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reesIdeal -- compute the defining ideal of the Rees Algebra

Synopsis

Description

The Rees algebra of a module M over a ring R is here defined, following the paper What is the Rees algebra of a module? David Eisenbud, Craig Huneke and Bernd Ulrich, Proc. Amer. Math. Soc. 131 (2003) 701--708, as follows: If h:F→M is a surjection from a free module, and g: M→G is the universal map to a free module, then the Rees algebra of M is the image of the induced map of Sym(gh): Sym(F)→Sym(G), and thus can be computed with symmetricKernel(gh). The paper above proves that if M is isomorphic to an ideal with inclusion g: M→R (or, in characteristic zero but not in characteristic >0 if M is a submodule of a free module and g’: M→G) is any injection), then the Rees algebra is equal to the image of g’h, so it is unnecessary to compute the universal embedding.

This package gives the user a choice between two methods for finding the defining ideal of the Rees algebra of an ideal or module M over a ring R: The call

reesIdeal(M)

computes the universal embedding g: M→G and a surjection f: F→M and returns the result of symmetricKernel(gf). On the other hand, if the user knows an non-zerodivisor a∈R such that M[a-1 is a free module (this is the case, for example, if a ∈M⊂R and a is a non-zerodivisor), then it is often much faster to call

reesIdeal(M,a)

which finds a surjection f: F→M and returns (J:a) ⊂Sym(F), the saturation of the ideal J:=(ker f)Sym(F). Note that this gives the correct answer even under the slightly weaker hypothesis that M[a-1] is “of linear type”. (See also isLinearType.)

Historical Background: The Rees Algebra of an ideal is the basic commutative algebra analogue of the blow up operation in algebraic geometry. It has many applications, and a great deal of modern work in commutative algebra has been devoted to it. The term "Rees Algebra" (of an ideal I in a ring R, say) is used here to refer to the ring R[It]⊂R[t] which is sometimes called the "blowup algebra" instead. (The origin of the name may be traced to a paper by David Rees, (On a problem of Zariski. Illinois J. Math. (1958)145–149) where Rees used the ring R[It,t-1, now also called the “extended Rees Algebra.”)
i1 : kk = ZZ/101;
i2 : S=kk[x_0..x_4];
i3 : i=monomialCurveIdeal(S,{2,3,5,6})

                          2                       3      2     2      2     2
o3 = ideal (x x  - x x , x  - x x , x x  - x x , x  - x x , x x  - x x , x x 
             2 3    1 4   2    0 4   1 2    0 3   3    2 4   1 3    0 4   0 3
     ------------------------------------------------------------------------
        2     2              3    2
     - x x , x x  - x x x , x  - x x )
        1 4   1 3    0 2 4   1    0 4

o3 : Ideal of S
i4 : time V1 = reesIdeal i;
     -- used 0.095304 seconds

o4 : Ideal of S[w , w , w , w , w , w , w , w ]
                 0   1   2   3   4   5   6   7
i5 : time V2 = reesIdeal(i,i_0);
     -- used 0.242945 seconds

o5 : Ideal of S[w , w , w , w , w , w , w , w ]
                 0   1   2   3   4   5   6   7
This example is particularly interesting upon a bit more exploration.
i6 : numgens V1

o6 = 15
i7 : numgens V2

o7 = 15
The difference is striking and, at least in part, explains the difference in computing time. Furthermore, if we compute a Grobner basis for both and compare the two matrices, we see that we indeed got the same ideal.
i8 : M1 = gens gb V1;

                                               1                                         84
o8 : Matrix (S[w , w , w , w , w , w , w , w ])  <--- (S[w , w , w , w , w , w , w , w ])
                0   1   2   3   4   5   6   7             0   1   2   3   4   5   6   7
i9 : M2 = gens gb V2;

                                               1                                         84
o9 : Matrix (S[w , w , w , w , w , w , w , w ])  <--- (S[w , w , w , w , w , w , w , w ])
                0   1   2   3   4   5   6   7             0   1   2   3   4   5   6   7
i10 : use ring M2

o10 = S[w , w , w , w , w , w , w , w ]
         0   1   2   3   4   5   6   7

o10 : PolynomialRing
i11 : M1 = substitute(M1, ring M2);

                                                1                                         84
o11 : Matrix (S[w , w , w , w , w , w , w , w ])  <--- (S[w , w , w , w , w , w , w , w ])
                 0   1   2   3   4   5   6   7             0   1   2   3   4   5   6   7
i12 : M1 == M2

o12 = true
i13 : numgens source M2

o13 = 84
Another example illustrates the power and usage of the code. We also show the output in this example. While a bit messy, the user can see how we handle the degrees in both cases.
i14 : S=kk[a,b,c]

o14 = S

o14 : PolynomialRing
i15 : m=matrix{{a,0},{b,a},{0,b}}

o15 = | a 0 |
      | b a |
      | 0 b |

              3       2
o15 : Matrix S  <--- S
i16 : i=minors(2,m)

              2        2
o16 = ideal (a , a*b, b )

o16 : Ideal of S
i17 : time reesIdeal i
     -- used 0.028007 seconds

                                        2
o17 = ideal (b*w  - a*w , b*w  - a*w , w  - w w )
                1      2     0      1   1    0 2

o17 : Ideal of S[w , w , w ]
                  0   1   2
i18 : res i

       1      3      2
o18 = S  <-- S  <-- S  <-- 0
                            
      0      1      2      3

o18 : ChainComplex
i19 : m=random(S^3,S^{4:-1})

o19 = | -9a-31b+36c -39a-46b+c -26a-17b-9c -17a-8b-16c |
      | 11a-45b+20c -a+27b+33c 50a-45b+42c 19a-19b+40c |
      | 46a-25b-45c 8a+5b+7c   42a-3b+20c  2a-49b+10c  |

