ideals to and from modules

from ideals to modules

An ideal I is also an R-submodule. In Macaulay 2 we distinguish between when we are thinking of I as as ideal or a module. If it is first defined as an ideal, it is easily turned into a module using the function module and for any submodule of the rank one free module R we can obtain an ideal of the generators using the function ideal.

i1 : R = ZZ/32003[x,y,z];

i2 : I = ideal(x^2,y*z-x);

o2 : Ideal of R

i3 : module I

o3 = image | x2 yz-x |

                             1
o3 : R-module, submodule of R

from modules to ideals

For any submodule of the rank one free module R we can obtain an ideal of the generators using the function ideal.

i4 : A = matrix{{x*y-z,z^3}};

             1       2
o4 : Matrix R  <--- R

i5 : M = image A

o5 = image | xy-z z3 |

                             1
o5 : R-module, submodule of R

i6 : ideal M

                      3
o6 = ideal (x*y - z, z )

o6 : Ideal of R

getting the quotient module corresponding to an ideal

We also often work with R/I as an R-module. Simply typing R/I at a prompt in Macaulay 2 constructs the quotient ring (see quotient rings). There are two ways to tell Macaulay 2 that we want to work with this as a module.

i7 : coker generators I

o7 = cokernel | x2 yz-x |

                            1
o7 : R-module, quotient of R

i8 : R^1/I

o8 = cokernel | x2 yz-x |

                            1
o8 : R-module, quotient of R

modules versus ideals for computations

Some functions in Macaulay 2 try to make an informed decision about ideal and modules. For example, if resolution is given an ideal I, it will compute a resolution of the module R^1/I, as in the following example.

i9 : resolution I

      1      2      1
o9 = R  <-- R  <-- R  <-- 0
                           
     0      1      2      3

o9 : ChainComplex

The functions dim and degree also operate on R^1/I if the input is I or R^1/I. However, the function hilbertPolynomial computes the Hilbert polynomial of the module I if the input is hilbertPolynomial I.

For basic information about working with modules see modules.