Triple Cohomology and the Galois Cohomology of Profinite Groups

Given a cotriple 𝔾 = (G, ε, δ) on a category X and a functor E:X ^Opp→A into an abelian category A, there exists the cohomology theory of Barr and Beck: Hⁿ(X, E) ε |A| (n ≥ 0, X ε |X|), ([1], p.249). Almost all the important cohomology theories in mathematics have been shown to be special instances of such a general theory (see [1], [2] and [3]). Usually E arises from an abelian group object Y in X in the following manner: it is the contravariant functor from X into the category Ab of abelian groups that associates to each object X in X the abelian group X(X, Y) of maps from X to Y. In such a situation we shall write Hⁿ(X, Y)_𝔾 instead of Hⁿ(X, E)_G. Barr and Beck [2] have shown that the Eilenberg-MacLane cohomology groups H?ⁿ(π, A), n ≥ 2, can be re-captured as follows. One considers the free group cotriple 𝔾′ on the category Gps of groups, which induces in a natural manner a cotriple 𝔾 on the category (Gps, π) of groups over a fixed group π.     

A Characterization of Group Rings

In this paper we prove that every coseparable involutory Hopf algebra over the ring of integers Z which is a free Z-module is the group ring of some group. This result was proved independently for Hopf algebras which are finitely generated Z-modules by H.-J. Schneider [6], using similar techniques. We then give some examples of coseparable Hopf algebras over number rings which are not group algebras, and give an example of a cocommutative coseparable coalgebra over a number ring which cannot be given a multiplicative structure making it into a Hopf algebra. The Hopf algebra structure theory required for this paper is found in [1], [4], and [5]. For completeness we give proofs here of the coalgebra analogues to some “well-known” facts about separable algebras.     

EPIREFLECTIVE SUBCATEGORIES AND SEMICLOSURE OPERATORS

ABSTRACT

Let C be a category of topological spaces and continuous functions which is full, hereditary and closed under homeomorphisms and products. If A is a subclass of C, let E(A) be the full subcategory of C whose objects are the subspaces in A. In this paper we characterize the epireflective subcategories of C containing A and contained in E(A) by introducing a “semiclosure” operator which is a generalization for the “idempotent semi-limit” operator introduced by S.S. Hong (see [5]) with respect to Top _o. In case A is extensive in C, so that E(A) = C, all the extensive subcategories of C containing A are thus characterized.     

TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS

Abstract

It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F _A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces.     

LIGHT FACTORIZATION STRUCTURES

Abstract

In this paper we investigate, for connection subcategories A of a topological category K, the concepts of A-monotone quotients and A-light sources, and characterize (1) those A, which give rise to an (A-monotone quotient, A-light)- factorization structure on K, (2) those factorization structures (C,D) on K, which are light, i.e. of the form (A-monotone quotient, A-light) for suitable A. It turns out that light factorization structures are rather rare in Top, but abundant and well-behaved in categories with hereditary quotients.     

DISCRETE FUNCTORS

Abstract

The concept of a T-discrete object is a generalization of the notion of discrete spaces in concrete categories. In this paper. T-discrete objects are used to define discrete functors. Characterizations of discrete functors are given and their relation to other important functors are studied. A faithful functor T: A → X is discrete iff the full subcategory B of A consisting of all T-discrete objects is (X-iso)-coreflective in A. It follows that the existence of bicoreflective subcategories is equivalent to the existence of suitable discrete functors. Finally, necessary and sufficient conditions are found such that for a given functor T: A → X, the full subcategory B of A consisting of all T-discrete A-objects is monocoreflective in A.     

ANOTHER CHARACTERIZATION OF BICOREFLECTIVE SUBCATEGORIES

Abstract

In this paper, the relation between the notion of a discrete functor (see [4]) and the notion of a fine functor (see [1]) is examined. As a generalization of the notion of a F-fine object (see [1]), discrete functors T: A → X are used to define K-fine objects, where K is a class of A-objects. It is shown that if T is in addition semi-topological, then (as for F-fine objects in a topological category, see [1]) the class of K-fine objects determines a bicoreflective subcategory of A. Moreover, it is shown that in co-complete, co-(well-powered) categories, the existence of bicoreflective subcategories is equivalent to the existence of functors that are both discrete and semi-topological.     

