Over a general base the answer is no, as pointed out to me by Matthieu Romagny in response to this question. He gives a reference to Lemma X.14 of Raynaud's book "Faisceaux amples sur les schémas en ...

You have probably already come up with the answer yourself, but I just thought the question shouldn't hang around unanswered in the forum. What you call "the sheaf of connected components", I would ...

For the first question the short answer is that the map $x$ factorises canonically through the classifying stack $BG$ with $G = Stab(x)$ and the category of quasi-coherent modules on $BG$ is ...

This has nothinging to do with finiteness. Let $f:X\to Y$ be affine and $L$ a line bundle on $Y$. Then the subschemes $Y_s$ for $s \in \Gamma(Y, L^n)$ pull back to subschemes $X_{f^*s}$. Hence $L$ is ...

EDIT I just realized that my previous version of the answer was totally misleading. Here is what really is problematic in your argument: Just because the images of $t$ and $t^2$ are distinct in $\...

Let me elaborate on Torstens comment. The map $GL_n(B) \to i_*F$ in your example is an fppf $D^\times$-torsor over $i_*F$, a fact which can be checked easily directly from the definition of torsor ...

If you have a presentation $s, t:R \to U$ of a stack $[U/R]$, the set of points of $[U/R]$ is just the equivalence classes of points in $|U|$ determined by the equivalence relation given by the image ...

Yes, it is the same concept since the fibre products of schemes DO coincide with the fibre product of the underlying topological spaces in the case of open immersions (Check, for instance, Hartshornes ...

I guess, to get a good theory, you would want to be able to have effective descent for certain classes maps. In Brian Conrads answer to this question: "Quasi-separatedness for Algebraic Spaces" he ...

Yes, provided that $X$ is a scheme (or, for instance, an algebraic stack). In particular, we need not assume that $X$ is locally noetherian. Let $(X, \mathcal{O}_X)$ be a locally ringed space (or a ...

I'm not sure if this is what you are after, but when I started to look at Grothendieck-topologies I thought of being an open immersion as a topological property; somehow it should be possible to ...