Timeline for When is the product of two ideals equal to their intersection?
Current License: CC BY-SA 2.5
8 events
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| Dec 13, 2010 at 18:00 | comment | added | t3suji | @Ariyan. Thanks! (I somehow always forget the "proper intersection" terminology, that's why my use of "purity" in place of... "propriety" or "properness".) | |
| Dec 13, 2010 at 17:49 | comment | added | Ariyan Javanpeykar |
@t3suji. Let V and W be closed integral subschemes of a nonsingular quasi-projective irreducible variety. Then, for any irreducible component Z of VcapW, it holds that codim Z <= codim V + codim W. (See Serre's Local Algebra.) We say that V and W intersect properly in Z if equality holds. A stronger condition is being in general position. If V and W are in general position all the higher Tor's vanish. The cycle [VcapW] associated to VcapW is then equal to the product cycle [V][W]. As far as I know, this is standard language in intersection theory for algebraic varieties.
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| Dec 13, 2010 at 15:38 | comment | added | t3suji | I agree that the condition with Tor is more geometric --- it can be viewed as a kind of `purity' of intersection (for instance, two smooth subvarieties of a smooth variety have this property if and only if their intersection has the expected dimension). Is there an accepted name for this condition? | |
| Dec 13, 2010 at 15:22 | comment | added | user6976 | I posted a followup question. | |
| Dec 13, 2010 at 14:47 | comment | added | user6976 | @Martin: So you think it is better than the question? If we have Groebner bases of $I$, and $J$, can we decide whether $IJ=I\cap J$? I think that can be an interesting question. In fact I am not sure that David's answer gives any algorithm to decide $IJ=I\cap J$. It must be decidable, though. | |
| Dec 13, 2010 at 14:35 | comment | added | Martin Brandenburg | But it is more "geometric" since only $V(I)$ and $V(J)$ are involved. | |
| Dec 13, 2010 at 14:22 | comment | added | user6976 | The condition with Tor is looking more complicated than the question. | |
| Dec 13, 2010 at 14:14 | history | answered | David E Speyer | CC BY-SA 2.5 |