Let L be a field which is a Galois extension of the field K with Galois group G. Greither and Pareigis [GP87] showed that for many G there exist K-Hopf algebras H other than the group ring KG which make L into an H-Hopf Galois extension of K (or a Galois H∗object in the sense of Chase and Sweedler [CS69]). Using Galois descent they translated the problem of determining the Hopf Galois structure...

(a) Find Galois points and the Galois groups for singular plane curves. – for smooth curves, the number of Galois points is at most three (resp. four) if they are outer (resp. inner). The Galois groups are cyclic. [46, 62] – (i) How is the structure of Galois group and how many Galois points do there exist? Is it true that the maximal number of outer (resp. inner) Galois points is three (resp. ...

Partial Galois extensions were recently introduced by Doku-chaev, Ferrero and Paques. We introduce partial Galois extensions for noncommutative rings, using the theory of Galois corings. We associate a Morita context to a partial action on a ring.

The smallest non-abelian p-groups play a fundamental role in the theory of Galois p-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups — as well as other closely related, larger p-groups — occur as Galois groups over given base fields. We show further how the appearance of some Galois...

Regular coverings of a graph or a map admit descriptions in terms of certain group-valued labellings on the graph. Controlling the properties of graphs resulting from such coverings requires to develop a certain type of ‘labelling calculus’. We give a brief outline of the related theory and survey the most important results.

We prove an analog of the classical Hartogs extension theorem for CR L2 functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a question formulated in the paper of Gromov, Henkin and Shubin [GHS] on holomorphic L2 functions on coverings of pseudoconvex manifolds.

The aim of this paper is to give a detailed proof of a comparison of Voevodsky’s categories of geometric motives with and without transfers, respectively. The latter category is defined by means of h-topology introduced by Voevodsky, a topology essentially given by Zariski coverings, finite coverings and blowups.

This paper is a report on the observation that some singular varieties admit Calabi-Yau coverings. We derive a formula for calculating the invariants of the coverings with degeneration methods. By applying these to Takagi’s Q -Fano examples([Ta1], [Ta2]), we construct several Calabi-Yau threefolds with Picard number one. It turns out that at least 22 of them are new.