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
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i2 : ff = matrix "a2,b2,c2"
o2 = | a2 b2 c2 |
1 3
o2 : Matrix S <-- S
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i3 : R = S/(ideal ff)
o3 = R
o3 : QuotientRing
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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
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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
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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
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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
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