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CompleteIntersectionResolutions :: EisenbudShamashTotal

EisenbudShamashTotal -- Precursor complex of total Ext

Synopsis

Description

Assume that M is defined over a ring of the form Rbar = R/(f0..fc-1), a complete intersection, and that M has a finite free resolution G over R. In this case M has a free resolution F over Rbar whose dual, F* is a finitely generated, Z/2-graded free module over a ring Sbar≅Rbar[s0..sc-1], where the degrees of the si are the negatives of the degrees of the fi. This resolution is is constructed from the dual of G, together with the duals of the higher homotopies on G defined by Eisenbud.

The function returns the differentials d0:F*even →F*odd and d1:F*odd→F*even.

The maps d0,d1 form a matrix factorization of sum(c, i->si*fi). The have the property that for any Rbar module N,

HH1 chainComplex {d0**N, d1**N}= ExtevenRbar(M,N)

HH1 chainComplex {d1**N, d0**N}= ExtoddRbar(M,N)

i1 : n = 3

o1 = 3
i2 : c = 2

o2 = 2
i3 : kk = ZZ/101

o3 = kk

o3 : QuotientRing
i4 : R = kk[x_0..x_(n-1)]

o4 = R

o4 : PolynomialRing
i5 : I = ideal(x_0^2, x_2^3)

             2   3
o5 = ideal (x , x )
             0   2

o5 : Ideal of R
i6 : ff = gens I

o6 = | x_0^2 x_2^3 |

             1       2
o6 : Matrix R  <--- R
i7 : Rbar = R/I

o7 = Rbar

o7 : QuotientRing
i8 : bar = map(Rbar, R)

o8 = map(Rbar,R,{x , x , x })
                  0   1   2

o8 : RingMap Rbar <--- R
i9 : Mbar = prune coker random(Rbar^1, Rbar^{-2})

o9 = cokernel | x_0x_1+24x_1^2+49x_0x_2+3x_1x_2+5x_2^2 |

                                  1
o9 : Rbar-module, quotient of Rbar
i10 : (d0,d1) = EisenbudShamashTotal Mbar

o10 = ({-2} | x_0x_1+24x_1^2+49x_0x_2+3x_1x_2+5x_2^2
       {-2} | x_0^2                                 
       {-3} | x_2^3                                 
       {-3} | x_1^3+27x_1^2x_2+24x_0x_2^2-18x_1x_2^2
       {-7} | 0                                     
      -----------------------------------------------------------------------
      s_1                         0                                     
      0                           s_1                                   
      0                           -s_0                                  
      -37s_1x_0-21s_1x_1+29s_1x_2 37s_1x_1-5s_1x_2+44s_0                
      -x_0^2                      x_0x_1+24x_1^2+49x_0x_2+3x_1x_2+5x_2^2
      -----------------------------------------------------------------------
      s_0x_0+24s_0x_1+39s_0x_2                   -30s_0                    |,
      0                                          0                         | 
      0                                          0                         | 
      37s_1x_2^3+s_0x_1^2-22s_0x_1x_2+20s_0x_2^2 -s_0x_0+24s_0x_1+39s_0x_2 | 
      x_0x_2^3+24x_1x_2^3+39x_2^4                -30x_2^3                  | 
      -----------------------------------------------------------------------
      {0}  | 0                      s_0       
      {-5} | x_2^3                  0         
      {-5} | 0                      x_2^3     
      {-3} | x_0-24x_1-39x_2        -x_1-49x_2
      {-4} | x_1^2-22x_1x_2+20x_2^2 24x_2^2   
      -----------------------------------------------------------------------
      s_1                                     0                0     |)
      -x_0x_1-24x_1^2-49x_0x_2-3x_1x_2-5x_2^2 0                -s_0  |
      -x_0^2                                  0                0     |
      -7                                      -30              0     |
      -44x_0-46x_1+x_2                        -x_0-24x_1-39x_2 37s_1 |

