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Let $K_1,K_2,k,F$ be fields and suppose we have a morphism, $$ K_1 \otimes_k K_2 \xrightarrow{\varphi} F$$

Is $\varphi$ determined by its kernel?

I was trying to give a description to the underlying set of the fibered product of schemes using the colimit of $\operatorname{Hom}(\operatorname{Spec}-,X)$ where the colimit is taken in the category of fields. To finish the description of the underlying set as described here Underlying set of the scheme theoretic fiber, I should be able to say that the only information I need about $\varphi$ is its kernel.

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Abellan
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1 Answers1

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Definitely not in general: Letting $k=K_2\subseteq K_1 \subseteq F$ be a tower of extension, one has $K_1 \otimes_k K_2 =K_1$ and $\varphi$ is an embedding of $K_1$ into $F$. Any such $\varphi$ is of course injective, but there are more than one of such embeddings in general.

Pavel Čoupek
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