The Stacks project

Example 65.14.2. Let $k$ be a field. Let $k'/k$ be a degree $2$ Galois extension with $\text{Gal}(k'/k) = \{ 1, \sigma \} $. Let $S = \mathop{\mathrm{Spec}}(k[x])$ and $U = \mathop{\mathrm{Spec}}(k'[x])$. Note that

\[ U \times _ S U = \mathop{\mathrm{Spec}}((k' \otimes _ k k')[x]) = \Delta (U) \amalg \Delta '(U) \]

where $\Delta ' = (1, \sigma ) : U \to U \times _ S U$. Take

\[ R = \Delta (U) \amalg \Delta '(U \setminus \{ 0_ U\} ) \]

where $0_ U \in U$ denotes the $k'$-rational point whose $x$-coordinate is zero. It is easy to see that $R$ is an étale equivalence relation on $U$ over $S$ and hence $X = U/R$ is an algebraic space by Theorem 65.10.5. Here are some properties of $X$ (some of which will not make sense until later):

  1. $X \to S$ is an isomorphism over $S \setminus \{ 0_ S\} $,

  2. the morphism $X \to S$ is étale (see Properties of Spaces, Definition 66.16.2)

  3. the fibre $0_ X$ of $X \to S$ over $0_ S$ is isomorphic to $\mathop{\mathrm{Spec}}(k') = 0_ U$,

  4. $X$ is not a scheme because if it were, then $\mathcal{O}_{X, 0_ X}$ would be a local domain $(\mathcal{O}, \mathfrak m, \kappa )$ with fraction field $k(x)$, with $x \in \mathfrak m$ and residue field $\kappa = k'$ which is impossible,

  5. $X$ is not separated, but it is locally separated and quasi-separated,

  6. there exists a surjective, finite, étale morphism $S' \to S$ such that the base change $X' = S' \times _ S X$ is a scheme (namely, if we base change to $S' = \mathop{\mathrm{Spec}}(k'[x])$ then $U$ splits into two copies of $S'$ and $X'$ becomes isomorphic to the affine line with $0$ doubled, see Schemes, Example 26.14.3), and

  7. if we think of $X$ as a finite type algebraic space over $\mathop{\mathrm{Spec}}(k)$, then similarly the base change $X_{k'}$ is a scheme but $X$ is not a scheme.

In particular, this gives an example of a descent datum for schemes relative to the covering $\{ \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)\} $ which is not effective.

Comments (2)

Comment #7684 by Mark on

Tiny typo in (4): where -> were

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  • 9 comment(s) on Section 65.14: Examples of algebraic spaces

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