Let $R$ be a commutative ring and let $M$ and $N$ be two $R$-modules. Suppose that for every $R$-module $P$, the modules $Hom_R(M,P)$ and $Hom_R(N,P)$ are isomorphic. Is it true that $M$ and $N$ are isomorphic?
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10$\begingroup$ This follows from Yoneda lemma if you assume this isomorphisms are natural. $\endgroup$– Piotr PstrągowskiCommented Jul 5, 2012 at 20:04
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11$\begingroup$ And if you do not assume the isomorphisms between Hom-sets to be natural, then for example over a field the question boils down to whether it is possible for two non-isomorphic vector spaces to have isomorphic duals. Over the field with two elements this is simply a question about the cardinality of power sets, which might very well be independant of ZFC. $\endgroup$– Piotr PstrągowskiCommented Jul 5, 2012 at 20:14
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2$\begingroup$ I suggest "Are modules isomorphic if their Hom-sets are all isomorphic?" (or something like that). $\endgroup$– Mark GrantCommented Jul 6, 2012 at 11:36
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$\begingroup$ @Piotr: could you please explain the "boils down" a bit further? The implication "isomorphic duals" $\implies$ "all hom spaces isomorphic" seems to require some implication of the sort $2^\kappa=2^\lambda\implies \alpha^\kappa=\alpha^\lambda$ for all cardinals $\alpha$. Is this true? $\endgroup$– user2035Commented Jul 6, 2012 at 11:52
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4$\begingroup$ I've cleaned up this comment thread; the removed comments are copied at tea.mathoverflow.net/discussion/1403/some-cleaned-up-comments. $\endgroup$– Anton GeraschenkoCommented Jul 6, 2012 at 17:11
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1 Answer
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K. Bongartz, "A generalization of a theorem of Auslander":
Let R be a commutative ring and A an abelian R-linear category such that each morphism set in A has finite length as an R-module. Let C be a full subcategory of A closed under direct sums and kernels. Then two objects M and N of C are isomorphic if and only if the lengths of Hom(M, X) and Hom(N, X) coincide for all X in C.
answered Jul 6, 2012 at 4:23