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Let $V$, $W$ be varieties (affine or projective) over an algebraically closed field $K$. Let $p \in V$ and $q \in W$. Is there a description of the local ring of $V\times W$ at $(p,q)$ in terms of the local ring of $V$ at $p$ and the local ring of $W$ at $q$?

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Let $k$ be a field. The fiber product of two $k$-schemes $X,Y$ (or even locally ringed spaces, see here) has as points triples $(x,y,\mathfrak{p})$, where $\mathfrak{p}$ is a prime ideal in $\kappa(x) \otimes_k \kappa(y)$, or equivalenty a prime ideal $\mathfrak{q}$ in $\mathcal{O}_{X,x} \otimes_k \mathcal{O}_{Y,y}$ which restricts to the maximal ideals in both factors. The stalk of the structure sheaf at $(x,y,\mathfrak{p})$ is $(\mathcal{O}_{X,x} \otimes_k \mathcal{O}_{Y,y})_\mathfrak{q}$. If $x,y$ are $k$-rational points, we have $\mathfrak{p}=0$ and therefore $\mathfrak{q}$ is the kernel of the natural map $\mathcal{O}_{X,x} \otimes_k \mathcal{O}_{Y,y} \to k$, $f \otimes g \mapsto f(x) \otimes g(y)$. So the stalk of $X \times_k Y$ at a rational point $(x,y)$ is a localization of the tensor product of the stalks, where those elements $\sum_i f_i \otimes g_i$ with $\sum_i f_i(x) \cdot g_i(y) \neq 0$ are inverted.

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