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Actions of reductive groups with finite stabilizers on quasi-projective varieties are often not proper. The simplest example I know is given by he action of $\mathrm{PGL}_2$ on the projective space $\ …

Suppose $G$ is an affine group scheme over an algebraically closed field $k$, and $V$ is a finite dimensional representation of $G$. Let $K$ be an extension of $k$, and assume that $V_K$ is reducible …

Here is a possible road to a solution (I am fairly sure that Milne had something more elementary in mind). Algebraic spaces satisfy fppf descent; hence $G$ is a group object is in the category of alge …

The map $G\to G/N$ is always a quotient in the étale topology. Since you are in characteristic $0$, the group scheme $N$ is smooth. Since $G \to G/N$ is an $N$-torsor, because the action of $N$ on $G$ …

Yes. The point is that $C$ is a $B$-torsor over $A$. Since being affine is a local property in the fpqc topology, $C$ is affine over $A$. [Edit]: Sorry I had not noticed grp's comment, or I wouldn't …

Consider images in $\mathrm{PGL}_n$ of the matrices $A$ and $B$, where $A$ is the diagonal matrix whose $i^{\rm th}$ diagonal entry is $\omega^i$, where $\omega$ is a primitive $n^{\rm th}$ root of $ …

If $P$ is a parabolic subgroup of a reductive group $G$ and $H$ is a closed subgroup of $G$ containing $P$, then $G/H$ is a quotient of $G/P$, so it is projective, and $H$ is parabolic. It follows tha …

This is true if you assume that $G$ is smooth. Consider the coaction $A \to A \otimes_k k[G]$; since $k[G]$ is a smooth $k$-algebra, the nilradical of $A \otimes_k k[G]$ is $N \otimes_k k[G]$; since $ …

Yes, $G/N$ is always an affine group scheme. This is explained, for example, in one of the last chapters of Waterhouse's book on affine group schemes (an excellent reference for these questions). [Ed …

No. For example, $\mathrm{PGL}_n$ contains a subgroup $G$ isomorphic to the product of two cyclic subgroups of order $n$, generated by the classes of the diagonal matrix whose entries are the powers o …

Linearly reductive groups, that is, linear algebraic groups, say over a field, in which the functor of invariants from finite dimensional representations to vector spaces is exact. I am not sure this …

I don't think this is literally true. For example, suppose that $G$ is a finite cyclic group of order 2 generated by $s$, suppose that $L$ is an non-trivial 2-torsion invertible $A$-module over a Dede …

Here is an example, which is in some sense the simplest one. Suppose that $k$ has characteristic $p > 0$; set $X := \mathop{\rm Spec} k[x,y]$. Let $G$ be a cyclic group of order $p$ acting via $(x,y) …

I think so. This is implied by the fact that the group of outer automorphisms of $H$ is finite. When $H$ is semisimple, then the group of outer automorphisms is contained in the group of automorphisms …

$\mathbb C^{n^2} \smallsetminus {0}$ is homotopy equivalent to $S^{2n^2-1}$, not $S^{n^2-1}$, and $\mathrm H^{2n^2-1}(GL_n(\mathbb C)) = 0$.