Local Noetherian Property

First note that these conditions are satisfied in case S= Spec A with A strict local, if we take for s the closed point. Next note also that (by the usual arguments like in EGA IV 8) if we start with an arbitrary locally noetherian S and S4S then we can find an 6tale neighbourhood of s in S and a point over s having the above properties. 

Let A be a noetherian local ring with the following properties ... 

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