given a ring $R$, a nilpotent ideal $I$ and a morphism $\phi$ of $R$-modules $M \to N$, such that $M/IM \to N/IN$ is an isomorphism.
It is easy to see that this implies $\phi$ surjective, but what about injectivity?
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The answer is no in general. E.g. if $M = R$ and $N = R/I$, then the natural surjection $M \to N$ will induce an isomorphism $M/I \to N/I$, which won't be an isomorphism unless $I = 0$.
Matt E
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Thank you. Now I have a question to Waterhouse, Group Schemes, Theorem 14.4, second paragraph: Why is it enough to show that $C/I_BC \to A/I_BA$ is an isomorphism? – user5262 Dec 31 '10 at 14:35
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@user5262: Dear 5262, You're welcome. Regarding Waterhous, I don't have a copy of the book at hand, so I can't tell you without more information. Regards, – Matt E Dec 31 '10 at 14:46
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I will start another thread and type the relevant pieces of Waterhouse up. – user5262 Dec 31 '10 at 14:49
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Here it is: http://math.stackexchange.com/questions/16000/question-regarding-waterhouse-affine-group-schemes – user5262 Jan 02 '11 at 10:52
*italics*) – kahen Dec 31 '10 at 14:29