EPIREFLECTIONS VERSUS BIREFLECTIONS FOR TOPOLOGICAL FUNCTORS

Abstract

In this paper it is proved that if T: A → X is a topological functor satisfying certain conditions, then there is a Galois Connection between the class of bireflective subcategories of A and the class of epireflective subcategories of A that are not bireflective and that are contained in the subcategory of separated objects of A. In general such a correspondence is not bijective.     

Topological Rings of Quotients and Rings Without Open Left Ideals

Simple locally compact rings without open left ideals were considered in [13] and general locally compact rings without open left ideals were studied extensively in [5] and [6]. We remove the hypothesis of local compactness and consider topological rings A without open left ideals but containing an open subring R. In section 4 we show that under these conditions A is completely determined by R. More precisely A can be identified with the topological ring of quotients C(R) introduced in [8]. As an R-module _RA is topologically isomorphic to I_*(_RR), the topological injective hull of _RR. The last statement was proved in [6] for A locally compact and R compact. Section 5 gives a characterization of those linearly topologized rings R that can be openly embedded into a ring A without open left ideals. In particular we shall show that the open left ideals form an idempotent ideal filter with quotient ring A. In section 6 we consider the class ? of all topological rings that can be openly embedded into a topological ring without open left ideals. If we restrict our attention to linearly topologized rings, then ? is Morita-invariant. In section 2 we construct a topological ring of quotients Q_*(R) and prove that it coincides with the ring C(R) of [8].     

On Going Down For Simple Overrings II

This paper deals with two themes involving the going down (GD) behavior of the overrings of a domain R. First, as illustrated by several results in [3] and [4], hypotheses about GD often replace stronger assumptions about flatness or one-dimensionality in sets of conditions implying that R is Prüfer. Second, as shown by the two-dimensional example in [4]. Remarks (iii)], some sort of finiteness condition is required in conjunction with GD hypotheses in order for R to be Prüfer.     

VARIETAL HULLS OF FUNCTORS

Abstract

The well known characterizations of equational classes of algebras with not necessaryly finitary operations by FELSCHER [6.7] and of categories of A-algebras for algebraic theories A in the sense of LINTON [10], esp., by means of their forgetful functors are the foundations of a concept of varietal functors U:K → L over arbitrary basecategories L. They prove to be monadic functors which satisfy an additional HOM-condition [17]. (In the case L = Set this condition is always fulfilled, see LINTON [11].)

Contrary to monadic functors, varietal functors are closed under composition. Pleasent algebraic properties of the base-category L can be ‘lifted’ along varietal functors, such as e.g. factorization properties, (co-) completeness, classical isomorphism theorems, etc.

By means of the well known EILENBERG-MOORE-algebras there is a universal monadic functor U^T:L ^T → L for any functor U: K → L, having a left adjoint F (T: = UF). But, in general, UT is not varietal. Under some suitable conditions, however it is possible, to construct a canonical varietal functor ?:R → L, the varietal hull of U. This hull has much more interesting (algebraic) properties than the EILENBERG-MOORE construction. Moreover, results of BANASCHEWSKI-HERRLICH [2] are extended.     

Galois Objects of Finitely Generated Projective Hopf Algebras

Let C be the category of cocommutative coalgebras over a commutative ring R and let H be a group object in C, i.e., let H be a cocommutative Hopf algebra. Assume that H is a finitely generated, projective R-module and that the integrals (of [4]) in H* ≡ Hom_R(H, R) are cocommutative elements. We will show that any Galois H-object (as defined in [3, Def. 1.2, p. 8]) is a finitely generated, projective R-module.     

