In general, the tensor product of two local rings is not local. For example, $\mathbb{C} \otimes\_{\mathbb{R}} \mathbb{C}\ $ is not a local ring. 


Let $\mathbb{F}\_{p}$ denote the finite field with $p$ elements. Let $A,B$ be two complete local noetherian $\mathbb{Z}\_p$-algebras with residue field $\mathbb{F}\_p$. Let $m\_A, m\_B$ denote the maximal ideals of $A,B$, respectively.


Question:


Is it true that $A \otimes\_{\mathbb{Z}\_p} B\ $ is a local ring?


Clearly, the ideal $m\_A \otimes B + A \otimes m\_B$ is a maximal ideal of $A \otimes\_{\mathbb{Z}\_p} B\ $ with the residue field $F\_p$. Is it the only maximal ideal of $A \otimes\_{\mathbb{Z}\_p} B\ $?



Is tensor product of local algebras local?
Let $A=\mathbb F\_p[[t]],B=\mathbb F\_p[[u]]$. Then, $1\otimes1-t\otimes u$ is neither in $\mathfrak m\_A\otimes B+A\otimes\mathfrak m\_B$ nor a unit, so it is contained in some other maximal ideal of $A\otimes B$. (Proof that $1\otimes1-t\otimes u$ is not a unit: An element of $\mathbb F\_p[[t]][[u]]$ coming from $A\otimes B$ has the property that its coefficients with respect to $u$ span a finite-dimensional subspace of $\mathbb F\_p[[t]]$, but this fails for the coefficients $t^k$ of $(1-tu)^{-1}=\sum t^ku^k$.)


