The Stacks project

Example 83.5.12. There exist reduced quasi-separated algebraic spaces $X$, $Y$ and a pair of morphisms $a, b : Y \to X$ which agree on all $k$-valued points but are not equal. To get an example take $Y = \mathop{\mathrm{Spec}}(k[[x]])$ and

\[ X = \mathbf{A}^1_ k \Big/ \big (\Delta \amalg \{ (x, -x) \mid x \not= 0\} \big ) \]

the algebraic space of Spaces, Example 65.14.1. The two morphisms $a, b : Y \to X$ come from the two maps $x \mapsto x$ and $x \mapsto -x$ from $Y$ to $\mathbf{A}^1_ k = \mathop{\mathrm{Spec}}(k[x])$. On the generic point the two maps are the same because on the open part $x \not= 0$ of the space $X$ the functions $x$ and $-x$ are equal. On the closed point the maps are obviously the same. It is also true that $a \not= b$. This implies that Lemma 83.5.11 does not hold with assumption (3) replaced by the assumption that $X$ be quasi-separated. Namely, consider the diagram

\[ \xymatrix{ Y \ar[d]_{-1} \ar[r]_1 & Y \ar[d]^ a \\ Y \ar[r]^ a & X } \]

then the composition $a \circ (-1) = b$. Hence we can set $R = Y$, $U = Y$, $s = 1$, $t = -1$, $\phi = a$ to get an example of a set-theoretically invariant morphism which is not invariant.

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