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isExactRes -- Test whether a chain complex is an exact resolution.

Synopsis

Description

Test whether a chain complex is an exact resolution, that is, it is exact everywhere except at the first non-zero module.

We consider cc as a doubly infinite complex extending it by adding trivial modules and homomorphisms.

i1 : R = QQ[x_1..x_4,z_1..z_4]

o1 = R

o1 : PolynomialRing
 
i2 : I =  ideal(z_2*z_3-z_1*z_4,x_4*z_3-x_3*z_4,x_2*z_2-x_1*z_4,x_4*z_1-x_3*z_2,x_2*z_1-x_1*z_3)

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

o2 : Ideal of R
 
i3 : cc= res I

      1      5      5      1
o3 = R  <-- R  <-- R  <-- R  <-- 0
                                  
     0      1      2      3      4

o3 : ChainComplex
 
i4 : isExactRes cc

o4 = true
 
i5 : isExactRes(cc[1])

o5 = true
 
i6 : isExactRes(cc[-1])

o6 = true

See also

Ways to use isExactRes :

For the programmer

The object isExactRes is a method function.