Examples of Auslander–Reiten Quivers

The previous chapter was mainly of a theoretical nature: we defined irreducible morphisms and almost split sequences and started to explore their use for the understanding of the radical of a module category. However, we did not say much about the explicit construction of almost split sequences, even though we pointed out that the proof of Theorem II.3.10 suggests the idea of a construction. Carrying out this construction in practice is quite difficult, and our objective in the present chapter is to explain how it can be done, at least in the easiest cases. In the first section, we prove that the indecomposable end terms of an almost split sequence are related by functors, which are called the Auslander–Reiten translations. In Section III.2, we derive the so-called Auslander–Reiten formulae, which lead us to a second existence proof for almost split sequences. Next, in Section III.3, we show how to apply these results to construct examples of almost split sequences. In the final Section III.4, we relate the Auslander–Reiten translates of a given module over an algebra to that over a quotient algebra.

\n Dada uma aplicação f: B -> A entre duas C*-álgebras, o cone dessa aplicação, Cf, é o conjunto formado pelos pares (b,g) pertencentes à soma direta da C*-álgebra B com o cone CA tais que f(b) = g(0), sendo CA o cone de A. Neste trabalho estudamos o funtor determinado pela associação da sequência exata curta 0 -> SA -> Cf -> B -> 0 para cada *-homomorfismo f: B -> A, e demonstramos que esse funtor é exato. Caracterizamos as aplicações de conexão associadas à sequência exata 0 -> SA -> Cf -> B -> 0, mostrando que a aplicação do índice é dada por tAK1(f) e que a aplicação exponencial é dada por bAK0(f), sendo tA o isomorfismo entre K1(A) e K0(SA) e bA a aplicação de Bott. Por fim, usando que toda sequência exata curta de C*-álgebras pode ser vista na forma 0 -> Ker f -> B -> A -> 0, mostramos que as aplicações de conexão d1 e d0 associadas a cada sequência exata curta podem ser dadas por dn = Kn+1(j)-1 Kn+1(i) hn, em que j é a inclusão do núcleo de f em Cf, i é a inclusão da suspensão SA também em Cf, hn = bA e h1 = tA .\n

Almost split sequences in dimension two

Almost split sequences and Auslander-Reiten quivers (species) have turned out to be an extremely useful tool in the representation theory of artinian algebras and integral representations of complete orders. Whereas the theory is well developed in the artinian situations, it is far behind for integral representations. The main reason is that the powerful tool of universal coverings and covering techniques — as developed by Riedtmann [14], Bongartz — Gabriel [4] — has not yet found its integral analogue. The main difficulty is that it is unclear to me, when a representation quiver — in the sense of C.Riedtmann [14] — is the Auslander-Reiten quiver of an order of finite lattice type; in particular, one obstacle is the multiplication with the parameter of the ground ring. A little progress has been made in this direction by Wiedemann. Apart from my general philosophy, that most of the results in the representation theory of artinian algebras have more or less obvious analoga in integral representation theory, there is a more solid connection between integral representations and some categories of torsionfree modules over artinian algebras as observed by C.M.Ringel and the author [15], such that in some instances integral representation theory might also influence the representation theory of artinian algebras — e.g. that some torsionfree categories of artinian modules have almost split sequences was — before its general treatment by M.Auslander and S.O.Smalo [1] — in secial cases — influenced by the results in integral representation theory — noted by the author [17] and independently by R.Bautista and R.Martinez [3].

G-semisimple algebras

Let $m \ge 2$ be an integer. By using $m$ submodules of a given module, we construct a certain exact sequence, which is a well known short exact sequence when $m=2$. As an application, we compute a minimal projective resolution of the Jacobson radical of

Let F be a number field, O the ring of algebraic integers in F, and let θ denote the inclusion O ↪ F. The localization exact sequence in algebraic K-theory splits into short exact sequences $$ 0 \to K_n O\xrightarrow{{\theta _\# }}K_n F \to \oplus _m K_{n - 1} \left( {O/m} \right) \to 0 $$ for all positive integers n, where θ # is the homomorphism induced by θ in K-theory and where m runs over the set of maximal ideals of O (see Section 5 of [Q1], Theorem 8 of [Q2] and Theoreme 1 of [S2]); in particular, θ # is always injective. On the other hand, G. Banaszak investigated the subgroup of divisible elements in K n F and explained the important role of these elements in relation with the Lichtenbaum-Quillen conjecture and etale K-theory (see [B1], [B2], [BG], [BZ]). If n is odd, K n F is a finitely generated abelian group and has therefore no non-trivial divisible elements. If n is even, K n F is a large torsion group but all its divisible elements belong to the image of θ # because ⊕ m K n-1 (O/m) is a direct sum of finite cyclic groups and hence contains no non-trivial divisible elements.

