i1 : R = ZZ/101[x,y,z] o1 = R o1 : PolynomialRing |
i2 : f = matrix{{x^2*y+1,x+y-2,2*x*y}} o2 = | x2y+1 x+y-2 2xy | 1 3 o2 : Matrix R <--- R |
i3 : isUnimodular f o3 = true |
i4 : P1 = coker transpose f -- Construct the cokernel of the transpose of f. o4 = cokernel {-3} | x2y+1 | {-1} | x+y-2 | {-2} | 2xy | 3 o4 : R-module, quotient of R |
i5 : isProjective P1 o5 = true |
i6 : rank P1 o6 = 2 |
i7 : phi1 = qsIsomorphism P1 o7 = {-3} | 50x 0 | {-1} | 0 1 | {-2} | -1 0 | o7 : Matrix |
i8 : isIsomorphism phi1 o8 = true |
i9 : image phi1 == P1 o9 = true |
i10 : P2 = ker f -- Construct the kernel of f. o10 = image {3} | 0 x+y-2 y2-2y | {1} | xy -x2y-xy2+2xy-1 -xy3+2xy2-y | {2} | 50x+50y+1 -50xy-50y2-x-2y+2 -50y3-2y2+2y-50 | 3 o10 : R-module, submodule of R |
i11 : isProjective P2 o11 = true |
i12 : rank P2 o12 = 2 |
i13 : phi2 = qsIsomorphism P2 o13 = {3} | 0 0 | {4} | 1 0 | {5} | 0 1 | o13 : Matrix |
i14 : isIsomorphism phi2 o14 = true |
i15 : image phi2 == P2 o15 = true |
i16 : P3 = image f -- Construct the image of f. o16 = image | x2y+1 x+y-2 2xy | 1 o16 : R-module, submodule of R |
i17 : isProjective P3 o17 = true |
i18 : rank P3 o18 = 1 |
i19 : phi3 = qsIsomorphism P3 o19 = {3} | -1 | {1} | 0 | {2} | -50x | o19 : Matrix |
i20 : isIsomorphism phi3 o20 = true |
i21 : image phi3 == P3 o21 = true |
i22 : P4 = coimage f -- Construct the coimage of f. o22 = cokernel {3} | 0 x+y-2 y2-2y | {1} | xy -x2y-xy2+2xy-1 -xy3+2xy2-y | {2} | 50x+50y+1 -50xy-50y2-x-2y+2 -50y3-2y2+2y-50 | 3 o22 : R-module, quotient of R |
i23 : isProjective P4 o23 = true |
i24 : rank P4 o24 = 1 |
i25 : phi4 = qsIsomorphism P4 o25 = {3} | -1 | {1} | 0 | {2} | -50x | o25 : Matrix |
i26 : isIsomorphism phi4 o26 = true |
i27 : image phi4 == P4 o27 = true |
The object qsIsomorphism is a method function with options.