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

## Synopsis

• Usage:
(E,T) = extVsCohomology(ff,N)
• Inputs:
• ff, , regular sequence in a regular ring S
• N, , graded module over R = S/ideal(ff) (usually a high syzygy)
• Outputs:
• E, ,
• T, , Ext and Tor as exterior modules

## 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