Higher homological algebra was introduced by Iyama. It is also known as $n$-homological algebra where $n \geq 2$ is a fixed integer, and it deals with $n$-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with $n+2$ objects. This was recently formalised by Jasso in the theory of $n$-abelian categories. There is also a derived version of $n$-homological algebra, formalised by Geiss, Keller, and Oppermann in the theory of $( n+2 )$-angulated categories (the reason for the shift from $n$ to $n+2$ is that angulated categories have triangulated categories as the base case). We introduce torsion classes and t-structures into the theory of $n$-abelian and $( n+2 )$-angulated categories, and prove several results to motivate the definitions. Most of the results concern the $n$-abelian and $( n+2 )$-angulated categories ${\mathcal M}( \Lambda )$ and ${\mathcal C}( \Lambda )$ associated to an $n$-representation finite algebra $\Lambda$, as defined by Iyama and Oppermann. We characterise torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in ${\mathcal M}( \Lambda )$ and intermediate t-structures in ${\mathcal C}( \Lambda )$ which is a category one can reasonably view as the $n$-derived category of ${\mathcal M}( \Lambda )$. We hint at the link to $n$-homological tilting theory.

The theory of representations of the Euclidean algebra and group in the planeE2 is investigated from a very general and basic point of view. A master representation is obtained for both the algebra as well as for the group. This master representation induces (subduces) all the representations ofE2 which are studied in this paper. The master representation is defined on the space of the universal enveloping algebra Ω ofE2, whose basis vectors are labelled by three (discrete) parameters. The finite-dimensional (linear, indecomposable) representations which are induced by the master representation are classified, both for the algebra and for the group. The bases for these representations are given and for the group the finite matrix elements. While the result regarding the classification of thefinite-dimensional indecomposable representations has been published, without proofs, we give here the proofs of the theorems. Moreover, the three- and four-dimensional representations of the Euclidean algebra and group, which are induced by the master representation, are obtained in this paper in explicit form.Infinite-dimensional indecomposable representations which are induced by the master representation are discussed at the level of both the algebra and the group. The familiar result of the infinite-dimensional irreducible representations is obtained as a final, and rather special, case of representations induced by the (indecomposable) master representation. The matrix elements for all representations of the groupE2 which are discussed here are given in explicit form. Due to the particular approach taken in this paper, relations among functions representing matrix elements in different representations are obtained.

A given R-module is called a right np-injective module if for any non-nilpotent element of , and any right R-homomorphism can be extended to , if is np-injective, then is a right np-injective ring. A given ring is called right weakly np-injective if for each non-nilpotent element of , there exists a positive integer such that any right R-homomorphism can be extended to . A given ring is a right weakly np-injective ring, if . In the matrix ring, we poved that is right np-injective, for some , if is right Weakly np-injective. So, We extended many of the known properties and characterizations of right np-injective rings and modules. Finally, the main result in np-injective module found that the R-module is np-injective module if and only if for any the short exact sequence of R-modules, is also a short exact sequence, where and .

In a preceding article~[7] the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above mentioned resolutions can be realized topologically. In fact there exists a family of cofibrations inducing short exact sequences in mod 2 cohomology. The resolutions above are obtained by splicing together these short exact sequences. Thus the injective resolutions are realizable in the best possible sense. In fact our construction appears to be in some sense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically (not only homotopically) certain dual Brown-Gitler spectra.

On minimal disjoint degenerations of modules over tame path algebras

Algebraic computation of short exact sequence and topological invariants using K-theory in general ring of classification spaces

Let A and B be finitely generated profinite groups. If G is isomorphic to A × B, then any short exact sequence 1 → A → G → B → 1 splits as a direct product.

A homological approach to the splitting theory of [formula omitted] spaces