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Results tagged with representable-functors
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user 7666
11
votes
Accepted
What if the base change of an algebraic space is representable
In general $X$ is not a scheme, even if it is proper. But if $X\otimes_k L$ is a quasiprojective scheme, so is $X$.
First, assume $X\otimes_k L$ quasiprojective. By standard techniques, this still …
- 19.6k
10
votes
Is an algebraic space over a DVR, whose special fibre and generic fibre are schemes, actuall...
Olivier's example is perfect, but let me just point out that counterexamples are easier to construct if you allow nonseparated spaces.
For example, start with two DVRs $R\hookrightarrow R'$ (with spe …
- 19.6k
7
votes
Accepted
Representability of sheaf of Ext^1 of a Néron model by $\mathbb{G}_m$
First question: no. Assume, to fix ideas, that $R$ is complete with uniformizer $\pi$, $k$ is algebraically closed, and $A$ is an elliptic curve with multiplicative reduction. Denote by $\mathscr{E}$ …
- 19.6k
1
vote
Accepted
representing base changes of the unit section
No. For one thing, it would imply that every group subsheaf $K$ of an (abelian) algebraic space in groups $H$ is an algebraic space (just take $G=H/K$). Now let me give a "concrete" example.
For an …
- 19.6k