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May 28, 2021 at 14:21 comment added Nataniel Marquis Yes, I thought at first of the theorem by Gabriel, but was not able to fulfuill the artinian assumption ( $k[[X,Y]][(XY)^{-1}]/k[[X,Y]]$ is not artinian over $k[[X,Y]]$).
May 28, 2021 at 6:49 comment added Antoine Ducros As far as I understand in the situation considered above by Nataniel the Artinian assumption is not fulfilled.
May 28, 2021 at 5:20 history answered Jochen Wengenroth CC BY-SA 4.0