bug in hook for (quotient, Matrix, Matrix) in local rings

[mikestillman]

@mahrud, @ggsmith: the following example is a boiled down version of a bug found by Hugh Geller, Tony Se, and Ela Celikbas. We suspect that the hook is somehow introducing permutations of the columns, but we haven't investigated that.

needsPackage "LocalRings"
R = localRing(QQ[a,b,c,u,v,w],ideal(a,b,c,u,v,w))
F = map(R^{{-4}, {-4}, {-5}, {-4}, {-4}, {-3}, {-3}, {-4}, {-4}, {-4}, {-4}, {-3}, {-4}},
        R^{{-5}, {-5}, {-5}, {-5}, {-5}, {-5}, {-4}, {-4}, {-4}, {-4}, {-4}, {-4}, {-4}, {-4}, {-4}, {-4}, {-4}, {-4}, {-4}, {-4}, {-4}},
        {{0, 0, 0, 0, 0, 0, (c^2*v)/(a+1), 0, -v, 0, (-b*c^2)/(a+1), 0, 0, (-c^3-a^2-a)/(a+1), 0, 0, -b, 0, 0, 0, 0}, 
         {0, 0, 0, 0, 0, 0, (b^2*v+v)/(a+1), -v, 0, 0, (-b^3-a^2-a-b)/(a+1), 0, 0, (-b^2*c-c)/(a+1), 0, 0, c, 0, 0, 0, 0},
         {0, 0, 0, 0, 0, 0, 0, 0, 0, (-b*v)/(u+1), (-b*w)/(u+1), 0, (-c*v)/(u+1), (-c*w)/(u+1), 0, 0, 0, 0, 0, 0, 0},
         {0, 0, 0, 0, 0, 0, 0, 0, 0, (v^3+u^2+u+v)/(u+1), (v^2*w+w)/(u+1), -w, 0, 0, 0, 0, 0, 0, (-c*v^2-c)/(u+1), c, 0},
         {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (v^3+u^2+u+v)/(u+1), (v^2*w+w)/(u+1), -w, 0, 0, 0, (b*v^2+b)/(u+1), -b, 0},
         {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (v^3+u^2+u+v)/(u+1), (v^2*w+w)/(u+1), -w, (-a*v^2-a)/(u+1), a, 0}, 
         {0, 0, 0, 0, 0, 0, 0, (-c^2*u)/(b^2+1), u, 0, 0, (a*c^2)/(b^2+1), 0, 0, -a, 0, 0, (-b^3-c^3-b)/(b^2+1), 0, 0, 0}, 
         {0, 0, 0, 0, 0, 0, (-b^2*w-w)/(a+1), w, 0, (-b^3-a^2-a-b)/(a+1), 0, 0, (-b^2*c-c)/(a+1), 0, 0, c, 0, 0, 0, 0, 0}, 
         {0, 0, 0, 0, 0, 0, (-c^2*w)/(a+1), 0, w, (-b*c^2)/(a+1), 0, 0, (-c^3-a^2-a)/(a+1), 0, 0, -b, 0, 0, 0, 0, 0}, 
         {0, 0, 0, 0, 0, 0, 0, 0, 0, (v*w^2)/(u+1), (w^3+u^2+u)/(u+1), v, 0, 0, 0, 0, 0, 0, (-c*w^2)/(u+1), 0, c}, 
         {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (v*w^2)/(u+1), (w^3+u^2+u)/(u+1), v, 0, 0, 0, (b*w^2)/(u+1), 0, -b}, 
         {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (v*w^2)/(u+1), (w^3+u^2+u)/(u+1), v, (-a*w^2)/(u+1), 0, a}, 
         {0, 0, 0, 0, 0, 0, u, (-a*u-u)/(b^2+1), 0, 0, 0, (b^3+a^2+a+b)/(b^2+1), 0, 0, c, 0, 0, (-a*c-c)/(b^2+1), 0, 0, 0}
         })
G = map(R^{{-4}, {-4}, {-5}, {-4}, {-4}, {-3}, {-3}, {-4}, {-4}, {-4}, {-4}, {-3}, {-4}},
    R^{{-5}, {-5}, {-6}, {-6}, {-5}, {-5}, {-6}, {-6}, {-5}, {-5}, {-5}, {-4}, {-5}},
    {{-v, 0, 0, 0, 0, 0, (b*c^2)/(a+1), (c^3+a^2+a)/(a+1), b, 0, 0, 0, 0}, 
     {0, 0, 0, 0, 0, 0, (b^3+a^2+a+b)/(a+1), (b^2*c+c)/(a+1), -c, 0, 0, 0, -v}, 
     {0, 0, (b*v)/(u+1), (c*v)/(u+1), 0, 0, (b*w)/(u+1), (c*w)/(u+1), 0, 0, 0, 0, 0}, 
     {0, c, (-v^3-u^2-u-v)/(u+1), 0, 0, 0, (-v^2*w-w)/(u+1), 0, 0, w, 0, 0, 0}, 
     {0, -b, 0, (-v^3-u^2-u-v)/(u+1), 0, 0, 0, (-v^2*w-w)/(u+1), 0, 0, w, 0, 0}, 
     {0, a, 0, 0, (-v^3-u^2-u-v)/(u+1), 0, 0, 0, (-v^2*w-w)/(u+1), 0, 0, w, 0}, 
     {u, 0, 0, 0, 0, 0, 0, 0, 0, (-a*c^2)/(b^2+1), a, (b^3+c^3+b)/(b^2+1), (-c^2*u)/(b^2+1)}, 
     {0, 0, (b^3+a^2+a+b)/(a+1), (b^2*c+c)/(a+1), -c, 0, 0, 0, 0, 0, 0, 0, w}, 
     {w, 0, (b*c^2)/(a+1), (c^3+a^2+a)/(a+1), b, 0, 0, 0, 0, 0, 0, 0, 0}, 
     {0, 0, (-v*w^2)/(u+1), 0, 0, c, (-w^3-u^2-u)/(u+1), 0, 0, -v, 0, 0, 0}, 
     {0, 0, 0, (-v*w^2)/(u+1), 0, -b, 0, (-w^3-u^2-u)/(u+1), 0, 0, -v, 0, 0}, 
     {0, 0, 0, 0, (-v*w^2)/(u+1), a, 0, 0, (-w^3-u^2-u)/(u+1), 0, 0, -v, 0}, 
     {0, 0, 0, 0, 0, 0, 0, 0, 0, (-b^3-a^2-a-b)/(b^2+1), -c, (a*c+c)/(b^2+1), (-a*u-u)/(b^2+1)}
     })

matrix{for i from 0 to numcols F - 1 list F_{i} % G}
F % G -- this should give the same answer as the previous line

matrix{for i from 0 to numcols F - 1 list F_{i} // G}
F // G -- this should give the same answer as the previous line

[mahrud]

We suspect that the hook is somehow introducing permutations of the columns, but we haven't investigated that.

It's definitely permuting the columns, because the columns of syz(F | G) seem to be sorted incorrectly for some reason. I recall asking about this on google groups, but there wasn't a solution, so instead I resorted to kludge ... which included a bug because I didn't reverse the permutation. Should be fixed by #2543.

If you happen to have time to take a fresh look at this, the relevant email is here: https://groups.google.com/g/macaulay2/c/h0IXFj3oMz4/m/LKFDWcN0BAAJ I'd love a better solution so I can remove this column reordering hack!