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Let $\phi: (R,m,k) \rightarrow (S,n,l)$ be a local homomorphism of Artinian rings, with $k,l$ being the corresponding residue fields. Let $E_R(k)$ be the injective hull of $k$ over $R$ and $E_S(l)$ the injective hull of $l$ over $S$. Suppose that we have shown that $Hom_R(S,E_R(k))$ is an injective $S$-module. Then by the decomposition theorem of injective modules (e.g. Theorem 18.5 in Matsumura) we must have that $Hom_R(S,E_R(k)) \cong E_S(l)^r$ for some $r>0$.

Question: An identical situation arises in the proof of Theorem 3.3.7 in the last line of page 112 in Bruns and Herzog and the authors write $Hom_R(S,E_R(k))\cong E_S(k)^r$. Should it not be $E_S(l)^r$?

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  • $\begingroup$ It looks like a typo unless one can make sense out of $k$ as an $S'$-module. $\endgroup$
    – Youngsu
    Nov 7, 2014 at 18:29
  • $\begingroup$ @Youngsu: Not only that, but we should have $E_{S'}(k) \cong E_{S'}(l)$, which is entirely unclear to me why it would be true. $\endgroup$
    – Manos
    Nov 7, 2014 at 18:52
  • $\begingroup$ @Manos: I don't have the book at hand, but assuming it is a typo and it should be $E_{S^{\prime}}(l)$ instead, can you follow the rest of the proof or does another problem arise? $\endgroup$
    – Hanno
    Nov 8, 2014 at 8:10
  • $\begingroup$ @Hanno: Yes, the proof follows through very smoothly with $l$ :) $\endgroup$
    – Manos
    Nov 8, 2014 at 21:40
  • $\begingroup$ @Manos: Good :) So is there anything unclear left then? $\endgroup$
    – Hanno
    Nov 8, 2014 at 21:40

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