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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 …

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

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 …

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 …

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 …

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 …

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 …

Besides the references already given, I like Dan Edidin's Notes on the construction of the moduli space of curves https://arxiv.org/abs/math/9805101 I quote : "In section 3 we return to curves and …

As a complement to the answers above : it is kind of well-known (at least I thought it was) that the natural morphism $$\operatorname{\mathbf{Hom}}_{gr} (G,H) \to \operatorname{\mathbf{Hom}}(BG,BH)$ …

The answer is yes (with mild hypothesis on the space). Moreover the topological situation is simpler, and this was very likely Grothendieck's inspiration. To see this you need two facts. First take …

If I remember correctly, this goes roughly as follows. Consider the category $\mathcal C=\operatorname{Rings}^{op}$, first endowed with the Zariski topology. You can consider sheaves on this site that …

This is a well known and well understood problem when the base field is $\mathbb C$. It was first studied by Chevalley and Weil (almost a century ago !) who were interested in modular curves (what els …