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

## Synopsis

• Usage:
isLinearComplex F
• Inputs:
• F, ,
• Outputs:
• , 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.