Let $M$ be a flat $A$-module, and $N$ a $A$-module isomorphic to $M$, what can we say about the flatness of $N$?
Module isomorphic to a flat module
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4$N$ is flat? Or what do you want to hear? – 2012-03-27
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0why is $N$ flat? – 2012-03-27
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4If $M\cong N$ as $A$-modules, then for any $A$-module $P$, $P\otimes_A M\cong P\otimes_A N$. – 2012-03-27
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5Dear Jr., here is a meta-rule for you. Whenever mathematicians define a property P that some objects in a category may or may not have, you can be sure that if an object has property P, then any isomorphic object also has property P. – 2012-03-27
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0Dear Georges , is there a formal proof of your statement? I mean, only using abstract category theory, does one could reach that conclusion? – 2012-03-27
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0@Randal'Thor Thanks! – 2012-03-27
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0@Randal'Thor Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868). – 2013-06-13
1 Answers
Let $f:M\to N$ be an isomorphism of $A$-modules, and let $P$ be an arbitrary $A$-module. Then $P\times M\to P\otimes_A N$, $(p,m)\mapsto p\otimes f(m)$ is $A$-bilinear, hence we get an induced well-defined homomorphism $$\operatorname{id}\otimes f:P\otimes_A M\to P\otimes_A N, p\otimes m\mapsto p\otimes f(m).$$ In the same way, we have an inverse homomorphism $\operatorname{id}\otimes f^{-1}$, such that $P\otimes_A M\cong P\otimes_A N$.
Now $M$ being flat means that if $g:P\to P'$ is an injective morphism of $A$-modules, $g\otimes\operatorname{id}:P\otimes_A M\to P'\otimes_A M$ is, too. But then the map $$P\otimes_AN\xrightarrow{\sim}P\otimes_AM\xrightarrow{g\otimes\operatorname{id}}P'\otimes_AM\xrightarrow{\sim}P'\otimes_A N$$ is an injective $A$-homomorphism, which proves the flatness of $N$.
As for your question regarding Georges' answer, I'm not exactly sure what you mean. In general, one defines these properties such that they stay invariant under isomorphism in the respective category. But looking for a proof of this, as Georges calls it, meta-rule, wouldn't really make sense to me.