subquotient modules -- the way Macaulay 2 represents modules

Not all modules arise naturally as submodules or quotients of free modules. As an example, consider the module M = I/I² in the example below.

i1 : R = QQ[x,y,z];

i2 : I = ideal(x*y,x*z,y*z)

o2 = ideal (x*y, x*z, y*z)

o2 : Ideal of R

i3 : M = I/I^2

o3 = subquotient (| xy xz yz |, | x2y2 x2yz xy2z x2z2 xyz2 y2z2 |)

                               1
o3 : R-module, subquotient of R

Macaulay2 represents each module (at least conceptually) as a subquotient module, that is, a submodule of a quotient of an ambient free module. A subquotient module is determined by two matrices f : R^m → Rⁿ and g : R^p → Rⁿ. The subquotient module with generators f and relations g is by definition the module M = ((image f) + (image g))/(image g).

If f is the identity map, M = coker g, and if g = 0, then M = image f. The class of subquotient modules is the smallest class containing free modules, which is closed under taking submodules and quotients.

One may create a subquotient module directly from matrices f and g having the same target free module.

i4 : f = matrix{{x,y}}

o4 = | x y |

             1       2
o4 : Matrix R  <--- R

i5 : g = matrix{{x^2,x*y,y^2,z^4}}

o5 = | x2 xy y2 z4 |

             1       4
o5 : Matrix R  <--- R

i6 : M = subquotient(f,g)

o6 = subquotient (| x y |, | x2 xy y2 z4 |)

                               1
o6 : R-module, subquotient of R

The same module can be constructed in the following manner.

i7 : N = (image f)/(image g)

o7 = subquotient (| x y |, | x2 xy y2 z4 |)

                               1
o7 : R-module, subquotient of R

i8 : N1 = (image f + image g)/(image g)

o8 = subquotient (| x y x2 xy y2 z4 |, | x2 xy y2 z4 |)

                               1
o8 : R-module, subquotient of R

i9 : M === N

o9 = true

Notice that Macaulay2 allows one to write (image f)/(image g), even though mathematically this really means: (image f + image g)/(image g). There is an important difference however. Modules in Macaulay2 always come with an ordered set of generators, and N1 has 4 more generators (all zero in the module!) than N. The modules M and N though are identical.

The two matrices f and g mentioned above are recovered using the routines generators(Module) and relations(Module).

i10 : generators M

o10 = | x y |

              1       2
o10 : Matrix R  <--- R

i11 : relations M

o11 = | x2 xy y2 z4 |

              1       4
o11 : Matrix R  <--- R

Submodules and quotients of free modules work as one would imagine.

i12 : N2 = R*M_0 + I*M

o12 = subquotient (| x x2y xy2 x2z xyz xyz y2z |, | x2 xy y2 z4 |)

                                1
o12 : R-module, subquotient of R

i13 : M/N2

o13 = subquotient (| x y |, | x x2y xy2 x2z xyz xyz y2z x2 xy y2 z4 |)

                                1
o13 : R-module, subquotient of R

i14 : prune(M/N2)

o14 = cokernel {1} | y x z4 |

                             1
o14 : R-module, quotient of R

Given a subquotient module M, there are several useful modules associated to M.The free module of which M is a subquotient is obtained using ambient(Module).

i15 : ambient M

       1
o15 = R

o15 : R-module, free

This is the same as the common target of the matrices of generators and relations.

i16 : ambient M === target relations M

o16 = true

i17 : ambient M === target generators M

o17 = true

M is a submodule of the module R^n/(image g). The routine super(Module) returns this quotient module.

i18 : super M

o18 = cokernel | x2 xy y2 z4 |

                             1
o18 : R-module, quotient of R

This may be obtained directly as the cokernel of the matrix of relations.

i19 : super M === cokernel relations M

o19 = true

Often the given representation of a module is not very efficient. Use trim(Module) to keep the module as a subquotient of the same ambient free module, but change the generators and relations to be minimal, or in the nonlocal or non-graded case, at least more efficient.

i20 : M + M

o20 = subquotient (| x y x y |, | x2 xy y2 z4 |)

                                1
o20 : R-module, subquotient of R

i21 : trim (M+M)

o21 = subquotient (| y x |, | y2 xy x2 z4 |)

                                1
o21 : R-module, subquotient of R

Use minimalPresentation(Module) to also allow the ambient free module to be improved. This currently returns a quotient of a free module, but in the future it might not.

i22 : minimalPresentation M

o22 = cokernel {1} | y 0 x 0 z4 0  |
               {1} | 0 y 0 x 0  z4 |

                             2
o22 : R-module, quotient of R

prune is a synonym for minimalPresentation.

i23 : prune M

o23 = cokernel {1} | y 0 x 0 z4 0  |
               {1} | 0 y 0 x 0  z4 |

                             2
o23 : R-module, quotient of R

For maps between modules, including between subquotient modules, see homomorphisms (maps) between modules.

subquotient modules -- the way Macaulay 2 represents modules

See also