Suppose $A$ and $B$ are two $\mathbb F_p$-algebras, and denote the Frobenius on $A$ by $\sigma$. Then $\sigma \otimes id$ is an endomorphism on $A\otimes_ {\mathbb F_p} B$. How to prove it induces a homeomorphism on $\operatorname{Spec}A\otimes_ {\mathbb F_p} B$?
Maybe the homomorphism on the Spetra is the idenetity? But I don't know how to prove it... Could you give some hints for me? Thanks!
It suffices to prove that the Frobenius on $\mathrm{Spec}(A)$ is a universal homeomorphism. This is proven in Stacks Project, Tag 0CC8. Specifically, it follows from the characterization of universal homeomorphisms from Tag 0BRA. A classical reference for this characterization is EGA IV$_4$, Cor. 18.12.11.