Timeline for Ext of cyclic module
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 9, 2012 at 0:45 | comment | added | Will Sawin | @Ralph: I'm not sure what you mean. (*) sounds like either the definition of locallly free or else the statement $R^n \cong I$, depending on whether the local maps could be any map or are localizations of the same map. Note than an ideal, to be locally free, must be locally free of rank one. | |
| Aug 8, 2012 at 23:07 | comment | added | Fred.Fred | Thanks everybody for answers, this is exactly what I needed. | |
| Aug 8, 2012 at 23:03 | vote | accept | Fred.Fred | ||
| Aug 8, 2012 at 23:03 | vote | accept | Fred.Fred | ||
| Aug 8, 2012 at 23:03 | |||||
| Aug 8, 2012 at 21:34 | comment | added | Ralph | () should be $(\ast)$. | |
| Aug 8, 2012 at 21:33 | comment | added | Ralph | @Will: Thanks. I was thinking about the following: Suppose $I$ is locally free and assume the localization $R_P^n \to I_P$ is an isomorphism for each max. P (). Then the localization of the inclusion $j: Ext_R^1(R/I,M) \to M^n/(...)$ is an isomorphism for each max. P and hence $j$ is an isomorphism. Do you know if () can be expected for Dedekind domains or other interesting categories of rings ? | |
| Aug 8, 2012 at 21:04 | comment | added | Will Sawin | In a Dedkenind domain, $(I^\vee \otimes M)/M=I^\vee \otimes (M/IM)=M/IM$, with the last identification non-canonical. | |
| Aug 8, 2012 at 20:47 | comment | added | David White | Clarifying the comment in light of the edit: Will was helping Ralph discover when you can get a closed form expression for $Hom_R(I,M)$. | |
| Aug 8, 2012 at 20:36 | history | edited | Ralph | CC BY-SA 3.0 |
Improved the previous result.
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| Aug 8, 2012 at 20:29 | comment | added | Will Sawin | $I$ being locally free is good enough, because then there is a dual module $I^{\vee}$ such that $Hom_R(I,M)=I^{\vee} \otimes M$. This occurs, for instane, in Dedekind domains, which is one case OP was interested in. | |
| Aug 8, 2012 at 19:52 | history | answered | Ralph | CC BY-SA 3.0 |