Timeline for Wikipedia's definition of 'locally free sheaf'
Current License: CC BY-SA 2.5
6 events
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| Nov 5, 2010 at 8:46 | comment | added | Georges Elencwajg | Dear roger123: Yes, everything you write is absolutely correct. For finitely presented modules, the equivalence of Locfree and Punctfree follows from Facts 1 and 3. Moreover your remark about the example is extremely interesting. Indeed, R/I can only be non finitely presented if I is non finitely generated. But I only assumed $I$ non-principal. So we are inexorably led to the conclusion that in a Von Neumann regular ring, every finitely generated ideal is principal. I did not know this so I checked five minutes ago in Rotman's book on Homological Algebra (3d ed.): it is stated in Lemma 4.8 ! | |
| Nov 5, 2010 at 7:43 | comment | added | roger123 | Thank you for your answer Georges! I think that for a finitely presented $R$ module Locfree is the same as Punctfree. If $R$ is noetherian, finitely presented is equivalent to finitely generated. So I think your example only works with a non-noetherian Ring, right? | |
| Nov 5, 2010 at 0:51 | history | edited | Georges Elencwajg | CC BY-SA 2.5 |
Added "the class of one". Changed "not finitely generated" to "non-principal"
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| Nov 5, 2010 at 0:37 | history | edited | Georges Elencwajg | CC BY-SA 2.5 |
Modified formulation of Fact 4. Added paragraph "Final irony"
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| Nov 4, 2010 at 22:08 | history | edited | Georges Elencwajg | CC BY-SA 2.5 |
replaced "pointfree" par "punctfree"
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| Nov 4, 2010 at 22:00 | history | answered | Georges Elencwajg | CC BY-SA 2.5 |