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Someone anonymous has changed the wording in coherent module. I think it is correct but it is a bit incoherent. They changed ’Noetherian ring’ to ’coherent ring’ but have not defined that concept. Can someone who knows the terminology better than me check this out? It probably just needs a line saying what a coherent ring is.
One of the papers in the Borel’s volume on D-modules has theory of coherent rings/algebras in elementary detail. I will check later…
The definition of a coherent module seems to differ from other sources such as Stacks Project. The other definitions only require finitely-generated submodules to be finitely presented, and not all of them. Can someone more familiar with modules confirm?
(Also, the usual definition says the module has to be finitely generated and finitely-generated submodules are finitely presented. That implies that the module itself is also finitely-generated, right?)
Yes, thanks for your note, I just fixed the entry. One should only require the finitely-generated submodules to be finitely presented. Else the condition would be the same as Noetherianity. Also the condition wouldn’t make any sense in constructive mathematics, where one can’t even show that all submodules of $\mathbb{Z}$ are finitely generated.
Indeed the module itself is finitely generated.
I have cleaned up and cross-linked a little more.
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