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This is of course true, for any semi-separated scheme (i.e. the diagonal is affine), or maybe you assume $X$ is separated if you like, and you can take any point (not necessarily closed). The reason t …

Here is a direct argument for $X$ satisfies the following condition: () there is a covering of $X$ by finitely many open affines $(U_i)$ such that each intersection $U_i\cap U_j$ is quasi-compact. ()h …

Let $C$ be a projective smooth connected curve over an algebraically closed field $k$. Let $(P,G,p)$ be a triple, where $G$ is a finite $k$-group scheme, $P$ is a $G$-torsor over $C$, $p\in P(k)$ a ra …

If $X$ is an algebraic space of finite type over a finite field $k$, then I think it is true that the set of $k$ rational points of $X$ is finite. This is of course true for $X$ is a scheme. I wish …

Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way. We know that if $X$ is a topological space, then there is an equivalence of categories between the categor …

It is well-know that the category of coherent D-modules over a smooth algebraic $k$-scheme is a Tannakian category. So it is equivalent to the category of finite representations of some affine group s …

If $\mathfrak{C}$ is a $k$-linear rigid abelian tensor category with End(1)=$k$(strictly speaking is isomorphic to $k$ as a $k$-algebra), and $k=\bar{k}$, and if $\omega_1$ and $\omega_2$ are two fibr …

Can I have some examples of finite non-commutative connected group schemes over a field $k$? I would like also to see some non-trivial torsors over a $k$-scheme $X$ under such group schemes. Thanks. …