A possible obstruction to the commutativity of the CI operators in codim c, even assymptotically, would be the non-triviality of the map M_{(k+4)} --> M_k \otimes \wedge^2(S^c) in the stable category of maximal Cohen-Macaulay modules.
In the following example, studied in the paper "Tor as a module over an exterior algebra" of Eisenbud, Peeva and Schreyer, the map is non-trivial...but it is stably trivial. The same goes for higher values of k (which take longer to compute). (note that in this case, with c = 3, two of the three alternating products are actually equal to 0, so we test only the third.)
Note that T is well-defined up to homotopy; so T^2 is well-defined mod mm^2.
i1 : kk = ZZ/101 o1 = kk o1 : QuotientRing |
i2 : S = kk[a,b,c] o2 = S o2 : PolynomialRing |
i3 : ff = matrix"a2,b2,c2" o3 = | a2 b2 c2 | 1 3 o3 : Matrix S <--- S |
i4 : R = S/ideal ff o4 = R o4 : QuotientRing |
i5 : M = R^1/ideal"a,bc" o5 = cokernel | a bc | 1 o5 : R-module, quotient of R |
i6 : k = 1 o6 = 1 |
i7 : m = k+5 o7 = 6 |
i8 : F = res(M, LengthLimit => m) 1 2 4 7 11 16 22 o8 = R <-- R <-- R <-- R <-- R <-- R <-- R 0 1 2 3 4 5 6 o8 : ChainComplex |
i9 : syzygies = apply(1..m, i->coker F.dd_i); |
i10 : t1 = makeT(ff,F,k+4); |
i11 : t2 = makeT(ff,F,k+2); |
i12 : T2Components = flatten for i from 0 to 1 list(for j from i+1 to 2 list map(F_k, F_(k+4), t2_i*t1_j-t2_j*t1_i)); |
i13 : g = map(syzygies_k, syzygies_(k+4), T2Components_2) o13 = {1} | 0 0 0 0 0 -c 0 0 b 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | o13 : Matrix |
i14 : isStablyTrivial g o14 = true |
The object isStablyTrivial is a method function.