              3       4
o19 : Matrix S  <--- S
i20 : i=minors(3,m)

                3    2         2      3      2                2         2  
o20 = ideal (17a  - a b + 16a*b  + 17b  - 44a c + 17a*b*c - 2b c + 22a*c  +
      -----------------------------------------------------------------------
           2     3     3     2        2      3      2                2   
      27b*c  - 2c , 18a  - 3a b + 3a*b  + 19b  + 46a c + 15a*b*c + 8b c +
      -----------------------------------------------------------------------
           2        2      3     3      2         2      3     2             
      13a*c  + 45b*c  - 41c , 24a  - 23a b + 29a*b  + 32b  + 9a c + 43a*b*c -
      -----------------------------------------------------------------------
        2         2        2     3     2         2      3      2            
      9b c + 47a*c  + 19b*c , 28a  - 7a b - 20a*b  - 40b  - 33a c + 5a*b*c +
      -----------------------------------------------------------------------
         2         2        2      3
      32b c + 29a*c  + 12b*c  - 29c )

o20 : Ideal of S
i21 : time I=reesIdeal (i,i_0);
     -- used 0.009253 seconds

o21 : Ideal of S[w , w , w , w ]
                  0   1   2   3
i22 : transpose gens I

o22 = {-1, -4} | w_0c+19w_1a-29w_1b-45w_1c-20w_2a+36w_2b-9w_2c+41w_3a-38w_3b-
      {-1, -4} | w_0b-48w_1a-11w_1b-50w_1c-17w_2a-24w_2b-16w_2c+31w_3a-24w_3b
      {-1, -4} | w_0a+21w_1a-22w_1b+41w_2a-8w_2b+10w_2c-24w_3a-32w_3b        
      {-3, -9} | w_0^3-35w_0^2w_1+45w_0w_1^2-14w_1^3+8w_0^2w_2-8w_0w_1w_2+23w
      -----------------------------------------------------------------------
      27w_3c                                                               
      +8w_3c                                                               
                                                                           
      _1^2w_2+8w_0w_2^2+w_1w_2^2+16w_2^3+26w_0^2w_3+37w_0w_1w_3-37w_1^2w_3-
      -----------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
      17w_0w_2w_3+16w_1w_2w_3+15w_2^2w_3+37w_0w_3^2+31w_1w_3^2+14w_2w_3^2-33w
      -----------------------------------------------------------------------
           |
           |
           |
      _3^3 |

                                4                         1
o22 : Matrix (S[w , w , w , w ])  <--- (S[w , w , w , w ])
                 0   1   2   3             0   1   2   3
i23 : i=minors(2,m);

o23 : Ideal of S
i24 : time I=reesIdeal (i,i_0);
     -- used 0.107399 seconds

o24 : Ideal of S[w , w , w , w , w , w , w , w , w , w , w  , w  , w  , w  , w  , w  , w  , w  ]
                  0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17
Investigating plane curve singularities
i25 : R = ZZ/32003[x,y,z]

o25 = R

o25 : PolynomialRing
i26 : I = ideal(x,y)

o26 = ideal (x, y)

o26 : Ideal of R
i27 : cusp = ideal(x^2*z-y^3)

               3    2
o27 = ideal(- y  + x z)

o27 : Ideal of R
i28 : RI = reesIdeal(I)

o28 = ideal(y*w  - x*w )
               0      1

o28 : Ideal of R[w , w ]
                  0   1
i29 : S = ring RI

o29 = S

o29 : PolynomialRing
i30 : totalTransform = substitute(cusp, S) + RI

                3    2
o30 = ideal (- y  + x z, y*w  - x*w )
                            0      1

o30 : Ideal of S
i31 : D = decompose totalTransform -- the components are the proper transform of the cuspidal curve and the exceptional curve

                            3    2             2       2      2
o31 = {ideal (y*w  - x*w , y  - x z, x*z*w  - y w , z*w  - y*w ), ideal (y,
                 0      1                 0      1     0      1
      -----------------------------------------------------------------------
      x)}

o31 : List
i32 : totalTransform = first flattenRing totalTransform

                3    2
o32 = ideal (- y  + x z, w y - w x)
                          0     1

                 ZZ
o32 : Ideal of -----[w , w , x, y, z]
               32003  0   1
i33 : L = primaryDecomposition totalTransform

                          3    2              2   2     2            2      
o33 = {ideal (w y - w x, y  - x z, w x*z - w y , w z - w y), ideal (y , x*y,
               0     1              0       1     0     1                   
      -----------------------------------------------------------------------
       2
      x , w y - w x)}
           0     1

o33 : List
i34 : apply(L, i -> (degree i)/(degree radical i))

o34 = {1, 2}

o34 : List
The total transform of the cusp contains the exceptional with multiplicity two. The proper transform of the cusp is a smooth curve but is tangent to the exceptional curve.
i35 : use ring L_0

        ZZ
o35 = -----[w , w , x, y, z]
      32003  0   1

o35 : PolynomialRing
i36 : singular = ideal(singularLocus(L_0));

                 ZZ
o36 : Ideal of -----[w , w , x, y, z]
               32003  0   1
i37 : SL = saturate(singular, ideal(x,y,z));

                 ZZ
o37 : Ideal of -----[w , w , x, y, z]
               32003  0   1
i38 : saturate(SL, ideal(w_0,w_1)) -- we get 1 so it is smooth.

o38 = ideal 1

                 ZZ
o38 : Ideal of -----[w , w , x, y, z]
               32003  0   1

Caveat

See also

Ways to use reesIdeal :

  • reesIdeal(Ideal)
  • reesIdeal(Ideal,RingElement)
  • reesIdeal(Module)
  • reesIdeal(Module,RingElement)