What is the tensor product of $\mathbb{C}(x)$ and $\mathbb{C}(x)$ over the structure sheaf $\mathcal{O}_{X,x}$?

Question

Let $X$ be a complex manifold with the structure sheaf $\mathcal{O}_X$. Let $\mathbb{C}(x)$ be the residue field at the point $x\in X$, i.e., $\mathbb{C}(X)=\mathcal{O}_{X,x}/m_x$, where $m_x$ is the maximal ideal of $\mathcal{O}_{X,x}$. Of course $\mathbb{C}(x)$ is exactly the usual complex number field $\mathbb{C}$.

Question: can we give a more precise description of $\mathbb{C}(x)\otimes_{\mathcal{O}_{X,x}}\mathbb{C}(x)=?$ Is it still $\mathbb{C}(x)$?

Answer 1

$\mathcal{O}_{X,x}/m_x \otimes_{\mathcal{O}_{X,x}} \mathbb{C}\cong \mathbb{C}/m_x\mathbb{C}\cong \mathbb{C}.$