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

extVsCohomology -- compares Ext_S(M,k) as exterior module with coh table of sheaf Ext_R(M,k)

Synopsis

Description

Given a matrix ff containing a regular sequence in a polynomial ring S over k, set R = S/(ideal ff). If N is a graded R-module, and M is the module N regarded as an S-module, the script returns E = Ext_S(M,k) and T = Tor^S(M,k) as modules over an exterior algebra.

The script prints the Tate resolution of E; and the cohomology table of the sheaf associated to Ext_R(N,k) over the ring of CI operators, which is a polynomial ring over k on c variables.

The output can be used to (sometimes) check whether the submodule of Ext_S(M,k) generated in degree 0 splits (as an exterior module

i1 : S = ZZ/101[a,b,c]

o1 = S

o1 : PolynomialRing
i2 : ff = matrix "a2,b2,c2"

o2 = | a2 b2 c2 |

             1       3
o2 : Matrix S  <--- S
i3 : R = S/(ideal ff)

o3 = R

o3 : QuotientRing
i4 : N = highSyzygy(R^1/ideal(a*b,c))

o4 = cokernel {4} | c -ab 0 0 0  0  0  0 0 0 0  0 0  0  0 0 |
              {5} | 0 c   b a 0  0  0  0 0 0 0  0 0  0  0 0 |
              {5} | 0 0   c 0 -b a  0  0 0 0 0  0 0  0  0 0 |
              {5} | 0 0   0 c 0  -b -a 0 0 0 0  0 0  0  0 0 |
              {5} | 0 0   0 0 c  0  0  b a 0 0  0 0  0  0 0 |
              {5} | 0 0   0 0 0  c  0  0 b 0 0  0 -a 0  0 0 |
              {5} | 0 0   0 0 0  0  c  0 0 0 0  0 b  0  a 0 |
              {5} | 0 0   0 0 0  0  0  c 0 b -a 0 0  0  0 0 |
              {5} | 0 0   0 0 0  0  0  0 c 0 b  a 0  0  0 0 |
              {5} | 0 0   0 0 0  0  0  0 0 0 0  b c  -a 0 0 |
              {5} | 0 0   0 0 0  0  0  0 0 0 0  0 0  b  c a |

                            11
o4 : R-module, quotient of R
i5 : E = extVsCohomology(ff,highSyzygy N);
Tate Resolution of Ext_S(M,k) as exterior module:
Note that maps go left to right
       -11 -10  -9 -8 -7 -6 -5 -4 -3 -2  -1
total: 198 146 102 66 38 18  9 16 36 64 100
    8: 106  79  56 37 22 11  4  1  1  1   1
    9:  92  67  46 29 16  7  2  .  .  .   .
   10:   .   .   .  .  .  .  .  5 14 27  44
   11:   .   .   .  .  .  .  3 10 21 36  55
---
Cohomology table of evenExtModule M:
   -5 -4 -3 -2 -1  0  1  2  3  4   5
2: 36 21 10  3  .  .  .  .  .  .   .
1:  .  .  .  .  .  .  .  .  .  .   .
0:  1  1  1  2  7 16 29 46 67 92 121
---
Cohomology table of oddExtModule M:
   -5 -4 -3 -2 -1  0  1  2  3   4   5
2: 28 15  6  1  .  .  .  .  .   .   .
1:  .  .  .  .  .  .  .  .  .   .   .
0:  1  1  1  4 11 22 37 56 79 106 137

See also

Ways to use extVsCohomology :

For the programmer

The object extVsCohomology is a method function.