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K3Carpets :: schreyerName

schreyerName -- get the names of generators in the (nonminimal) Schreyer resolution according to Schreyer's convention

Synopsis

Description

We name the generators of the syzygies by the list of the monomial parts of the leadTerm with position m recursively:

schreyerName(F,i,n) = append(schreyerName(F,i-1,m),mon)

where mon denotes the monomial part and m the position in F_(i-1) of leadTerm F.dd_i_n.

i1 : (a,b)=(5,4)

o1 = (5, 4)

o1 : Sequence
i2 : I = carpet(a,b);

                ZZ
o2 : Ideal of -----[x ..x , y ..y ]
              32003  0   5   0   4
i3 : F = res(I, FastNonminimal =>true)

        ZZ                  1        ZZ                  28        ZZ                  127        ZZ                  286        ZZ                  385        ZZ                  323        ZZ                  166        ZZ                  48        ZZ                  6
o3 = (-----[x ..x , y ..y ])  <-- (-----[x ..x , y ..y ])   <-- (-----[x ..x , y ..y ])    <-- (-----[x ..x , y ..y ])    <-- (-----[x ..x , y ..y ])    <-- (-----[x ..x , y ..y ])    <-- (-----[x ..x , y ..y ])    <-- (-----[x ..x , y ..y ])   <-- (-----[x ..x , y ..y ])  <-- 0
      32003  0   5   0   4         32003  0   5   0   4          32003  0   5   0   4           32003  0   5   0   4           32003  0   5   0   4           32003  0   5   0   4           32003  0   5   0   4           32003  0   5   0   4          32003  0   5   0   4         
                                                                                                                                                                                                                                                                                      9
     0                            1                             2                              3                              4                              5                              6                              7                             8

o3 : ChainComplex
i4 : betti F

            0  1   2   3   4   5   6  7 8
o4 = total: 1 28 127 286 385 323 166 48 6
         0: 1  .   .   .   .   .   .  . .
         1: . 28 105 184 185 110  36  5 .
         2: .  .  22 101 195 203 120 38 5
         3: .  .   .   1   5  10  10  5 1

o4 : BettiTally
i5 : i=3,n=10

o5 = (3, 10)

o5 : Sequence
i6 : schreyerName(F,3,10)

o6 = {x x , x , x }
       3 4   3   1

o6 : List
i7 : h=schreyerName F;
i8 : h#8

              2                                   2                          
o8 = {{y y , x , y , y , x , x , x , x }, {y y , x , y , y , x , x , x , x },
        1 3   1   2   1   5   4   3   2     2 3   1   2   1   5   4   3   2  
     ------------------------------------------------------------------------
       2                                   2                                
     {y , x y , y , y , x , x , x , x }, {y , x y , y , y , x , x , x , x },
       3   2 0   2   1   4   3   2   1     3   3 0   2   1   4   3   2   1  
     ------------------------------------------------------------------------
       2                                   2         2
     {y , x y , y , y , x , x , x , x }, {y , x y , x , y , y , x , x , x }}
       3   4 0   2   1   4   3   2   1     3   5 0   1   2   1   4   3   2

o8 : List
i9 : h#7_20

       2
o9 = {y , x x , y , y , x , x , x }
       3   1 4   2   1   3   2   1

o9 : List
i10 : h#7_20 == schreyerName(F,7,20)

o10 = true

Ways to use schreyerName :

For the programmer

The object schreyerName is a method function.