This question comes from me reading Van den Bergh's paper(page 12 middle of the proof).<https://arxiv.org/abs/math/0211064>


Let $X=Spec R$ where $R$ is a complete local ring over an algebraically closed field $k$,$x$ its unique closed point. $Y$ an $X$-scheme. Define $\hat{Y\times Y}=(Y\times Y)\times\_{X\times X} Spec \mathcal{\hat{O}}\_{X\times X,(x,x)}$ (here $\times$ without subscript means over $k$).


It is claimed in the paper that 



> 
> $Y\times\_X Y$ is a closed subscheme of $\hat{Y\times Y}$.
> 
> 
> 


Here are my questions:


1.How do we understand $\hat{Y\times Y}$? What is its geometric meaning? 


2.Why is the above statement true?


Thanks for the help!



$Y\times_X Y$ a closed subscheme of $\hat{Y\times Y}$?
Note that we can equivalently describe $Y \times\_X Y$ as $(Y \times Y) \times\_{X \times X} X$, where $X \to X \times X$ is the diagonal map $\Delta\_X$ (this can easily be seen from the functor of points point of view, or by a quick diagram-theoretic argument). Note that $\Delta\_X$ is a closed immersion since $X$ is affine.


Thus (by base change) it suffices to show that $\operatorname{Spec} \hat{\mathcal O}\_{X \times X, (x,x)} \to X \times X$ induces a closed immersion $\Delta\_X \colon X \longrightarrow \operatorname{Spec} \hat{\mathcal O}\_{X \times X,(x,x)}$. On the level of rings, this is the map
\begin{align\*}
(R \otimes\_k R)^\hat{} &\to R,\label{1}\tag{1}\\
r\_1 \otimes r\_2 &\mapsto r\_1r\_2,
\end{align\*}
where the completion is along the maximal ideal corresponding to $R \otimes R \twoheadrightarrow R \twoheadrightarrow k$. But the map (\ref{1}) is already surjective before completing, so it will remain so after. $\square$


Thus, the geometric intuition behind $Y \hat{\times} Y$ is that it is the base change of the $(X\times X)$-scheme $Y \times Y$ along the completion $\operatorname{Spec} \hat{\mathcal O}\_{X \times X, (x,x)} \to X \times X$.


**Example.** Let $R = k[[x]]$. It is instructive to think about the difference between $k[[x]] \otimes k[[y]]$ and $k[[x,y]]$ (see e.g. [this post](https://math.stackexchange.com/a/1773613)). The above procedure takes the variety $Y \times Y$ over $k[[x]] \otimes k[[y]]$ and spits out the variety $Y \hat{\times} Y$ over $k[[x,y]]$.


