Macaulay2 » Documentation
Packages » OldChainComplexes » GradedModule » betti(GradedModule)
next | previous | forward | backward | up | index | toc

betti(GradedModule) -- Betti diagram showing the of degrees in a graded module or chain complex

Description

The diagram can be used to determine the degrees of the entries in the matrices of the differentials in a chain complex (which is a type of graded module) provided they are homogeneous maps of degree 0.

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

            0  1  2 3 4
o5 = total: 2 10 14 7 1
         0: 2  4  6 4 1
         1: .  6  8 3 .

o5 : BettiTally

Column j of the top row of the diagram gives the rank of the free module C_j. The entry in column j in the row labelled i is the number of basis elements of (weighted) degree i+j in the free module C_j. When the chain complex is the resolution of a module the entries are the total and the graded Betti numbers of the module.

Menu

Ways to use this method:

  • betti(ChainComplex)
  • betti(GradedModule) -- Betti diagram showing the of degrees in a graded module or chain complex

The source of this document is in OldChainComplexes/docs/betti-doc.m2:108:0.