Example of an Algebraic Space ("false" affine line with different tangents at origin)

Question

I have a question about the following example from the Algebraic spaces and quotients by equivalence relation of schemes by Roy Mikael Skjelnes (page 12) of a presheaf quotient, which has associated sheaf which can not be a scheme. It's originally an example from Knutson's book on Algebraic spaces, but there my concern also isn't answered. Here the example:

Example 2.15. . The following example is based on ([Knu71], page 9). Let $X = Spec(k[x, y]/(xy))$ again be the coordinate cross. Let $R$ be the disjoint union of $X$ and $X_0$, where $X_0 = X \ (0, 0)$ is the complement of the origin. We consider the two morphisms $\pi_i: R \to X$, where $\pi_1$ is the natural inclusion on both components of $R$. The morphism $\pi_2$ is the identity on the component $X$, but on the other component $X_0$ the morphism $\pi_2$ switches the axes. Note that the two maps $\pi_1$ and $\pi_1$ are Zariski local open immersions.

The glueing of $X$ along $R$ will give the affine line, but the quotient sheaf $A$, of $R \rightrightarrows X$ is not a scheme, in any topology. To see this note that the presheaf quotient has, over the origo, different tangent directions sticking out. Since the presheaf quotient is separated (Proposition (1.13)), these tangent directions will also appear in the sheaf quotient. Hence, since the sheaf quotient is different from the affine line, it follows that the sheaf quotient can not be a scheme.

Thus, the non-scheme like points, as the tangent directions in the above example, are not particular for algebraic spaces being etale quotient sheaves by ´etale equivalence relations. One encounters these nonscheme points when taking Zariski quotient sheaves by Zariski equivalence relations.

By consruction of the quotient of $R \rightrightarrows X$ the resulting presheaf quotient has, over the origin, two different tangent directions sticking out.

Then it is clamed that since the presheaf quotient is separated (Proposition (1.13)), these tangent directions will also appear in the sheaf quotient. I do not understand. The Proposition (1.13) states:

Proposition 1.13. Let $\operatorname{Cov}$ be a pretopology, and let $R$ and $X$ be two sheaves. Assume that we have an equivalence relation $R \rightrightarrows X$, and let $X_R$ denote its presheaf quotient. Then the presheaf $X_R$ is separated. In particular we have that $LX_R$ is a sheaf, and that the equivalence relation is effective, i.e. $R = X \times{LX_R} X$. (Here: $LF$ is defined by $LF(S)\mathrel{:=} \lim_{T \to S \in \operatorname{Cov}(S)} F(T)$; see also page 5.)

Question: Proposition (1.13) states only that the presheaf (of the quotient) is separated, there is no statement about the separateness of the sheaf itself. So I not see why all tangent directions, which apear in the presheaf, must also apear without any loss in hypothetically existing associated sheaf?

Secondly: I know that separatedness means that it 'separates' points (= $\operatorname{Spec}(k) \to X$-morphisms) in algebro-geometric setting. Does it also 'separate' tangent directions?

Answer 1

You are confusing two completely different notions of "separated".

A presheaf of sets $F$ on a site is called "separated" if whenever $\{U_i \to U\}$ is a covering family, $F(U) \to \prod F(U_i)$ is injective. This is the notion that Skjelnes is talking about. The sheafification of a presheaf of sets is often constructed by iterating twice an endofunctor $L$ on presheaves such that $L$ takes arbitrary presheaves to separated presheaves and separated presheaves to sheaves.

There is also the algebro-geometric notion of separated, which corresponds (for schemes over the complex numbers) to being Hausdorff. This is completely unrelated. In particular the line with doubled tangents along the origin is not separated in the algebro-geometric sense. But it is of course separated as a presheaf of sets, since it is an algebraic space, and in particular a sheaf of sets.