+ M2 --no-readline --print-width 104 Macaulay2, version 1.6 with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : M = R^5 5 o2 = R o2 : R-module, free i3 : M = R o3 = R o3 : PolynomialRing i4 : M = R^1 1 o4 = R o4 : R-module, free i5 : M = R^3 3 o5 = R o5 : R-module, free i6 : m = matrix{ {x, x^2, y*z}, {y*z, 1, z^2}} o6 = | x x2 yz | | yz 1 z2 | 2 3 o6 : Matrix R <--- R i7 : L = coker m o7 = cokernel | x x2 yz | | yz 1 z2 | 2 o7 : R-module, quotient of R i8 : transpose m o8 = {-2} | x yz | {-2} | x2 1 | {-2} | yz z2 | 3 2 o8 : Matrix R <--- R i9 : ker m o9 = image {2} | -x2z2+yz -x3z2+xyz x3yz2-xy2z | {2} | -y2z2+xz2 -xy2z2+x2z2 xy3z2-x2yz2 | {2} | x2yz-x x3yz-x2 -x3y2z+x2y | 3 o9 : R-module, submodule of R i10 : presentation o9 o10 = {6} | -x xy | {7} | 1 0 | {8} | 0 1 | 3 2 o10 : Matrix R <--- R i11 : m o11 = | x x2 yz | | yz 1 z2 | 2 3 o11 : Matrix R <--- R i12 : I = ideal(x,y,z) o12 = ideal (x, y, z) o12 : Ideal of R i13 : Hom(I, I) o13 = subquotient (| 1 |, | -y 0 -z 0 0 0 0 0 0 |) | 0 | | x -z 0 0 0 0 0 0 0 | | 0 | | 0 y x 0 0 0 0 0 0 | | 0 | | 0 0 0 -y 0 -z 0 0 0 | | 1 | | 0 0 0 x -z 0 0 0 0 | | 0 | | 0 0 0 0 y x 0 0 0 | | 0 | | 0 0 0 0 0 0 -y 0 -z | | 0 | | 0 0 0 0 0 0 x -z 0 | | 1 | | 0 0 0 0 0 0 0 y x | 9 o13 : R-module, subquotient of R i14 : module I o14 = image | x y z | 1 o14 : R-module, submodule of R i15 : Hom(o14, o14) o15 = subquotient (| 1 |, | -y 0 -z 0 0 0 0 0 0 |) | 0 | | x -z 0 0 0 0 0 0 0 | | 0 | | 0 y x 0 0 0 0 0 0 | | 0 | | 0 0 0 -y 0 -z 0 0 0 | | 1 | | 0 0 0 x -z 0 0 0 0 | | 0 | | 0 0 0 0 y x 0 0 0 | | 0 | | 0 0 0 0 0 0 -y 0 -z | | 0 | | 0 0 0 0 0 0 x -z 0 | | 1 | | 0 0 0 0 0 0 0 y x | 9 o15 : R-module, subquotient of R i16 : prune o15 1 o16 = R o16 : R-module, free i17 : Hom(ideal(x,y), R^1) o17 = image {-1} | x | {-1} | y | 2 o17 : R-module, submodule of R i18 : prune o17 1 o18 = R o18 : R-module, free i19 : Hom(R/ideal(x,y), R^1) stdio:19:1:(3): error: expected modules over the same ring i20 : Hom(R^1/ideal(x,y), R^1) o20 = image 0 1 o20 : R-module, submodule of R i21 : prune o20 o21 = 0 o21 : R-module i22 : J = ideal(x,y) o22 = ideal (x, y) o22 : Ideal of R i23 : J ** J stdio:23:3:(3): error: no method for binary operator ** applied to objects: -- ideal (x, y) (of class Ideal) -- ** ideal (x, y) (of class Ideal) i24 : (module J) ** (module J) o24 = cokernel {2} | -y 0 -y 0 | {2} | x 0 0 -y | {2} | 0 -y x 0 | {2} | 0 x 0 x | 4 o24 : R-module, quotient of R i25 :