[Macaulay2] localization

[Baptiste Calmès]

Hi everybody,

How far is the process of localizing a ring at a multiplicative subset implemented in M2? I can't find anything apart from the total fraction field in the documentation. Am I missing something, or is it just not implemented? It seems to be such an important operation in commutative algebra, that one would like to have it in M2.

Baptiste


--
You received this message because you are subscribed to the Google Groups "Macaulay2" group.
To post to this group, send email to maca...@googlegroups.com.
To unsubscribe from this group, send email to macaulay2+...@googlegroups.com.
For more options, visit this group at http://groups.google.com/group/macaulay2?hl=en.

[David Cook II]

A special case of what you want is already implemented: The LocalRings
package implements localisation of a ring at a maximal ideal.

On May 7, 6:52 am, "Baptiste Calmès" <baptiste.cal...@gmail.com>
wrote:

[Baptiste Calmès]

David Cook II wrote:
> A special case of what you want is already implemented: The LocalRings
> package implements localisation of a ring at a maximal ideal.

Thanks for pointing this out. I hadn't noticed this package.

Unfortunately, it seems that this package does not have a very detailed documentation, so I'm not really sure how to use it. It seems that the local ring obtained by localizing at a maximal ideal is more or less encoded simply by keeping the original ring and the maximal ideal. It is not completely clear to me how to compute in this ring.

I was actually trying to use M2 while teaching some intersection theory to students. I wanted to compute multiplicities of intersections for curves. If A and B are polynomials defining curves in k[x,y], one can always compute the dimension of k[x,y]/(a,b) by using degree(k[x,y]/ideal(a,b)), but it only gives the sum of multiplicities of intersections of the various points of intersection. One would like to be able to isolate each point, and compute the multiplicity of intersection at this point. Localizing would be great for that...

[Daniel R. Grayson]

You might also try using the function "primaryDecomposition" for that,
if the points are rational.

On May 8, 2010, at 2:05 PM, Baptiste Calmès wrote:

>
> I was actually trying to use M2 while teaching some intersection
> theory to students. I wanted to compute multiplicities of
> intersections for curves. If A and B are polynomials defining curves
> in k[x,y], one can always compute the dimension of k[x,y]/(a,b) by
> using degree(k[x,y]/ideal(a,b)), but it only gives the sum of
> multiplicities of intersections of the various points of
> intersection. One would like to be able to isolate each point, and
> compute the multiplicity of intersection at this point. Localizing
> would be great for that...

[Baptiste Calmès]

Le 8 mai 2010 à 21:20, Daniel R. Grayson a écrit :

> You might also try using the function "primaryDecomposition" for that, if the points are rational.

Yes, you are right, of course, this is a good way of dealing with multiplicities.

But nevertheless, it would be nice to be able to localize more often ;-)

Baptiste

[Susobhan Mazumdar]

Is there any improvement of localizing a ring at a multiplicative system?

[Prajwal Samal]

Is it possible to localize a polynomial ring at a single variable in Macaulay2?

[Irena Swanson]

To localize at an element f in a ring R, adjoin a new variable, say Y, and for the new ring R[Y]/ideal(Y*f-1).  This ring is just R_f.

About implementation in Macaulay2, however, depending on what you need to do with it, if R = k[x_1..x_5], your new ring may have to be k[X_1..X_5,Y]/ideal(Y*f - 1), with appropriately mapped variables.  I hope that a better M2 programmer chimes in with a slick solution.

Irena

To unsubscribe from this group and stop receiving emails from it, send an email to macaulay2+...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/macaulay2/897e6c54-1e89-4376-800e-bd110b969d55n%40googlegroups.com.