Timeline for Criterion for being reflexive via Ext
Current License: CC BY-SA 3.0
13 events
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| Jul 21, 2016 at 17:28 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Jul 21, 2016 at 14:33 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Jul 21, 2016 at 14:26 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Jul 21, 2016 at 14:21 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Jul 21, 2016 at 13:54 | comment | added | Jason Starr | @HailongDao: You are correct, and I will edit my answer. For $M=\mathfrak{p}=\langle x,y,z \rangle \subset k[x,y,z,t]$, the module does have depth $2$ and one regular sequence of length $2$ is $(t,x)$. Of course the localization $M_{\mathfrak{p}}$ as a $k[x,y,z,t]_{\mathfrak{p}} = k(t)[x,y,z]_{\langle x,y,z \rangle}$ does not have depth $2$. | |
| Jul 21, 2016 at 13:42 | comment | added | Hailong Dao | It is not true that $M$ is reflexive if depth is at least $2$. Over a normal domain you need it to be $S_2$, a more subtle condition. For example, take $M=(x,y,z)$ in $k[x,y,z,t]$. It has depth 2 but not reflexive, since a reflexive ideal would have to have height one. | |
| Jul 18, 2016 at 16:20 | vote | accept | Hans | ||
| Jul 18, 2016 at 15:31 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Jul 18, 2016 at 15:30 | comment | added | Jason Starr | The definition that I wrote is correct if $R$ is normal. If $R$ is not normal, then you will find commonly used at least two inequivalent definitions, cf. stacks.math.columbia.edu/tag/0AY1. If $R$ is normal, then every finitely generated, torsion-free $R$-module is reflexive at all points of codimension $0$ and codimension $1$. Then one reference relating reflexive to depth $\geq 2$ is stacks.math.columbia.edu/tag/0AY5. | |
| Jul 18, 2016 at 15:18 | comment | added | Hans | Hi Jason, thanks for this answer! Just two quick question: What happens for rings of dimension 1? Is it true that reflexive is equivalent to projective? And do you have a reference for your first statement, namely that the module is reflexive iff its depth is at least two? | |
| Jul 17, 2016 at 18:10 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| S Jul 17, 2016 at 17:58 | history | answered | Jason Starr | CC BY-SA 3.0 | |
| S Jul 17, 2016 at 17:58 | history | made wiki | Post Made Community Wiki by Jason Starr |