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(This question is related to this one)

Let $k$ be a field and consider the category $Sch/k$ of schemes over $k$, say also separable and of finite type. The Yoneda embedding $$ Y:Sch/k \to Pre(Sch/k) $$ does not respect colimits but if you factorize $Y$ through $$ Y:Sch/k \to Shv(Sch/k) $$ with respect to the Zariski Grothendieck topology (given by the Zariski open immersions), it does respect some colimits (if you want you may replace $Sch/k$ by the category of commutative algebras over $k$ of finite type).

In particular a pushout of the form $$ U~\xleftarrow{f}~ U\cap V ~\xrightarrow{g}~ V $$ in $Sch/k$ where $f$ and $g$ are open immersions is also a pushout in $Shv(Sch/k)$.

The pushout of two closed immersions $$ B\leftarrow A \rightarrow C $$ in $Sch/k$ exists in general but let's consider the situation where $A,B,C$ are affine because in this case the existence is 'immediate' (whatever that is) by the antiequivalence to $k$-algebras. For example the coordinate cross $Spec k[X,Y]/(XY)$ is the pushout of $ \mathbb{A^1}\leftarrow Spec k\to \mathbb{A^1}$.

My question is

What happens to these 'closed' pushouts under the Yoneda embedding into Shv(Sch/k)? Are they preserved or is there an affine counterexample?

Edit: What happens if one takes sheaves with respect to the etale topology?

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I think the title is a little too general, since the Yoneda embedding is defined much more generally than for sheaves. – arsmath Dec 16 '10 at 16:19
Ok, I've edited it but it sounds even worse now. Maybe someone likes to change it. – roger123 Dec 16 '10 at 17:01
I think Johnathan's argument still works without change if you specify $k$ to be separably closed. – S. Carnahan Mar 25 '11 at 8:25

No, the embedding of schemes in the big Zariski site does not preserve colimits. It is possible to see this in the example you suggest by computing the "tangent space" at the origin. Let $X$ be the colimit in the category of schemes; the tangent space of $X$ at the origin is a $2$-dimensional vector space. Let $Y$ be the colimit in the category of sheaves; I claim that the "tangent space" of $Y$ at the origin is the union of the coordinate axes inside of the tangent space of $X$.

The tangent space of $Y$ is the space of sections of $Y$ over $Z := \mathrm{Spec}\: k[\epsilon] / \epsilon^2$. The Zariski topology is trivial on $Z$, so sections of $Y$ over $Z$ are the same as sections of the presheaf colimit over $Z$. The "tangent space" of the presheaf colimit is the union of the tangent spaces of the two copies of $\mathbf{A}^1$.

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Thanks Jonathan. I am still struggling with the argument. Please give me a little more time until I'll accept your answer. – roger123 Mar 24 '11 at 23:04

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