irreducible Morphisms and Auslander-Reiten Triangles in the Stable Category of Modules over Repetitive Algebras

Let ${\Bbbk }$ be an algebraically closed field, let Λ be a finite dimensional ${\Bbbk }$ -algebra, and let $\widehat {\Lambda }$ be the repetitive algebra of Λ. For the stable category of finitely generated left $\widehat {\Lambda }$ -modules $\widehat {\Lambda }$ -mod, we prove that every Auslander-Reiten triangle in $\widehat {\Lambda }$ -mod is induced from an Auslander-Reiten sequence of finitely generated left $\widehat {\Lambda }$ -modules. We use this fact to show that the irreducible morphisms in $\widehat {\Lambda }$ -mod fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. Finally, we use this classification to describe the shape of the Auslander-Reiten triangles in $\widehat {\Lambda }$ -mod.

A convenient category of locally stratified spaces

In this thesis we define the notion of a locally stratified space. Locally stratified spaces are particular kinds of streams and d-spaces which are locally modelled on stratified spaces. We construct a locally presentable and cartesian closed category of locally stratified spaces that admits an adjunction with the category of simplicial sets. Moreover, we show that the full subcategory spanned by locally stratified spaces whose associated simplicial set is an ∞-category has the structure of a category with fibrant objects. We define the fundamental category of a locally stratified space and show that the canonical functor θ_A from the fundamental category of a simplicial set A to the fundamental category of its realisation is essentially surjective. We show that the functor θ_A sends split monomorphisms to isomorphisms, in particular we show that θ_A is not necessarily an equivalence of categories. On the other hand, we show that the fundamental category of the realisation of the simplicial circle is equivalent to the monoid of the natural numbers. To conclude, we define left covers of locally stratified spaces and we show that, under suitable assumptions, the category of representations of the fundamental category of a simplicial set is equivalent to the category of left covers over its realisation.

The hearts of weight structures are the weakly idempotent complete categories

This paper proves that additive categories that occur as hearts of weight structures are precisely the weakly idempotent complete categories, that is, the categories where all split monomorphisms give direct sum decompositions. The work also gives several other conditions equivalent to weak idempotent completeness (some of them are completely new) and discusses weak idempotent completions of additive categories

Regular morphisms in abelian categories

We establish some properties involving regular morphisms in abelian categories. We show a decomposition theorem on the image of a regular sum of morphisms, a characterization of regular morphisms in terms of consecutive pairs of morphisms, and a description of certain equivalent morphisms. We also generalize Ehrlich’s Theorem on one-sided unit regular morphisms by showing that if [Formula: see text] is an [Formula: see text]-regular object, then a morphism [Formula: see text] is left (right) unit regular if and only if there exists a split monomorphism (epimorphism) [Formula: see text]. We also study regular morphisms determined by generalized inverses in additive categories.

Symmetric cohomology of groups

We investigate the relationship between the symmetric, exterior and classical cohomologies of groups. The first two theories were introduced respectively by Staic and Zarelua. We show in particular, that there is a map from exterior cohomology to symmetric cohomology which is a split monomorphism in general and an isomorphism in many cases, but not always. We introduce two spectral sequences which help to explain the relationship between these cohomology groups. As a sample application we obtain that symmetric and classical cohomologies are isomorphic for torsion free groups.

English

In this paper, we prove that any pure submodule of a strict Mittag-Leffler module is a locally split submodule. As applications, we discuss some relations between locally split monomorphisms and locally split epimorphisms and give a partial answer to the open problem whether Gorenstein projective modules are Ding projective.

Irreducible Morphisms Between Modules over a Repetitive Algebras

We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra.

English

We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category F such that the homotopy category of this model structure is equivalent to the stable category F as triangulated categories. This seems to be well-accepted by experts but we were unable to find a complete proof for it in the literature. When F is a weakly idempotent complete (i.e., every split monomorphism is an inflation) Frobenius category, the model structure we constructed is an exact (closed) model structure in the sense of Gillespie (2011).

K-theory for the tame C⁎-algebra of a separated graph

A separated graph is a pair ( E , C ) consisting of a directed graph E and a set C = ⨆ v ∈ E 0 C v , where each C v is a partition of the set of edges whose terminal vertex is v . Given a separated graph ( E , C ) , such that all the sets X ∈ C are finite, the K-theory of the graph C ⁎ -algebra C ⁎ ( E , C ) is known to be determined by the kernel and the cokernel of a certain map, denoted by 1 C − A ( E , C ) , from Z ( C ) to Z ( E 0 ) . In this paper, we compute the K-theory of the tame graph C ⁎ -algebra O ( E , C ) associated to ( E , C ) , which has been recently introduced by the authors. Letting π denote the natural surjective homomorphism from C ⁎ ( E , C ) onto O ( E , C ) , we show that K 1 ( π ) is a group isomorphism, and that K 0 ( π ) is a split monomorphism, whose cokernel is a torsion-free abelian group. We also prove that this cokernel is a free abelian group when the graph E is finite, and determine its generators in terms of a sequence of separated graphs { ( E n , C n ) } n = 1 ∞ naturally attached to ( E , C ) . On the way to showing our main results, we obtain an explicit description of a connecting map arising in a six-term exact sequence computing the K-theory of an amalgamated free product, and we also exhibit an explicit isomorphism between ker ⁡ ( 1 C − A ( E , C ) ) and K 1 ( C ⁎ ( E , C ) ) .

