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LocalRings > LocalRing

LocalRing -- The class of all local rings

Description

Currently only localizations at prime ideals of a polynomial ring are supported.

i1 : S = QQ[x,y,z,w];
 
i2 : I = ideal"xz-y2,yw-z2,xw-yz"; -- The twisted cubic curve

o2 : Ideal of S
 
i3 : R = S_I

o3 = R

                                  2           2
o3 : LocalRing, maximal ideal (- y  + x*z, - z  + y*w, - y*z + x*w)
 
i4 : K = frac(S/I)

o4 = K

o4 : FractionField

The maximal ideal and a residue map to the residue field are stored in the ring.

i5 : max R

               2           2
o5 = ideal (- y  + x*z, - z  + y*w, - y*z + x*w)

o5 : Ideal of R
 
i6 : R.maxIdeal

               2           2
o6 = ideal (- y  + x*z, - z  + y*w, - y*z + x*w)

o6 : Ideal of S
 
i7 : R.residueMap

o7 = map (K, R, {x, y, z, w})

o7 : RingMap K <--- R

Objects over the base ring can be localized easily.

i8 : I ** R

               2           2
o8 = ideal (- y  + x*z, - z  + y*w, - y*z + x*w)

o8 : Ideal of R

See also

Functions and methods returning a local ring :

Methods that use a local ring :

For the programmer

The object LocalRing is a type, with ancestor classes EngineRing < Ring < Type < MutableHashTable < HashTable < Thing.

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