inverse of matrix is not well defined

[mahrud]

I have a well defined, surjective map of non-free modules, but its inverse is not well defined. What is causing this?

n = 4
S = ZZ/11[x_0..x_(n-1)]
R = quotient minors_2 matrix { S_*_{0..n-2}, S_*_{1..n-1}}

g = map(cokernel(map(R^{3:{-3}},R^{6:{-4}},{{x_2-4*x_3, x_1, 5*x_3, x_0, 0, 0}, {-x_3, 0, x_2+4*x_3, 0, x_1, x_0}, {0,
       3*x_3, 3*x_3, 3*x_2, 3*x_2-x_3, 3*x_1-x_2}})),cokernel(map(R^{6:{-3}},R^{12:{-4}},{{2*x_2, 2*x_0+4*x_1-x_2, -5*x_2,
       2*x_2, 2*x_1+4*x_2, 2*x_1+4*x_2, 4*x_3, -5*x_1-4*x_2, -5*x_0-4*x_1, 5*x_2, 5*x_1-3*x_2, 5*x_0-3*x_1+2*x_2},
       {5*x_2+2*x_3, 5*x_0-4*x_1+5*x_2-x_3, 3*x_2-5*x_3, 5*x_2+x_3, 5*x_1-4*x_2+4*x_3, 5*x_1-4*x_2+4*x_3, 4*x_3,
       3*x_1+2*x_2-4*x_3, 3*x_0+2*x_1-5*x_2, 5*x_3, -2*x_2-3*x_3, -2*x_1-x_2+2*x_3}, {x_1, 0, 0, 0, x_0, 0, x_2, 0, 0, 0, 0,
       0}, {0, x_0, 0, x_2, 0, x_1, 4*x_3, 0, 0, 0, 0, 0}, {2*x_3, x_3, x_2+5*x_3, -2*x_3, -4*x_3, -4*x_3, 4*x_3, x_1+4*x_3,
       x_0, -5*x_3, 3*x_3, -2*x_3}, {-x_3, -x_3, -3*x_3, -5*x_3, -5*x_3, 5*x_3, 5*x_3, 4*x_3, x_3, x_2, x_1-4*x_3,
       x_0-2*x_3}})),{{4, -5, 1, 0, 0, 0}, {0, -4, 0, 1, 0, 0}, {2, -3, 0, 0, -3, 1}})

isWellDefined g and isSurjective g -- g should be invertible
(g * inverse g) == id_(target g) -- and it can be inverted
isWellDefined inverse g -- but the inverse is not well defined

isWellDefined (id_(target g) // g) -- this one works, but is much slower

cc: @Devlin-Mallory

[Devlin-Mallory]

I think inverse g doesn't work for a map where the target is non-free, although no warning is given. In the code above, inverse g calls quotientRemainder(id_(target g), g); the first line of quotientRemainder(f, g) is L := source f; -- result may not be well defined if L is not free.

In your example, inverse g is the same matrix as inverse cover g, so I suspect it's essentially treating g as a map of free modules; the "naive" choice of inverse then may or may not respect the defining relations of the module M.

(Also, just a note that in general isWellDefined g and isSurjective g is not enough to guarantee g is invertible - this is sufficient in general only when target g is a projective module. I think maybe id_(target g) // g is the way to go.)

[mahrud]

I agree with you in general, but in this case g is a split surjection. I'm not sure what the algorithm does exactly, but I would expect that at least in this case there should be an easier way to compute the inverse.

[mahrud]

As you mention, id_(target g) // g works fine, but I think g \\ id_(source g) should also work, except it seems to be failing in this example:

R = ZZ/32003[x,y,z]/(x^3, x^2*y, x*y^2, y^4, y^3*z)
M = cokernel(map(R^{5:{1}, 6:{0}, 2:{-1}}, R^{7:{0}, 20:{-1}, 9:{-2}}, {{0, z, 0, y, 0, x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, z, 0, y, 0, x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {-x, 0, 0, 0, 0, 0, 0, x*y, 0, 0, 0, x^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y^2*z, 0, 0, 0, 0, y^3, 0, 0, 0}, {z, 0, 0, 0, 0, 0, 0, -y*z, y^2, 0, 0, -x*z, x*y, 0, 0, x^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {y, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, z, -y, x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y*z, 0, 0, 0, 0, y^2, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y, z, 0, x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, y, 0, z, x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z, 0, 0, y, 0, 0, x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z, 0, 0, y, 0, 0, x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z, 0, 0, y, 0, 0, x, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -z, 0, y, 0, x, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, z, 0, 0, 0, y, x}}))
N = cokernel(map(R^{2:{1}}, R^6, {{z, y, x, 0, 0, 0}, {0, 0, 0, z, y, x}}))
g = map(N, M, {{10821, -12119, 15659, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {-7155, -15464, 7853, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}})

g * (id_N // g) -- identity
g * (g \\ id_M) -- zero!!
isWellDefined (g \\ id_M) -- true

[Devlin-Mallory]

Maybe I'm missing something, but I think the difference here is between being a left and right inverse.

Given g: N -> M, id_N // g asks for a map h: N -> M such that g * h is id_N, while g \ id_M asks for a map h': N -> M such that h' * g is id_M. In the example, g is a surjection with nontrivial kernel, so a right inverse can (and does) exist, while a left inverse never exists, so in particular (g \ id_M) returns 0.

(If you were trying to split an injection g, then g \ id_(source g) would be the one to use.)

[mahrud]

Ah, of course, I drew my diagram wrong, thanks!

(The original issue still stands)