Timeline for When are two ideals in a regular local ring generated by a regular sequence?
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| Sep 30, 2010 at 16:19 | comment | added | Karl Schwede | One should point out that in the geometric situation, if one additionally assumes that $x_1, \dots, x_n$ form a set of generators for the maximal ideal, then this means that $\Spec R/I$ and $\Spec R/J$ intersect (only at the origin) as transversally as possible. | |
| Sep 30, 2010 at 12:50 | answer | added | Hailong Dao | timeline score: 3 | |
| Sep 30, 2010 at 8:58 | comment | added | Ariyan Javanpeykar | This means that I and J are in general position. This is a quite strong restriction which implies that I and J are both complete intersections. You could think about linearly independent linear forms. Basically, what happens algebraically is that Serre´s tor formula for the intersection of two subvarieties simplifies because the higher Tor´s vanish (as you mentioned). (Note that I and J intersect properly if they are in general position. Krull's hauptidealsatz or something..) | |
| Sep 30, 2010 at 8:23 | history | asked | Hanno | CC BY-SA 2.5 |