# isHomogeneous(Complex) -- whether a complex is homogeneous

## Synopsis

• Function: isHomogeneous
• Usage:
isHomogeneous C
• Inputs:
• C, ,
• Outputs:
• , that is true when C is a homogeneous (i.e. graded) complex

## Description

A complex is homogeneous (graded) if the base ring is graded, all of the component objects are graded, and all the component maps are graded of degree zero.

 i1 : S = ZZ/101[a,b,c,d]; i2 : I = minors(2, matrix{{a,b,c},{b,c,d}}) 2 2 o2 = ideal (- b + a*c, - b*c + a*d, - c + b*d) o2 : Ideal of S i3 : C = freeResolution (S^1/I) 1 3 2 o3 = S <-- S <-- S 0 1 2 o3 : Complex i4 : isHomogeneous C o4 = true i5 : J = minors(2, matrix{{a,b,c},{b,c,d^2}}) 2 2 2 2 o5 = ideal (- b + a*c, a*d - b*c, b*d - c ) o5 : Ideal of S i6 : D = freeResolution (S^1/J) 1 3 2 o6 = S <-- S <-- S 0 1 2 o6 : Complex i7 : isHomogeneous D o7 = false