Is there any source in which I can find the exact relation between the injective hulls of $k[x_1,...,x_n]/m$ and $k[|x_1,...,x_n|]/m$ where $m$ is the maximal ideal $m=(x_1,...,x_n)$?
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I wouldn't know about injective hulls, but since both quotients are just $k$ it seems reasonable to guess that the injective hulls are the same... – David C. Ullrich Apr 18 '18 at 18:54
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@DavidC.Ullrich Well, the injective hull of the trivial representation of a group tends to depend quite a lot on the group, so I don't see why. – Tobias Kildetoft Apr 18 '18 at 18:58
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@TobiasKildetoft Because just looking at the definition it's clear that isomorphic modules have isomorphic injective hulls. – David C. Ullrich Apr 18 '18 at 20:55
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2@David But they are modules over different rings. – Tobias Kildetoft Apr 18 '18 at 21:24
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The problem is that the OP denotes by $m$ two ideals of two different rings. – Pierre-Yves Gaillard Apr 18 '18 at 22:52
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The injective hulls are the same, though the rings are different. Let $A$ denote either of the rings and let $A_p=A/(x_1^p,\ldots,x_n^p)$. Then you have a natural inclusion $A_p\to A_{p+1}$ given by $1\mapsto \prod x_i$. Take the direct limit to get the injective hull.
Mohan
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So, are you saying that the injective hull over the polynomial ring and formal power series ring are the same ? let me ask you one more thing. when we are in local complete case an endomoprhism on the injective hull is just a multiplication by an element of the ring. what do we get if we consider the polynomial case ? – greenspider Apr 21 '18 at 22:21
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@greenspider Just the localization of the polynomial ring at the maximal ideal. – Mohan Apr 22 '18 at 02:18