Let $A$ be a semiperfect noetherian ring. A module $M$ is called reflexive in case the canonical map $f_M: M^{**} \cong M$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. This is equivalent to say that $Ext_A^i(Tr(M),A)=0$ for $i=1,2$ when $Tr$ denotes the Auslander-Bridger duality.

Question: Assume $M$ is finitely presented. Is $M$ reflexive iff $M^{**} \cong M$? (You are welcome to give example for any kind of ring, I know none)

This should be true for Artin algebras: Assume $M^{**} \cong M$. We have $M^{**} \cong \Omega^2 Tr \Omega^2 Tr(M)$ and thus $M \in \Omega^1(mod-A)$ and thus $M$ is torsionfree, which is equivalent to $f_M$ being injective. But $M^{**} \cong M$ gives us that the modules have the same length and thus $f_M$ is even an isomorphism and $M$ is relfexive. In the book of Auslander and Bridger I found that this should also be true in case the ring is additionally commutative Gorenstein (we dont need semiperfect here). Remark with regards to the previous (deleted) thread: I decided to split up the bigger confusing thread into smaller questions to make things less confusing.


1 Answer 1


This is true even with weaker assumptions (finitely generated modules for Noetherian rings, or for non-Noetherian semiperfect rings).

If $M\cong M^{**}$ then $M$ is a dual, and for any dual the natural map $M\to M^{**}$ is a split monomorphism, so if $M$ is not reflexive then $M\cong M\oplus N$ for some non-zero $N$.

This is not possible if $M$ is a Noetherian module, since $M$ would have an increasing chain of submodules $N< N\oplus N<N\oplus N\oplus N<\dots$.

It's also not possible if $M$ is finitely generated and $A$ is semiperfect, since then $M/MJ(A)\cong M/MJ(A)\oplus N/NJ(A)$ are finite direct sums of simple modules, so Krull-Schmidt can be applied.

  • $\begingroup$ Thanks, is there an example for general rings where this fails? Maybe this is worth an extra question in case it is non-trivial? $\endgroup$
    – Mare
    Dec 20, 2017 at 10:48
  • $\begingroup$ @Mare I believe there are examples of non-reflexive infinitely generated abelian groups that are isomorphic to their double dual. I don't know about finitely generated/presented modules for general rings. $\endgroup$ Dec 20, 2017 at 10:53
  • $\begingroup$ @Mare See mathoverflow.net/questions/76000/… for the fact about abelian groups. $\endgroup$ Dec 20, 2017 at 11:06
  • $\begingroup$ @Jeremyy Rickard I noted that I can not write @ and your name in a comment anymore in this thread. Is this a bug or normal for accepted answers? I had to write your name with two yy to make it work. Looks like a strange bug? $\endgroup$
    – Mare
    Dec 20, 2017 at 11:11
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    $\begingroup$ @Mare As it’s my answer, I get notified anyway, so you don’t need to @ me (unlike if I’d commented on your question or somebody else’s answer). $\endgroup$ Dec 20, 2017 at 12:10

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