declareCouple -- initializes generating classes for an exact couple

Synopsis

• Usage:

  declareCouple(Q, pageClasses, auxClasses)

• Inputs:
  • Q, a ring, a couple ring
  • pageClasses, a list, of the form {... , generatorName => degree, ...} with generatorName a Symbol and degree a List. The degrees should be external degrees for the ring Q. Since these classes appear on the page of the spectral sequence, their degrees should be even, i.e., Q.isEvenDegree(degree) should return true.
  • auxClasses, a list, also of the form {... , generatorName => degree, ...}; the degrees should be odd external degrees for Q.
• Outputs:
  • a module, a free exact couple with the specified generators
• Consequences:

  • assigns appropriate values to the symbols supplied as generator names

Description

i1 : Q = coupleRing(ZZ,1,e,f,Degrees=>{{-1,0},{2,-2}})

o1 = Q

o1 : PolynomialRing

i2 : declareCouple(Q, {z => {4,0}}, {x => {1,0}, y => {1,2}, w => {5,2}})

o2 = cokernel {5, 2} | e_1^2 e_1f_1 0     0      0     0      0     0   |
              {1, 0} | 0     0      e_1^2 e_1f_1 0     0      0     0   |
              {1, 2} | 0     0      0     0      e_1^2 e_1f_1 0     0   |
              {4, 0} | 0     0      0     0      0     0      e_1^3 f_1 |

                            4
o2 : Q-module, quotient of Q

i3 : C = cospan(e_1*z-f_1*y)

o3 = cokernel {5, 2} | 0    e_1^2 e_1f_1 0     0      0     0      0     0   |
              {1, 0} | 0    0     0      e_1^2 e_1f_1 0     0      0     0   |
              {1, 2} | -f_1 0     0      0     0      e_1^2 e_1f_1 0     0   |
              {4, 0} | e_1  0     0      0     0      0     0      e_1^3 f_1 |

                            4
o3 : Q-module, quotient of Q

i4 : isHomogeneous C

o4 = true

i5 : expectExactCouple C

Caveat

See also

• enforceCoupleRelations -- mods out by tautological relations satisfied by every exact couple