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quotient.m2
M = matrix {{1},{0}} N = matrix {{0},{1}} assert(M // N == 0) M = sub(matrix {{1},{0}}, QQ) N = sub(matrix {{0},{1}}, QQ) assert(M // N == 0) M = sub(matrix {{1},{0}}, ZZ/32003) N = sub(matrix {{0},{1}}, ZZ/32003) assert(M // N == 0) K = GF(32003^2) M = sub(matrix {{1},{0}}, K) N = sub(matrix {{0},{1}}, K) assert(M // N == 0) -- Example from Mike Stillman for factoring maps of non-free modules -- see https://github.com/Macaulay2/M2/pull/3567#pullrequestreview-2638871494 S = ZZ/101[a..d]/(a^2+b^2+c^2+d^2) F = res(coker vars S, LengthLimit => 10) A = coker F.dd_4 B = coker F.dd_3 C = coker F.dd_5 HAB = Hom(A,B) basis(0, HAB) f = homomorphism random HAB HBC = Hom(B,C) basis(3, HBC) gd = homomorphism random(3, HBC) g = map(target gd ** S^{3}, source gd, gd) assert(degree g == {0} and isHomogeneous g) h = g*f -- So: f : A --> B -- g : B --> C ** S^{3} -- h : A --> C ** S^{3} assert(g * (h//g) == h) assert(h // g == f) assert(f \\ h == g) assert((f\\h) * f == h) assert(g // h == 0) -- does it always happen that this is zero if lift can't occur? Probably not assert(h \\ f == 0)
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