Complex _ ZZ -- access individual object in a complex

Synopsis

Operator: _
Usage:

C_i

C^i
Inputs:
- C, a complex,
- i, an integer, either the homological or cohomological index
Outputs:
- a module, the i-th object

Description

Complexes can be either chain complexes or cochain complexes. Subscripts refer to homological indices, and superscripts refer to cohomological indices.

In this package homological indices are used by default. For example, the concentration references homological indices. Nevertheless, we always have the equation $C^i = C_{-i}$.

i1 : S = ZZ/101[a..c]

o1 = S

o1 : PolynomialRing

i2 : C = freeResolution coker vars S

      1      3      3      1
o2 = S  <-- S  <-- S  <-- S
                           
     0      1      2      3

o2 : Complex

i3 : C_2

      3
o3 = S

o3 : S-module, free, degrees {3:2}

i4 : C^(-2)

      3
o4 = S

o4 : S-module, free, degrees {3:2}

i5 : C_2 == C^(-2)

o5 = true

Indices that are outside of the concentration automatically return the zero object.

i6 : C_-7

o6 = 0

o6 : S-module

Ways to use this method:

Complex _ ZZ -- access individual object in a complex