next | previous | forward | backward | up | top | index | toc | Macaulay2 website
CompleteIntersectionResolutions :: hfModuleAsExt

hfModuleAsExt -- predict betti numbers of moduleAsExt(M,R)

Synopsis

Description

Given a module M over the ring of operators $k[x_1..x_c]$, the call $N = moduleAsExt(M,R)$ produces a module N over the ring R whose ext module is the exterior algebra on n=numgensR generators tensored with M. This script computes numValues values of the Hilbert function of $$ M \otimes \wedge k^n, $$ which should be equal to the total betti numbers of N.

i1 : kk = ZZ/101;
i2 : S = kk[a,b,c];
i3 : ff = matrix{{a^4, b^4,c^4}};

             1       3
o3 : Matrix S  <--- S
i4 : R = S/ideal ff;
i5 : Ops = kk[x_1,x_2,x_3];
i6 : MM = Ops^1/(x_1*ideal(x_2^2,x_3));
i7 : N = moduleAsExt(MM,R);
i8 : betti res( N, LengthLimit => 10)

             0  1  2  3  4  5  6  7  8  9 10
o8 = total: 36 27 29 31 33 35 37 39 41 43 45
        -6: 18  6  .  .  .  .  .  .  .  .  .
        -5:  .  .  .  .  .  .  .  .  .  .  .
        -4: 18 21 21  7  .  .  .  .  .  .  .
        -3:  .  .  .  .  .  .  .  .  .  .  .
        -2:  .  .  8 24 24  8  .  .  .  .  .
        -1:  .  .  .  .  .  .  .  .  .  .  .
         0:  .  .  .  .  9 27 27  9  .  .  .
         1:  .  .  .  .  .  .  .  .  .  .  .
         2:  .  .  .  .  .  . 10 30 30 10  .
         3:  .  .  .  .  .  .  .  .  .  .  .
         4:  .  .  .  .  .  .  .  . 11 33 33
         5:  .  .  .  .  .  .  .  .  .  .  .
         6:  .  .  .  .  .  .  .  .  .  . 12

o8 : BettiTally
i9 : hfModuleAsExt(12,MM,3)

o9 = (23, 25, 27, 29, 31, 33, 35, 37, 39, 41)

o9 : Sequence

See also

Ways to use hfModuleAsExt :

For the programmer

The object hfModuleAsExt is a method function.