Lefschetz Motives and the Tate Conjecture

A Lefschetz class on a smooth projective variety is an element of the Q-algebra generated by divisor classes. We show that it is possible to define Q-linear Tannakian categories of abelian motives using the Lefschetz classes as correspondences, and we compute the fundamental groups of the categories. As an application, we prove that the Hodge conjecture for complex abelian varieties of CM-type implies the Tate conjecture for all Abelian varieties over finite fields, thereby reducing the latter to a problem in complex analysis.     

Almost split sequences in tri-exact categories

We study the existence of almost split sequences in tri-exact categories, that is, extension-closed subcategories of triangulated categories. Our results unify and extend a number of existence theorems for almost split sequences in abelian categories and exact categories (that is, extension-closed subcategories of abelian categories), and those for almost split triangles in triangulated categories by numerous researchers. As applications, we obtain some new results on the existence of almost split triangles in the derived categories of all modules over an algebra with a unity or a locally finite dimensional algebra given by a quiver with relations.     

Differential graded versus simplicial categories

We establish a connection between differential graded and simplicial categories by constructing a three-step zig-zag of Quillen adjunctions relating the homotopy theories of the two. In an intermediate step, we extend the Dold-Kan correspondence to a Quillen equivalence between categories enriched over non-negatively graded complexes and categories enriched over simplicial modules. As an application, we obtain a simple calculation of Simpson's homotopy fiber, which is known to be a key step in the construction of a moduli stack of perfect complexes on a smooth projective variety.     

A Predictive Approach to Tail Probability Estimation

One of the issues contributing to the success of any extreme value modeling is the choice of the number of upper order statistics used for inference, or equivalently, the selection of an appropriate threshold. In this paper we propose a Bayesian predictive approach to the peaks over threshold method with the purpose of estimating extreme quantiles beyond the range of the data. In the peaks over threshold (POT) method, we assume that the threshold identifies a model with a specified prior probability, from a set of possible models. For each model, the predictive distribution of a future excess over the corresponding threshold is computed, as well as a conditional estimate for the corresponding tail probability. The unconditional tail probability for a given future extreme observation from the unknown distribution is then obtained as an average of the conditional tail estimates with weights given by the posterior probability of each model.     

Two acyclicizations of a complex and an explicit recollement

By assigning to each complex over a semi-simple ring two acyclicizations, we construct an explicit recollement for homotopy categories of a certain triangular matrix ring such that all the six triangle functors of the recollement preserve acyclic complexes.     

Bjarni Jónsson’s contributions in algebra: an addendum

Normal categories are pointed categorical counterparts of 0-regular varieties, i.e., varieties where each congruence is uniquely determined by the equivalence class of a fixed constant 0. In this paper, we give a new axiomatic approach to normal categories, which uses self-dual axioms on a functor defined using subobjects of objects in the category. We also show that a similar approach can be developed for 0-regular varieties, if we replace subobjects with subsets of algebras containing 0.     

Cross-connections of linear transformation semigroups

Cross-connection theory developed by Nambooripad is the construction of a regular semigroup from its principal left (right) ideals using categories. We use the cross-connection theory to study the structure of the semigroup \(\textit{Sing}(V)\) of singular linear transformations on an arbitrary vector space V over a field K. There is an inbuilt notion of duality in the cross-connection theory, and we observe that it coincides with the conventional algebraic duality of vector spaces. We describe various cross-connections between these categories and show that although there are many cross-connections, upto isomorphism, we have only one semigroup arising from these categories. But if we restrict the categories suitably, we can construct some interesting subsemigroups of the variants of the linear transformation semigroup.     

Merging experts’ opinions: A Bayesian hierarchical model with mixture of prior distributions

In this paper, a general approach is proposed to address a full Bayesian analysis for the class of quadratic natural exponential families in the presence of several expert sources of prior information. By expressing the opinion of each expert as a conjugate prior distribution, a mixture model is used by the decision maker to arrive at a consensus of the sources. A hyperprior distribution on the mixing parameters is considered and a procedure based on the expected Kullback–Leibler divergence is proposed to analytically calculate the hyperparameter values. Next, the experts’ prior beliefs are calibrated with respect to the combined posterior belief over the quantity of interest by using expected Kullback–Leibler divergences, which are estimated with a computationally low-cost method. Finally, it is remarkable that the proposed approach can be easily applied in practice, as it is shown with an application.     

A Construction of Relatively Pure Submodules

In this paper we reconsider a classical theorem by Bican and El Bashir, which guarantees the existence of non-trivial relatively pure submodules in a module category over a ring with unit. Our aim is to generalize the theorem to module categories over rings with several objects. As an application, we then consider the special case of α-pure objects in such module categories.     

Three results on Frobenius categories

This paper consists of three results on Frobenius categories: (1) we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an extension-closed exact subcategory of the Frobenius category formed by Cohen–Macaulay modules over some additive category; this is an analogue of Gabriel–Quillen’s embedding theorem for Frobenius categories; (3) we show that under certain conditions an exact category with enough projective and enough injective objects allows a natural new exact structure, with which the given category becomes a Frobenius category. Several applications of the results are discussed.