The direct product theorem for profinite groups

La structure profinie du groupe des unités d'un anneau de séries entières à coefficients dans un anneau fini

\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

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.

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.

In this chapter we introduce the differential module for a group homomorphism and show the Crowell exact sequence associated to a short exact sequence of groups. Applying these constructions to the Abelianization map of a link group, we obtain the Alexander module of a link and the exact sequence relating the Alexander module with the link module. We then discuss analogous constructions for pro-finite (pro-l) groups to obtain the complete differential module and the complete Crowell exact sequence. Applying these constructions to a homomorphism from a Galois group with restricted ramification, we obtain the complete Alexander module for primes and the exact sequence relationg the complete Alexander module with a Galois (Iwasawa) module.

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

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

In this chapter we investigate to what extent the closed range theorem and its consequences can be carried over to Fréchet spaces. This question is of great interest in view of the duality of subspaces and quotients, as well as for many applications in analysis. We begin with a useful surjectivity criterion which generalizes corollary 9.5.

Unitary extensions of Hilbert <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>A</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>-modules split

Models of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:msub></mml:math> over a discrete valuation ring

In a recent paper [7], Erdmann has calculated Ext1G between Weyl modules for SL2. In this paper we generalize this result to solve the corresponding problem for quantum GL2 as defined by Dipper and Donkin in [2]. We also show how our result also holds for the Manin quantization. To apply the methods of [7], it is necessary to determine the block structure of quantum GL2, so the first main result of this paper is a description of this, derived from the analysis of the subcomodule structure of the symmetric powers in [10].After an initial section of generalities, the next section consists of the determination of the block structure. We also need a quantum analogue of two short exact sequences from [11], which we give in the following section. With these results, the argument now follows much as in [7]; we consider the infinitesimal case, and then use the Lyndon–Hochschild–Serre spectral sequence to obtain the desired result. Finally we show how the result also holds for the Manin quantization.It should be noted that the result here uses the classical case, so is not independent of that in [7]. The only real difference in the arguments used occurs in Lemma 4·8 where the original methods do not generalize, so we use a more direct argument. There is also an unfortunate typographical error in the statement of the main result in [7].