Galois extensions as functors of comodules

Let A be a finite Hopf algebra over a commutative ring k. We show a one-to-one correspondence between the A-Galois extensions of k and certain functors from the category of A-comodules to the category of k-modules.     

A note on coinduction functors between categories of comodules for corings

In this note we consider different versions of coinduction functors between categories of comodules for corings induced by a morphism of corings. In particular we introduce a new version of the coinduction functor in the case oflocally projective corings as a composition of suitable “Trace” and “Hom” functors and show how to derive it from a moregeneral coinduction functor between categories of type σ[M]. In special cases (e.g. the corings morphism is part of a morphism of measuringa-pairings or the corings have the same base ring), a version of our functor is shown to be isomorphic to the usual coinduction functor obtained by means of the cotensor product. Our results in this note generalize previous results of the author on coinduction functors between categories of comodules for coalgebras over commutative base rings.     

On subgroup functors of finite soluble groups

The principal aim of this paper is to study the regular and transitive subgroup functors in the universe of all finite soluble groups. We prove that they form a complemented and non-modular lattice containing two relevant sublattices. This is the answer to a question (Question 1.2.12) proposed by Skiba (1997) in the finite soluble universe.     

Fitting functors and radicals of finite groups

We develop methods for recognizing the Fitting classes and radicals of finite groups by Fitting functors and the prescribed properties of Hall π-subgroups.     

Certain homotopic properties of functors with finite supports

It is proved that Shchepin-normal functors with finite supports in the category of compacta are continuous in the topology of homotopically n-regular Kuratowski convergence in the space of LCⁿ-subsets of a metric compactum. These functors also preserve the property of mappings of polyhedra to induce an isomorphism of homotopic groups of dimension not greater than n.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1618–1620, November, 1992.     

The property O n for finite seminormal functors

We give a finite combinatorial test for finite seminormal functors to possess the property O _n and use it in establishing that in some cases this property leads to some well-known functors. For example, if some functor F possesses the property O ₂ then F ₂ coincides with either exp₂ or the squaring functor. Hence we conclude that if F(D ^ω ₁) and D ^ω ₁ are homeomorphic then F ₂ is either exp₂ or (·)².     

Characterization of modules of finite projective dimension via Frobenius functors

Let M be a finitely generated module over a local ring R of characteristic p > 0. If depth(R) = s, then the property that M has finite projective dimension can be characterized by the vanishing of the functor Extⁱ_R(M, ^fnR){{\rm Ext}^i_R(M, ^{f^n}R)} for s + 1 consecutive values i > 0 and for infinitely many n. In addition, if R is a d-dimensional complete intersection, then M has finite projective dimension can be characterized by the vanishing of the functor Extⁱ_R(M, ^fnR){{\rm Ext}^i_R(M, ^{f^n}R)} for some i ≥ d and some n > 0.     

Decomposition of comodules

In this note we develop a decomposition theory of comodules over a general coalgebra. We relate it with other known decomposition theories and apply it to the decomposition of categories.     

Decompositions of comodules

In this paper we study the relations among the different decomposition theories of comodules.     

Constructing unconditional finite dimensional decompositions

Primariness of a Banach space is almost always obtained through the use of the Pelczynski decomposition method. In this paper we show that it is possible to directly construct UFDD’s in many cases from which the primariness can be deduced. We give applications tol _p andX _p. Research supported in part by NSF grant DMS-8602395.     

Groups acting on finite dimensional spaces with finite stabilizers

It is shown that every H -group G of type admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X ^H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a -module which is projective as a -module for all finite then M has finite projective dimension. Received: April 30, 1997     

Strict polynomial functors and coherent functors

We build an explicit link between coherent functors in the sense of Auslander [Coherent functors. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965). Springer, Berlin, pp. 189–231, 1966] and strict polynomial functors in the sense of Friedlander and Suslin (Invent. Math. 127(2), 209–270, 1997). Applications to functor cohomology are discussed. V. Franjou is partially supported by the LMJL—Laboratoire de Mathématiques Jean Leray, CNRS: Université de Nantes, école Centrale de Nantes, and acknowledges the hospitality and support of CRM Barcelona where the final corrections on this paper were implemented.     

On gorenstein injective and projective comodules

Gorenstein injective, projective and coflat comodules were introduced and studied by Asensio and Enochs et al. We further investigate these comodules and introduce n-Gorenstein injective, projective and coflat comodules, which give a new characterization of Gorenstein comodules.     

Coinduction functor and simple comodules

Consider a coring with exact rational functor, and a finitely generated and projective right comodule. We construct a functor (coinduction functor) which is right adjoint to the hom-functor represented by this comodule. Using the coinduction functor, we establish a bijective map between the set of representative classes of torsion simple right comodules and the set of representative classes of simple right modules over the endomorphism ring. A detailed application to group-graded modules is also given.     

Separable polynomials over finite dimensional algebras

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality n^d or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly n^d distinct zeros in [Ktilde] ?_k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ?_k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with coefficients in A there is a non empty Zariski open subset consisting of separable polynomials; in other polynomials with coefficients in a finite dimensional algebra are “generically” separable.