o10 : Sequence
i11 : d0*d1

o11 = {-2} | s_1x_2^3+s_0x_0^2 0                 0                
      {-2} | 0                 s_1x_2^3+s_0x_0^2 0                
      {-3} | 0                 0                 s_1x_2^3+s_0x_0^2
      {-3} | 0                 0                 0                
      {-7} | 0                 0                 0                
      -----------------------------------------------------------------------
      0                 0                 |
      0                 0                 |
      0                 0                 |
      s_1x_2^3+s_0x_0^2 0                 |
      0                 s_1x_2^3+s_0x_0^2 |

                                     5                              5
o11 : Matrix (kk[s , s , x , x , x ])  <--- (kk[s , s , x , x , x ])
                  0   1   0   1   2              0   1   0   1   2
i12 : d1*d0

o12 = {0}  | s_1x_2^3+s_0x_0^2 0                 0                
      {-5} | 0                 s_1x_2^3+s_0x_0^2 0                
      {-5} | 0                 0                 s_1x_2^3+s_0x_0^2
      {-3} | 0                 0                 0                
      {-4} | 0                 0                 0                
      -----------------------------------------------------------------------
      0                 0                 |
      0                 0                 |
      0                 0                 |
      s_1x_2^3+s_0x_0^2 0                 |
      0                 s_1x_2^3+s_0x_0^2 |

                                     5                              5
o12 : Matrix (kk[s , s , x , x , x ])  <--- (kk[s , s , x , x , x ])
                  0   1   0   1   2              0   1   0   1   2
i13 : S = ring d0

o13 = S

o13 : PolynomialRing
i14 : phi = map(S,R)

o14 = map(S,R,{x , x , x })
                0   1   2

o14 : RingMap S <--- R
i15 : IS = phi I

              2   3
o15 = ideal (x , x )
              0   2

o15 : Ideal of S
i16 : Sbar = S/IS

o16 = Sbar

o16 : QuotientRing
i17 : SMbar = Sbar**Mbar

o17 = cokernel | x_0x_1+24x_1^2+49x_0x_2+3x_1x_2+5x_2^2 |

                                   1
o17 : Sbar-module, quotient of Sbar

Hom(d0,Sbar) and Hom(d1,Sbar) together form the resolution of Mbar; thus the homology of one composition is 0, while the other is Mbar

i18 : prune HH_1 chainComplex{dual (Sbar**d0), dual(Sbar**d1)} == 0

o18 = true
i19 : Mbar' = Sbar^1/(Sbar_0, Sbar_1)**SMbar

o19 = cokernel | x_0x_1+24x_1^2+49x_0x_2+3x_1x_2+5x_2^2 s_0 s_1 |

                                   1
o19 : Sbar-module, quotient of Sbar
i20 : ideal presentation prune HH_1 chainComplex{dual (Sbar**d1), dual(Sbar**d0)} == ideal presentation Mbar'

o20 = true

As a second example we compute Ext(Mbar, Rbar):

i21 : prune HH_1 chainComplex {Sbar**d0,Sbar**d1}

o21 = 0

o21 : Sbar-module
i22 : prune HH_1 chainComplex {Sbar**d1,Sbar**d0}

o22 = cokernel {-2} | s_1 s_0 x_0x_1+24x_1^2+49x_0x_2+3x_1x_2+5x_2^2 |

                                   1
o22 : Sbar-module, quotient of Sbar
i23 : prune Ext(Mbar, Rbar^1)

o23 = cokernel {-1, -2} | X_2 x_0x_1+24x_1^2+49x_0x_2+3x_1x_2+5x_2^2 x_0^2 X_1 x_2^3 |

                                                                         1
o23 : kk[X , X , x , x , x ]-module, quotient of (kk[X , X , x , x , x ])
          1   2   0   1   2                           1   2   0   1   2

See also

Ways to use EisenbudShamashTotal :