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109.49 A non-separated group algebraic space over a field

Every group scheme over a field is separated, see Groupoids, Lemma 39.7.3. This is not true for group algebraic spaces over a field (but see end of this section for positive results).

Let $k$ be a field of characteristic zero. Consider the algebraic space $G = \mathbf{A}^1_ k/\mathbf{Z}$ from Spaces, Example 64.14.8. By construction $G$ is the fppf sheaf associated to the presheaf

\[ T \longmapsto \Gamma (T, \mathcal{O}_ T) / \mathbf{Z} \]

on the category of schemes over $k$. The obvious addition rule on the presheaf induces an addition $m : G \times G \to G$ which turns $G$ into a group algebraic space over $\mathop{\mathrm{Spec}}(k)$. Note that $G$ is not separated (and not even quasi-separated or locally separated). On the other hand $G \to \mathop{\mathrm{Spec}}(k)$ is of finite type!

Lemma 109.49.1. There exists a group algebraic space of finite type over a field which is not separated (and not even quasi-separated or locally separated).

Proof. See discussion above. $\square$

Positive results: If the group algebraic space $G$ is either quasi-separated, or locally separated, or more generally a decent algebraic space, then $G$ is in fact separated, see More on Groupoids in Spaces, Lemma 78.9.4. Moreover, a finite type, separated group algebraic space over a field is in fact a scheme by More on Groupoids in Spaces, Lemma 78.10.2. The idea of the proof is that the schematic locus is open dense, see Properties of Spaces, Proposition 65.13.3 or Decent Spaces, Theorem 67.10.2. By translating this open we see that every point of $G$ has an open neighbourhood which is a scheme.

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