Down-Up Algebras From Trees

It is shown that the global dimension of any n-ary down-up algebra A _n  = A(n,α, β,γ) is less than or equal to n + 2, and when γ _i  = 0 for all i (A _n is graded by total degree in the generators), then the global dimension of A _n is n + 2. Furthermore, a sufficient condition for A _n to be prime is given; when γ _i  = 0 for all i this condition is also necessary. An example is given to show that the condition is not always necessary.     

Preprojective Cluster Variables of Acyclic Cluster Algebras

It is proved that any cluster-tilted algebra defined in the cluster category 𝒞(H) has the same representation type as the initial hereditary algebra H. For any valued quiver (Γ, Ω), an injection from the subset 𝒫?(Ω) of the cluster category 𝒞(Ω) consisting of indecomposable preprojective objects, preinjective objects, and the first shifts of indecomposable projective modules to the set of cluster variables of the corresponding cluster algebra 𝒜_Ω is given. The images are called “preprojective cluster variables”. It is proved that all preprojective cluster variables other than u_i have denominators u ^dim M in their irreducible fractions of integral polynomials, where M is the corresponding preprojective module or preinjective module. In case the valued quiver (Γ, Ω) is of finite type, the denominator theorem holds with respect to any cluster. Namely, let x = (x₁,…, x_n) be a cluster of the cluster algebra 𝒜_Ω, and V the cluster tilting object in 𝒞(Ω) corresponding to x, whose endomorphism algebra is denoted by Λ. Then the denominator of any cluster variable y other than x_i is x ^dim M, where M is the indecomposable Λ-module corresponding to y. This result is a generalization of the corresponding result of Caldero–Chapoton–Schiffler to the non-simply-laced case.     

THE LARGEST NON-TRIVIAL TORSION CLASS OF MODULES

ABSTRACT

In this paper we investigate the following two classes of left R-modules: N(P) ={A|A has no non-zero direct summand P ε P} and H(p) = {A} if B ? A with B ε N(P), then B = 0}, where P is a class of projective R-modules. We demonstrate that N(p) is, in general, not a torsion class but that H(P) is always a torsionfree class. We also investigate those classes P and rings R for which N(P) is the largest non-trivial torsion class of R-modules.     

Twisted flag varieties of trialitarjan groups

For (A, σ) a central simple algebra of even degree with orthogonal involution, we present a method for constructing isotropic right ideals in the even Clifford algebra (C ₀(A, σ)σ) from isotropic right ideals in (A, σ). We then use this construction to fully describe the twisted flag varieties associated to algebraic groups of type D ₄ (including the trialitarian groups).     

Endomorphism Algebras of 2-term Silting Complexes

We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra A whose global dimension gl.dim A ≤ 2 and any 2-term silting complex P in the bounded derived category D ^b (A) of A, the global dimension of \(\text {End}_{{D^b(A)}}(\mathbf {P})\) is at most 7. We also show that for each n > 2, there is an algebra A with gl.dim A = n such that D ^b (A) admits a 2-term silting complex P with \(\mathrm {gl. dim~}\text {End}_{{D^b(A)}}(\mathbf {P})\) infinite.     

On the Factoriality of Local Rings of Small Embedding Codimension

Let A be a local ring of dimension d. If A is a quotient of a regular local ring of dimension n = d+r, then we say that A has embedding codimension ≤ r. This paper investigates some special properties of local rings of small embedding codimension. The main idea is to exploit a result in [17], which says that local rings of small embedding codimension and depth ≥ 3 are parafactorial. This tells us, with suitable additional hypotheses, that the ring is factorial, or Gorenstein, or even a complete intersection.     

An enumeration algorithm for combinatorial problems of the reliability analysis of binary coherent systems

In this paper the problem of the generation of all elements of a system of sets is investigated and a backtrack algorithm (Al) is given solving this problem.The algorithm is applied to combinatorial problems of reliability:

A t-graph is interpreted as a binary coherent system S and the system function of S is represented as the sum of Boolean orthogonal products. This approach is a simple and efficient alternative to a method suggested by Satyanarayana and N. Hagstbom [9].

A directed p-graph is also interpreted as a binary coherent system S and both the minimal cuts of S and the cocycles of G are determined.