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Yes, this is true. A group scheme over a field is smooth if and only if it is geometrically reduced, so the hypotheses ensure that $N$ is smooth. You can even allow $G$ and $G'$ to be arbitrary group …

There is an exercise (p. 88) in Beauville's book Complex Algebraic Surfaces that claims that: For $X$ a smooth complex projective variety, if the Kodaira dimension (defined in this book as the maxi …

Does anyone know an example of an integral scheme $X$ over a field $k$ such that $X_{\overline{k}}$ is connected but reducible? Does it make a difference if $k$ is perfect, or if we ask for $X_{\overl …

A useful criterion for triviality of a line bundle $\mathscr{L}$ on an integral curve $C$ is that the trivial line bundle is the unique line bundle of degree $0$ which admits a global section. This is …

This question is a follow-up to this question which I asked on MSE. Let $f: X \rightarrow Y$ be a surjective morphism of schemes, and $\mathscr{F}$ a coherent sheaf on $Y$. Are there conditions we c …

Let $X$ be a geometrically integral (or geometrically reduced and geometrically connected) proper scheme over a finite field $k = \mathbb{F}_q$, so its Picard scheme exists and $\mathrm{Pic}^0_{X/k}$ …

Let $G$ be a reductive group scheme over some base $X$ and $P \subseteq G$ a parabolic subgroup. To a $P$-torsor $\mathscr{E}_P$, we may associate a $G$-torsor $\mathscr{E} = G \times^P \mathscr{E}_P$ …

Thanks to Laurent Moret-Bailly for pointing out that I missed a crucial hypothesis! Now I can construct the quasi-inverse, which I'll record below in case some future person is confused by the same pr …

In a (necessarily non-Noetherian) integral domain $A$ of (Krull) dimension $1$, is it possible that there is an infinite collection of prime ideals $\mathfrak{p}_i$ such that $\cap_i \mathfrak{p}_i \n …

There are lots of ways in which the complex singular (co-)homology of a smooth projective variety over $\mathbb{C}$ is "special" among complex manifolds - the hard Lefschetz theorem, the Hodge decompo …

Consider a hypersurface $V(f) \subseteq \mathbb{C}^{n+1}$ with an isolated singularity at the origin. If $L := V(f) \cap S^{2n+1}_\epsilon$ is the link of $V(f)$ (with $S^{2n+1}_\epsilon$ a sufficient …

Let $\mathscr{X}$ be a smooth proper DM stack over a field $k$ (perhaps assumed to be separably closed and/or of char. $0$) and let $\pi \colon \mathscr{X} \rightarrow X$ be its coarse moduli space. …

I think this is a counterexample. Let $V = \mathbf P^2 \times \mathbf P^1$, $L= \mathscr{O}_{P^2} \boxtimes \mathscr{O}_{P^1}(1)$. Let $\ell_1, \ell_2$ be two distinct lines in $\mathbf{P}^2$ and let …

First of all, there's no need to use flat cohomology here. By Theorem III.3.9 in Milne's Etale Cohomology, the canonical map $H^i_{\mathrm{et}}(X, G) \rightarrow H^i_{\mathrm{fppf}}(X, G)$ is an isomo …

As pointed out in the comments, this is false for general bases. Let $k$ be a field, $S = \mathrm{Spec}(k[t])$, let $\overline{X} = \mathbb{P}^1 \times_{\mathrm{Spec} k} S$ with projection $\overline{ …