isLinearComplex -- tests whether a complex of graded modules is linear

Synopsis

Usage:

isLinearComplex F
Inputs:
- F, a chain complex,
Outputs:
- a Boolean value, true if F is a linear complex, false if it is not

Description

We say that a (nonzero) $\ZZ^r$-graded complex F is linear if its first nonzero module is generated in a single multidegree and all of its maps (including the zero maps) have degree at most 1. For instance, if F_i is zero for $i<0$ and F_0 is generated in degree $0\in\ZZ^r$, then F_i should be generated in multidegrees with sum at most $i$ for $i>0$. If F is zero then the function returns true.

i1 : S = ZZ/101[x_1..x_4]

o1 = S

o1 : PolynomialRing

i2 : I = ideal(x_1*x_2, x_1*x_3,x_1*x_4, x_2*x_3, x_3*x_4)

o2 = ideal (x x , x x , x x , x x , x x )
             1 2   1 3   1 4   2 3   3 4

o2 : Ideal of S

i3 : M = S^1/I

o3 = cokernel | x_1x_2 x_1x_3 x_1x_4 x_2x_3 x_3x_4 |

                            1
o3 : S-module, quotient of S

i4 : F = res M

      1      5      6      2
o4 = S  <-- S  <-- S  <-- S  <-- 0
                                  
     0      1      2      3      4

o4 : ChainComplex

i5 : betti F

            0 1 2 3
o5 = total: 1 5 6 2
         0: 1 . . .
         1: . 5 6 2

o5 : BettiTally

i6 : isLinearComplex F

o6 = false

i7 : F' = res truncate(2,M)

      5      14      13      4
o7 = S  <-- S   <-- S   <-- S  <-- 0
                                    
     0      1       2       3      4

o7 : ChainComplex

i8 : betti F'

            0  1  2 3
o8 = total: 5 14 13 4
         2: 5 14 13 4

o8 : BettiTally

i9 : isLinearComplex F'

o9 = true

Caveat

It is possible for a complex to have differential matrices containing only linear entries yet be nonlinear by the definition above.

Ways to use isLinearComplex :

isLinearComplex(ChainComplex)

For the programmer