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The examples I have are: $S$ is equal to the spectrum of a global field; or a proper non-empty open subscheme of the spectrum of the ring of integers $\mathcal O_{K}$ of a number field $K$ (proper mea …

Let $F$ be a field and let $G$ be a smooth, commutative and connected $F$-group scheme of finite type. Let $K/F$ be a finite Galois extension of fields. Then there exists a canonical norm (or trace, i …

Let $k$ be a field of characteristic 0 and let $X$ and $Y$ be smooth, projective and geometrically integral $k$-schemes of finite type. Assume that both $X$ and $Y$ have 0-cycles of degree 1. Does $X\ …

I need information or directions to the literature regarding the structure of the group of units of $B\otimes_{A}B$, where $B/A$ is a finite Galois extension of rings of integers, say associated to a …

Let $k$ be a field with separable algebraic closure $k^{\rm s}$ and corresponding absolute Galois group $\varGamma={\rm Gal}(k^{\rm s}/\,k)$ and let $X$ and $Y$ be geometrically connected and geometri …

The referee for one of my papers gave the following argument: Let $k$ be any field and let $X$ and $Y$ be smooth, proper and geometrically integral k-schemes of finite type. Let $x$ be a $0$-cycle on …

There exists information on the Picard (and Brauer) group of a reductive algebraic group over a number field k. For example, Sansuc shows (in his big Crelle paper of 1980) that if G is connected and …

I need either a proof or a reference in modern (scheme-theoretic) language. According to Sansuc, this result can be gleaned from Borel's book on linear algebraic groups, but the old-style algebraic ge …

Let $k$ be a field of positive characteristic and let $G$ be a connected semisimple algebraic group over $k$ with fundamental group $\mu$. Note that $\mu$ can be non-smooth. It is stated in Sansuc's 1 …

First of all, the version that we would like people to read (and hopefully check for more mistakes, if any remain) is the Arxiv version of our paper. We worked on that for 4 years and with great care. …

Quadratic twists of elliptic curves (or, more generally, abelian varieties) are familiar objects in arithmetic geometry. I would like to extend that definition to the category of 1-motives over global …

Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ b …