The isomorphism between coimage and image is not always obvious, as the following example shows.
i1 : R = QQ[a..d]; |
i2 : M = matrix{{a^3,b^3-c^3,a*b*c,a*(b^2-c^2)}} o2 = | a3 b3-c3 abc ab2-ac2 | 1 4 o2 : Matrix R <--- R |
i3 : image M o3 = image | a3 b3-c3 abc ab2-ac2 | 1 o3 : R-module, submodule of R |
i4 : coimage M o4 = cokernel {3} | 0 -bc 0 -b2+c2 | {3} | a 0 0 0 | {3} | b-c a2 b2-c2 0 | {3} | -b-c 0 -bc a2 | 4 o4 : R-module, quotient of R |
i5 : kernel M o5 = image {3} | 0 -bc 0 -b2+c2 | {3} | a 0 0 0 | {3} | b-c a2 b2-c2 0 | {3} | -b-c 0 -bc a2 | 4 o5 : R-module, submodule of R |