Related papers: On Galois descent of complete intersections

We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals…

We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of…

We introduce perfect cones over topological spaces and study their connection with absolute $B_1$-retracts

We obtain criteria for detecting complete intersections in projective varieties. Motivated by a conjecture of Hartshorne concerning subvarieties of projective spaces, we investigate situations when two-codimensional smooth subvarieties of…

The Marden theorem of geometry of polynomials and the great Poncelet theorem from projective geometry of conics by their classical beauty occupy very special places. Our main aim is to present a strong and unexpected relationship between…

We prove $p$-complete arc-descent results for finite projective modules and perfect complexes over integral perfectoid rings. Using our results, we clarify a reduction argument in the proof of the classification of $p$-divisible groups over…

We prove that any quadratic complete intersection with certain action of the symmetric group has the strong Lefschetz property over a field of characteristic zero. As a consequence of it we construct a new class of homogeneous complete…

We propose a definition of when a triangulated category should be considered a complete intersection. We show (using work of Avramov and Gulliksen) that for the derived category of a complete local Noetherian commutative ring R, the…

We present a formula relating the set of left descents of an element of a Coxeter group with the sets of left descents of its projections on maximal quotients indexed by simple right descents. This formula is an instance of a general result…

We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…

In this paper we will give the calculus, the criterion, and the existence of the arithmetic Galois covers of higher relative dimensions.

We prove existence of conjugate Galois representations, and we use it to derive a simple method of weight reduction. As a consequence, an alternative proof of the level 1 case of Serre's conjecture follows.

We prove a Galois correspondence theorem for groupoids acting orthogonally and partially on commutative rings. We also consider partial actions that are not orthogonal, presenting two correspondences in this case: one for strongly Galois…

We define deformation rings for potentially semi-stable deformations of fixed discrete series inertial type in dimension $2$. In the case of representations of the Galois group of $\mathbf{Q}_p$, we prove an analogue of the Breuil-M\'ezard…

Let $S$ be a Shimura variety with reflex field $E$. We prove that the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the…

Let $F$ be a number field and $p\geq7$ a rational prime. We obtain a simple descent criterion characterising those projective Galois representations $\overline\rho:G_F\to\mathrm{PGL}_2(\mathbb{F}_p)$ for which the corresponding twist…

For a partial Galois extension of commutative rings we give a seven terms sequence, which is an analogue of the Chase-Harrison-Rosenberg sequence.

These are the notes for an undergraduate course at the University of Edinburgh, 2021-2023. Assuming basic knowledge of ring theory, group theory and linear algebra, the notes lay out the theory of field extensions and their Galois groups,…

This is essentially an erratum, with some example to indicate inconsistencies. Suppose $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$. The Complete Intersection conjecture states that, for any ideal $I$ in $A$,…

We prove the finiteness of the kernel of the localization map in the Galois cohomology of a connected reductive group over a global field