constantStrand -- a constant strand of a chain complex

Synopsis

Usage:

Cd = constantStrand(C, deg)
Inputs:
- C, a chain complex, a chain complex created using (the unsupported) res(I, Strategy=>4.1)
- deg, an integer, the degree of the strand of the complex
Outputs:
- Cd, a chain complex, a chain complex over kk, consisting of the submatrices of C of degree deg

Description

Warning! This function is very rough currently. It works if one uses it in the intended manner, as in the example below. But it should be much more general, handling other rings with grace, and also it should handle arbitrary (graded) chain complexes.

i1 : R = ZZ/101[a..d]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(a^3, b^3, c^3, d^3, (a+3*b+7*c-4*d)^3)

             3   3   3   3   3     2         2      3      2             
o2 = ideal (a , b , c , d , a  + 9a b + 27a*b  + 27b  + 21a c + 25a*b*c -
     ------------------------------------------------------------------------
        2         2        2      3      2                2             
     13b c + 46a*c  + 37b*c  + 40c  - 12a d + 29a*b*d - 7b d + 34a*c*d +
     ------------------------------------------------------------------------
                2         2        2        2      3
     b*c*d + 18c d + 48a*d  + 43b*d  + 33c*d  + 37d )

o2 : Ideal of R

i3 : C = res(I, Strategy=>4)

      1      9      25      31      18      4
o3 = R  <-- R  <-- R   <-- R   <-- R   <-- R
                                            
     0      1      2       3       4       5

o3 : ChainComplex

i4 : betti C

            0 1  2  3  4 5
o4 = total: 1 9 25 31 18 4
         0: 1 .  .  .  . .
         1: . .  .  .  . .
         2: . 5  1  .  . .
         3: . 1  3  1  . .
         4: . 3 17 13  4 .
         5: . .  4 13 10 3
         6: . .  .  4  3 1
         7: . .  .  .  1 .

o4 : BettiTally

i5 : CR = constantStrand(C, 5)

             ZZ 3       ZZ 3
o5 = 0 <-- (---)  <-- (---)  <-- 0 <-- 0 <-- 0
            101        101                    
     0                           3     4     5
           1          2

o5 : ChainComplex

i6 : CR.dd_2

o6 = | -33 48  0   |
     | 2   -15 5   |
     | 2   0   -43 |

              ZZ 3       ZZ 3
o6 : Matrix (---)  <-- (---)
             101        101

Caveat

This function should be defined for any graded chain complex, not just ones created using res(I, Strategy=>4).

Ways to use constantStrand :

constantStrand(ChainComplex,ZZ)

For the programmer

The object constantStrand is a method function.