Let G be a finite group. Suppose that Y is a projective algebraic variety over Z (i.e an integral scheme which is projective and flat over Spec (Z)) of relative dimension d. In this paper, we consider finite Galois covers π : X → Y with group G which are everywhere unramified, i.e “G-torsors”. Let F be a G-equivariant coherent sheaf on X. Consider the value of the right derived global section f...

Weak coalgebra-Galois extensions are studied. A notion of an invertible weak entwining structure is introduced. It is proven that, within an invertible weak entwining structure, the surjectivity of the canonical map implies bijectivity provided the structure coalgebra C is either coseparable or projective as a C-comodule.

Let E be a cyclic extension of pth-power degree of a field F of characteristic p. For all m, s ∈ N, we determine KmE/p sKmE as a (Z/pZ)[Gal(E/F )]-module. We also provide examples of extensions for which all of the possible nonzero summands in the decomposition are indeed nonzero. Let F be a field of characteristic p. Let KmF denote the mth Milnor K-group of F and kmF = KmF/pKmF . (See, for ins...

Résumé : Dans beaucoup d’applications il est utile de représenter une grande quantité de données en les regroupant en une hiérarchie de classes décrites à un niveau d’abstraction adéquat. La représentation générique que nous utilisons ici est un treillis de Galois, i.e. un treillis correspondant au partitionnement des termes d’un langage de représentation en classes d’équivalence relativement à...

We have developed a probabilistic method that uses numerical polynomial homotopy continuation to compute Galois groups of Schubert problems. Our implementation of the method produced answers to the questions considered inaccessible by other techniques. The problem of computing all solutions of a system of polynomial equations may be solved using numerical polynomial homotopy continuation [10]. ...

James Ax showed that, in each characteristic, there is a natural bijection from the space of complete theories of pseudo-finite fields, in first order logic, to the set of conjugacy classes of procyclic subgroups of the absolute Galois group of the prime field. I show that when the set of subgroups of a profinite group is considered to have the Vietoris (a.k.a. hyperspace, finite, exponential, ...

In the paper, we investigate the duality of categories of complete lattices and maps preserving suprema or infima according to [15, p. 179–183; 1.1–1.12]. The duality is based on the concept of the Galois connection. Let S, T be complete lattices. Note that there exists a connection between S and T which is Galois. One can prove the following proposition (1) Let S, T , S , T be non empty relati...

For an odd prime number p, we consider degree p extensions L/K of p-adic fields with normal closure L˜ such that the Galois group L˜/K is dihedral order 2p. We shall prove a complete characterization freeness ring integers OL over its associated AL/K in unique Hopf-Galois structure on L/K, which analogous to one already known for cyclic fields. derive positive and negative results criteria as A...