Timeline for Locally, the minimal number of ideal generators is the codimension
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 25, 2022 at 21:20 | history | became hot network question | |||
| Aug 25, 2022 at 19:16 | vote | accept | Felix Lungu | ||
| Aug 25, 2022 at 19:08 | answer | added | Jason Starr | timeline score: 7 | |
| Aug 25, 2022 at 19:01 | comment | added | Jason Starr | Sorry, I missed that. Every smooth subvariety is a local complete intersection. | |
| Aug 25, 2022 at 17:24 | comment | added | Felix Lungu | @JasonStarr Thanks! Just to make sure: did you see that I'm assuming $X$ to be smooth? | |
| Aug 25, 2022 at 15:38 | comment | added | Jason Starr | What is true is that you can cover $X$ by open affines $X_f$ each of which is a union of some of the irreducible components in such a set-theoretic local complete intersection. In other words, after forming the union of $X$ with other irreducible subvarieties (whose dimension equals the dimension of $X$), the union is a set-theoretic local complete intersection. This fact is one of the cornerstones of dimension theory. | |
| Aug 25, 2022 at 15:35 | comment | added | Jason Starr | No, that is not even possible locally. There is a connectedness theorem, usually attributed to Hartshorne, that says that such varieties (set-theoretic local complete intersections, or even set-theoretic Cohen-Macaulay varieties) are "locally connected away from codimension $>1$", i.e., remove a codimension 2 closed subset does not (locally) disconnect the variety. So a surface in $4$-space that has a unique singular point, and where that point has two local branches, gives a counterexample. | |
| Aug 25, 2022 at 13:16 | history | asked | Felix Lungu | CC BY-SA 4.0 |