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I would like to have some geometric intuition for - Noetherian rings/modules - Local rings - Projective modules - Injective modules

As an illustration of what I am looking for, I was told once that that for a local ring to be a domain roughly means that locally a curve has only one component.


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I think this question is too vague. Suppose X is an algebraic variety - by which I mean "something glued from finitely many affine algebraic sets". To it you can associate a sheaf of rings (the structure sheaf). These rings are naturally Noetherian and reduced. If X is connected, it is irreducible iff all its affine sections are integral domains. At any point, you can construct the "local ring of the point". This is a noetherian local ring; its minimal prime ideals are in bijection with the irreducible components of X passing through it. In particular, it is a domain iff there is only one ... – Tom Bachmann Dec 7 '12 at 11:23
... component passing through it. In this sense, geometry gives you intuition for Noetherian rings. Algebraic vector bundles on varieties correspond to locally free coherent sheaves - over affine varieties, these are locally free Noetherian modules. – Tom Bachmann Dec 7 '12 at 11:26

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