## Extendiendo morfismos con el lema de Zorn

Bueno, esto era una pregunta mía en math.stackexchange, que a medida que la fui escribiendo me fui respondiendo solo, entonces quedó meramente en una observación interesante. Se trata de dos aplicaciones del lema de Zorn en contextos bien diferentes, pero a efectos muy similares. Después me di cuenta que es simplemente porque en ambos casos hay que extender morfismos, y resulta que con el lema de Zorn el mismo truco sirve en situaciones diferentes.

Primero, recordemos que un $R$-módulo $E$ es inyectivo si para todo $B$ módulo y $A\subset B$ submódulo, todo morfismo $A\to E$ se extiende a un morfismo $B \to E$. En otras palabras, los morfismos hacia $E$ definidos en submódulos se extienden a todo el módulo. Mi post original fue:

I have stumbled upon a remarkable similarity between the proof of Baer’s criterion and an extension theorem in field theory. Here are the statements:

Baer’s criterion: Let $R$ be a ring. A left $R$-module $E$ is injective iff every $R$-map $f:I \to E$, where $I$ is a left ideal of $R$, can be extended to a map $R\to E$.

Extension theorem: let $F\subset K$ be an algebraic field extension, and $L$ an algebraically closed field. Then every field homomorphism $\sigma: F \to L$ can be extended to a field homomorphism $K\to L$.

Now, these two theorems don’t seem to have anything to do with each other, but the proofs are strikingly similar. Here’s how they go:

For Baer’s criterion (the non-trivial statement, i.e. the sufficiency of the ideal condition): to prove injectivity, we let $A\subset B$ be a submodule and $f:A\to E$ an $R$-map, and define $X=\{(A',g'): A\subset A'\subset B,\, g': A' \to E,\, g'|_A=f\}$.

Let $\leq$ be the partial order in $X$ given by $(A',g')\leq (A'',g'') \iff A'\subset A'' \text{ and } g''|_{A'}=g'$.

By Zorn’s lemma there’s $(A_0,g_0)$ a maximal element. If $A_0=B$ we’re done, if not, let $b\in B\setminus A_0$, define $I=\{r\in R: rb\in A_0\}$ and $h:I\to E$, $r \mapsto g_0(rb)$. Apply the hypothesis to find an extension $\tilde{h}$ of $h$ to $R$. Let $A_1=A_0 + Rb$ and $g_1:A_1 \to E$, $a_0+rb \mapsto g_0(a_0)+r\tilde{h}(1)$, and $(A_1,g_1)$ contradict the maximality of $(A_0,g_0)$.

For the extension theorem: define $M=\{(A,\tau):F\subset A \subset K,\, \tau:A\to L,\, \tau|_F=\sigma\}$. Let $\leq$ be the partial order on $M$ given by $(A,\tau)\leq (A',\tau') \iff A\subset A' \text{ and } \tau'|_A=\tau$.

By Zorn’s lemma there’s $(A,\tau)$ a maximal element. If $A=K$ we’re done, if not let $\alpha \in K\setminus A$, I claim $\tau$ extends to $A(\alpha)\to L$, contradiction.

Let $p$ be the minimal polynomial for $\alpha$ over $A$. The polynomial $\tau p\in L[X]$ has a root $r\in L$ by hypothesis. Since $A(\alpha)= \frac{A[X]}{\langle p \rangle}$, define

$\tilde{\tau}: \frac{A[X]}{\langle p \rangle} \to L$ as $\tilde{\tau}|_A=\tau$, $\tilde{\tau}(X)=r$, and $(A(\alpha),\tilde{\tau})$ contradicts the maximality of $(A,\tau)$.

The resemblance of both proofs shouldn’t come as such a surprise since both are about *extending morphisms*, so maybe the only observation to make is “it’s a useful technique, remember it’s useful for extending morphisms”. If that’s it, then this question is useless. But I’m intrigued. Is this technique used to prove other extending theorems? Is it possible to generalize it and write a single categorical proof for situations of this kind? Any other observations are appreciated (or perhaps it’s just a dumb observation that doesn’t serve any purpose).

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### Una respuesta a Extendiendo morfismos con el lema de Zorn

1. bstonek dijo:

Otro teorema que se demuestra igual: el teorema de Hahn-Banach de extensión de funcionales…