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In the french school, un torseur sert à tordre, a torsor is used to twist. More precisely, let $\eta$ be an object in a topos, and $G=\operatorname{Aut}(\eta)$. If $\nu$ is a form of $\eta$ (another …

update: there is now a complete reference [2005.09690] Invariance of the tame fundamental group under base change between algebraically closed fields. I think it is generally admitted as folklore that …

As Keerthi Madapusi Pera points out in his comments, it is certainly reasonable to define a unipotent flat vector bundle as a flat vector bundle that is a successive extension of the trivial one $(\m …

As the previous answer points out you have to consider local systems for a finer topology than the Zariski topology. It is natural to consider the étale topology. The category of étale local systems o …

I suggest having a look at Beĭlinson, A.; Bernstein, J. A proof of Jantzen conjectures. MR: Matches for: MR=1237825 §1.2 . https://people.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf

It seems that your question is not well defined unless $K$ is finitely generated over its prime field. See for instance Jannsen, Uwe Weights in arithmetic geometry. Jpn. J. Math. 5 (2010), no. 1, 73–1 …

Yes they are. In fact the formation of ${\rm Aut}_S(\omega)$ commutes by definition with base change, so you can reduce to $S=k$, and $S\to k={\rm id}$. Then the fact that the $k$-group morphism $G\to …

If $G$ is an affine groupe scheme over some base $S$, you can consider the groupoid $G\rightrightarrows S$. The corresponding prestack $[G\rightrightarrows S]^{pre}$ is (equivalent to) the prestack of …

Parabolic bundles were introduced in the 70's by Mehta and Seshadri in the set up of a Riemann surface with cusps. They were trying to generalize the Narasimhan-Seshadri correspondence on a compact Ri …

I recently attented a nice online talk by Pengfei Huang and he indicated two sources: the first chapter of his own phd Non-abelian Hodge theory and some specializations - TEL - Thèses en ligne Intro …

Another option is to proceed as follows : show that there exists a unique connection $f^*\nabla$ on $f^*\mathcal F$ verifying : $$ (f^*\nabla)(f^*s) = f^*(\nabla(s))$$ where on the right-hand side yo …

About 1. : no, smoothness isn't essential. "Flat is enough" : De Jong's slogan to express this result due to M.Artin. https://www.math.columbia.edu/~dejong/wordpress/?p=1584 I quote : "Given a flat, f …

See Angelo Vistoli Intersection theory on algebraic stacks and on their moduli spaces Inventiones mathematicae (1989) Volume: 97, Issue: 3, page 613-670 EUDML  |  Intersection theory on algebraic stac …

Cartier descent is historically important, since together with Galois descent, it was Grothendieck's source of inspiration for fppf descent. As far as I remember, and with all due respect, Katz's proo …

This is more a comment than an answer: a few years back, in 2011, while working with some friends on SGA1, we also found out that we could not prove this statement without the hypothesis that $X$ is q …