Irreducible morphisms in the bounded derived category

We study irreducible morphisms in the bounded derived category of finitely generated modules over an Artin algebra Λ , denoted D b ( Λ mod ) , by means of the underlying category of complexes showing that, in fact, we can restrict to the study of certain subcategories of finite complexes. We prove that as in the case of modules there are no irreducible morphisms from X to X if X is an indecomposable complex. In case Λ is a selfinjective Artin algebra we show that for every irreducible morphism f in C b ( Λ proj ) either f j is split monomorphism for all j ∈ Z or split epimorphism, for all j ∈ Z . Moreover, we prove that all the non-trivial components of the Auslander–Reiten quiver of C b ( Λ proj ) are of the form Z A ∞ .

Semidirect products of internal groupoids

We give a characterization of those finitely complete categories with initial object and pushouts of split monomorphisms that admit categorical semidirect products. As an application we examine the case of groupoids with fixed set of objects. Further, we extend this to the internal case.

Pure morphisms in pro-categories

Pure epimorphisms in categories pro- C , which essentially are just inverse limits of split epimorphisms in C , were recently studied by J. Dydak and F.R.R. del Portal in connection with Borsuk’s problem of descending chains of retracts of ANRs. We prove that pure epimorphisms are regular epimorphisms whenever C has weak finite limits, or pullbacks, or copowers. This improves the results of the above paper, and the results of the present authors on pure subobjects in accessible categories. We also turn to pure monomorphisms in pro- C , essentially just inverse limits of split monomorphisms in C , and prove that they are regular monomorphisms whenever C has finite products or pushouts.

On Pure Quotients and Pure Subobjects

In the theory of accessible categories, pure subobjects, i.e. filtered colimits of split monomorphisms, play an important role. Here we investigate pure quotients, i.e., filtered colimits of split epimorphisms. For example, in abelian, finitely accessible categories, these are precisely the cokernels of pure subobjects, and pure subobjects are precisely the kernels of pure quotients.

On pure morphisms in accessible categories

Pure morphisms in locally presentable categories were characterized as directed colimits of split monomorphisms, and they are regular monomorphisms. We prove that both results hold in accessible categories with pushouts, but not in general accessible categories.

Surgery up to homotopy equivalence for nonpositively curved manifolds

Let M' be a smooth closed manifold which admits a metric of nonpositive curvature. We show, using a theorem of Farrell and Hsiang, that if n + k ? 6, then the surgery obstruction map [M X D , a; G/TOP] Lh+k(7TIM, w1(M)) is injective, where Lh are the obstruction groups for surgery up to homotopy equivalence. Farrell and Hsiang have shown that if Mn is a closed nonpositively curved Riemannian manifold and n + k > 6, then the surgery map [Mn X Dk , a; G/TOP, *] -*Ln+k(7T1M, wI(M)) for topological surgery in a split monomorphism [1]. This note considers the implications of their result of h-surgery, i.e. surgery up to homotopy equivalence rather than simple homotopy equivalence. LEMMA. Let QP be a compact manifold such that Os: [QP, a; G/TOP, *] Ls(7T,Q, wl(Q)) is monic, let Nn be a compact, connected submanifold of QP which has a normal microbundle vQ(N), and let f: UP -* QP be a simple homotopy equivalence which is a homeomorphism near the boundary. Then any normal map f Iv: V= f -(N) N induced by making f transverse to N is normally cobordant to a homeomorphism. PROOF. Recall that the long exact sequence of topological surgery sa(QP x Dka)i DQP X Dk a; G/TOP, *] -4LP+k(lQ, WI(Q)) 8a(QP X Dk ) -Q, _) [Q a; G/TOP, * ] -LP(7T1Q, WI(Q)) (where a = s or h denotes surgery up to simple homotopy equivalence or up to homotopy equivalence, respectively) is a long exact sequence of groups and group homomorphisms (see [4]; also see [2] for surgery in the topological category). Let x = [ f: U Q] E Ss(Q, a) and let i: N Q be the inclusion map. Then the element [f Iv: V N] E [N, a; G/TOP, * ] defined above is i*(,qs(x)), where i*: [Q, a; G/TOP, * ] -* [N, a; G/TOP, * ]. Since Os is a monomorphism, image('q5) = 0 in [Q, a; G/TOP, *], and since i* is a homomorphism of groups we see that Received by the editors December 15, 1982 and, in revised form, August 15, 1983. 1980 Mathematics Subject Classification. Primary 57R67, 57R65. C1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page

Finite dimensional hereditary algebras of wild representation type

Let R be a finite dimensional hereditary algebra. We are concerned with the problem of determining the indecomposable R-modules of finite length. This problem completely has been solved in the case when R is of finite or of tame representation type, but seems to be rather hopeless in the case of wild representation type. In this situation, the only known classes of modules are the socalled pre-projective and the pre-injective ones. The remaining indecomposable modules are called regular. In this paper, we want to initiate the study of the regular modules. The result we obtain seems to be rather surprising: we will show that the regular modules behave rather similar to modules over a serial algebra. In order to state the main theorem, we need the notion of an irreducible homomorphism, which was introduced by Auslander and Reiten [-3]. Let X and Y be two non-zero R-modules. A homomorphism f : X ~ Y is said to be irreducible, if it is neither a split monomorphism, nor a split epimorphism, and if for any factorisation X ~ I ~ Y o f f , either f is a split monomorphism o r f is a split epimorphism. Note that an irreducible homomorphism is always either a monomorphism or an epimorphism. A non-zero R-module S will be called quasisimple, ifS is regular, and there is no irreducible monomorphism of the form U ~ S with U non-zero. In this case, we will call the map 0 ~ S irreducible.