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Complexes :: isHomogeneous(Complex)

isHomogeneous(Complex) -- whether a complex is homogeneous

Synopsis

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

See also