intclMonIdeal(Ideal,allComputations=>...) -- normalization of Rees algebra

Synopsis

Usage:

intclMonIdeal I
Inputs:
- an ideal, the leading monomials of the elements of the ideal are considered as generators of a monomial ideal
Outputs:
- an ideal, the integral closure of the input ideal
- an instance of the type MonomialSubalgebra, the normalization of the Rees algebra of I

Description

The leading monomials of the elements of I are considered as generators of a monomial ideal. This function computes the integral closure of I\subset R in the polynomial ring R[t] and the normalization of its Rees algebra. If f_1,\ldots,f_m are the monomial generators of I, then the normalization of the Rees algebra is the integral closure of K[f_1t,\ldots,f_nt] in R[t]. For a definition of the Rees algebra (or Rees ring) see Bruns and Herzog, Cohen-Macaulay rings, Cambridge University Press 1998, p. 182. The function returns the integral closure of the ideal I and the normalization of its Rees algebra. Since the Rees algebra is defined in a polynomial ring with an additional variable, the function creates a new polynomial ring with an additional variable. If the option allComputations is set to true, all data that has been computed by Normaliz is stored in a RationalCone in the CacheTable of the monomial subalgebra returned. This method can also be used with the option grading.

i1 : R=ZZ/37[x,y];

i2 : I=ideal(x^3, x^2*y, y^3, x*y^2);

o2 : Ideal of R

i3 : (intCl,normRees)=intclMonIdeal(allComputations=>true,I)

              3     2   2    3
o3 = (ideal (y , x*y , x y, x ),
                                
                                
                                
                                
                                
     ------------------------------------------------------------------------
     MonomialSubalgebra{cache => CacheTable{...1...}                })
                                           3         2    2      3
                        generators => {y, y a, x, x*y a, x y*a, x a}
                                ZZ
                        ring => --[x..y, a]
                                37

o3 : Sequence

i4 : normRees.cache#"cone"

o4 = RationalCone{"cgr" => | 0 |                                          }
                           | 4 |
                  "equ" => | 0 |
                           | 3 |
                  "gen" => | 0 1 0 |
                           | 0 3 1 |
                           | 1 0 0 |
                           | 1 2 1 |
                           | 2 1 1 |
                           | 3 0 1 |
                  "inv" => HashTable{"" => (1, 1, 1)                     }
                                     "class group" => 1 : (1)
                                     "degree 1 elements" => 6
                                     "dim max subspace" => 0
                                     "embedding dim" => 3
                                     "external index" => 1
                                     "graded" => true
                                     "grading denom" => 1
                                     "grading" => (1, 1, -2)
                                     "hilbert basis elements" => 6
                                     "hilbert quasipolynomial denom" => 1
                                     "hilbert series denom" => (1, 1, 1)
                                     "hilbert series num" => (1, 3)
                                     "ideal multiplicity" => 9
                                     "inhomogeneous" => false
                                     "integrally closed" => true
                                     "internal index" => 1
                                     "multiplicity denom" => 1
                                     "multiplicity" => 4
                                     "number extreme rays" => 4
                                     "number support hyperplanes" => 4
                                     "primary" => true
                                     "rank" => 3
                                     "size triangulation" => 4
                                     "sum dets" => 4
                  "sup" => | 0 0 1  |
                           | 0 1 0  |
                           | 1 0 0  |
                           | 1 1 -3 |

o4 : RationalCone

Further information

Default value: false
Method: intclMonIdeal(Ideal) -- normalization of Rees algebra
Option key: allComputations

Functions with optional argument named allComputations :

diagInvariants(Matrix,Matrix,Ring,allComputations=>...) -- see diagInvariants -- ring of invariants of a diagonalizable group action
ehrhartRing(List,allComputations=>...) -- Ehrhart ring
ehrhartRing(List,RingElement,allComputations=>...) -- Ehrhart ring
finiteDiagInvariants(Matrix,Ring,allComputations=>...) -- see finiteDiagInvariants -- ring of invariants of a finite group action
intclMonIdeal(Ideal,allComputations=>...) -- normalization of Rees algebra
intclMonIdeal(Ideal,RingElement,allComputations=>...) -- normalization of Rees algebra
intclToricRing(List,allComputations=>...) -- integral closure of a toric ring
intclToricRing(MonomialSubalgebra,allComputations=>...) -- integral closure of a toric ring
intersectionValRingIdeals(Matrix,Ring,allComputations=>...) -- see intersectionValRingIdeals -- intersection of valuation ideals
intersectionValRings(Matrix,Ring,allComputations=>...) -- see intersectionValRings -- intersection of ring of valuations
normaliz(List,allComputations=>...) -- see normaliz(List) -- calls Normaliz with several input matrices
normaliz(Matrix,String,allComputations=>...) -- see normaliz(Matrix,String) -- calls Normaliz
normaliz(Matrix,ZZ,allComputations=>...) (missing documentation)
normalToricRing(Ideal,Thing,allComputations=>...) -- see normalToricRing(Ideal,Thing) -- normalization of a toric ring given by a binomial ideal
normalToricRing(List,allComputations=>...) -- normalization of a toric ring
normalToricRing(MonomialSubalgebra,allComputations=>...) -- normalization of a toric ring
torusInvariants(Matrix,Ring,allComputations=>...) -- see torusInvariants -- ring of invariants of torus action