resolution(Module) -- compute a free resolution of a module

Synopsis

• Function: resolution

• Usage:

  resolution M

  res M

• Inputs:
  • M, a module
• Optional inputs:
  • DegreeLimit => ..., default value null, compute only up to this degree
  • FastNonminimal => ..., default value false, compute a non-minimal graded free resolution
  • HardDegreeLimit => ..., default value {},
  • LengthLimit => ..., default value infinity, stop when the resolution reaches this length
  • PairLimit => ..., default value infinity, stop when this number of pairs has been handled
  • ParallelizeByDegree (missing documentation) => ..., default value false,
  • SortStrategy => ..., default value 0,
  • StopBeforeComputation => ..., default value false, whether to stop the computation immediately
  • Strategy => ..., default value null,
  • SyzygyLimit => ..., default value infinity, stop when this number of syzygies is reached
• Outputs:
  • a chain complex, a free resolution of M

Description

The given generators and relations are used to determine a presentation of M to serve as the first matrix of the free resolution; if the presentation is not minimal, and a minimal resolution is desired, use resolution minimalPresentation M instead.

Warning: the resolution can have free modules with unexpected ranks when the module M is not homogeneous. Here is an example where even the lengths of the resolutions differ. We compute a resolution of the kernel of a ring map in two ways. The ring R is constructed naively, but the ring S is constructed with variables of the right degrees so the ring map g will turn out to be homogeneous.

i1 : k = ZZ/101; T = k[v..z];

i3 : m = matrix {{x,y,z,x^2*v,x*y*v,y^2*v,z*v,x*w,y^3*w,z*w}}

o3 = | x y z vx2 vxy vy2 vz wx wy3 wz |

             1      10
o3 : Matrix T  <-- T

i4 : n = rank source m

o4 = 10

i5 : R = k[u_1 .. u_n]

o5 = R

o5 : PolynomialRing

i6 : S = k[u_1 .. u_n,Degrees => degrees source m]

o6 = S

o6 : PolynomialRing

i7 : f = map(T,R,m)

                             2            2               3
o7 = map (T, R, {x, y, z, v*x , v*x*y, v*y , v*z, w*x, w*y , w*z})

o7 : RingMap T <-- R

i8 : g = map(T,S,m)

                             2            2               3
o8 = map (T, S, {x, y, z, v*x , v*x*y, v*y , v*z, w*x, w*y , w*z})

o8 : RingMap T <-- S

i9 : res ker f

      1      17      57      76      46      12      1
o9 = R  <-- R   <-- R   <-- R   <-- R   <-- R   <-- R  <-- 0
                                                            
     0      1       2       3       4       5       6      7

o9 : ChainComplex

i10 : res ker g

       1      14      35      35      15      2
o10 = S  <-- S   <-- S   <-- S   <-- S   <-- S  <-- 0
                                                     
      0      1       2       3       4       5      6

o10 : ChainComplex

i11 : isHomogeneous f

o11 = false

i12 : isHomogeneous g

o12 = true

i13 : R = ZZ/32003[a..d]/(a^2+b^2+c^2+d^2);

i14 : M = coker vars R

o14 = cokernel | a b c d |

                             1
o14 : R-module, quotient of R

i15 : C = resolution(M, LengthLimit=>6)

       1      4      7      8      8      8      8
o15 = R  <-- R  <-- R  <-- R  <-- R  <-- R  <-- R
                                                 
      0      1      2      3      4      5      6

o15 : ChainComplex

A manually constructed resolution can be installed as the resolution of a module, bypassing the call to the engine when a resolution is requested, as follows.

i16 : A = QQ[x,y]

o16 = A

o16 : PolynomialRing

i17 : C = chainComplex(
      	       map(A^1,A^{3:-2},{{x^2,x*y,y^2}}),
      	       map(A^{3:-2},A^{2:-3},{{y,0},{ -x,y},{0,-x}}),
      	       map(A^{2:-3},0,0))

       1      3      2
o17 = A  <-- A  <-- A  <-- 0
                            
      0      1      2      3

o17 : ChainComplex

i18 : M = HH_0 C

o18 = cokernel | x2 xy y2 |

                             1
o18 : A-module, quotient of A

i19 : res M = C;

i20 : res M

       1      3      2
o20 = A  <-- A  <-- A  <-- 0
                            
      0      1      2      3

o20 : ChainComplex

For an overview of resolutions, in order of increasing detail, see:

• Hilbert functions and free resolutions
• free resolutions of modules
• computing resolutions -- most detailed

• betti(GradedModule) -- display or modify a Betti diagram
• status(Resolution) -- status of a resolution computation