betti ChainComplex -- display of degrees in a chain complex

betti ChainComplex -- display of degrees in a chain complex

Synopsis:

Usage: n = betti C

Function: betti -- display degrees

Input:

C, an instance of class ChainComplex.

Output:

n, an instance of class Net: a net displaying the degrees of the generators of the modules in C.

The display can be used to determine the degrees of the entries in the matrices of the differentials in the chain complex, provided they are homogeneous maps of degree 0.

Here is a sample display:

i1 : R = ZZ/101[a..h]



o1 = R



o1 : PolynomialRing

i2 : p = genericMatrix(R,a,2,4)



o2 = | a c e g |

     | b d f h |



             2       4

o2 : Matrix R  <--- R

i3 : q = generators gb p



o3 = | g e c a 0     0     0     0     0     0     |

     | h f d b fg-eh dg-ch bg-ah de-cf be-af bc-ad |



             2       10

o3 : Matrix R  <--- R

i4 : C = resolution cokernel leadTerm q



      2      10      14      7      1

o4 = R  <-- R   <-- R   <-- R  <-- R  <-- 0

                                           

     0      1       2       3      4      5



o4 : ChainComplex

i5 : betti C



o5 = total: 2 10 14 7 1

         0: 2  4  6 4 1

         1: .  6  8 3 .

The top row of the display indicates the ranks of the free module C_j in column j. The entry below in row i column j gives the number of basis elements of degree i+j.

If these numbers are needed in a program, one way to get them is with tally.

i6 : degrees C_2



o6 = {{2}, {2}, {2}, {2}, {2}, {2}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}}



o6 : List

i7 : t2 = tally degrees C_2



o7 = Tally{{2} => 6}

           {3} => 8



o7 : Tally

i8 : peek t2



o8 = Tally{{2} => 6}

           {3} => 8

i9 : t2_{2}



o9 = 6

i10 : t2_{3}



o10 = 8

Code:

     -- ../../../Macaulay2/m2/chaincomplexes.m2:659
     betti ChainComplex := C -> bettiDisplay rawbetti C