poincare(ChainComplex) -- assemble degrees of a chain complex into a polynomial

Function: poincare
Usage:

poincare C
Inputs:
- C, a chain complex,
Outputs:
- a ring element, in the Laurent polynomial ring degrees ring, whose variables correspond to the degrees of the ambient ring

Description

We compute poincare for a chain complex.

i1 : R = ZZ/32003[a..h];

i2 : C = res ideal(a*b, c*d, e*f)

      1      3      3      1
o2 = R  <-- R  <-- R  <-- R  <-- 0
                                  
     0      1      2      3      4

o2 : ChainComplex

i3 : poincare C

           2     4    6
o3 = 1 - 3T  + 3T  - T

o3 : ZZ[T]

Note that since the Hilbert series is additive in exact sequences, for a free resolution this only depends on the Betti numbers of the resolution. For more details, see Hilbert functions and free resolutions.

i4 : b = betti C

            0 1 2 3
o4 = total: 1 3 3 1
         0: 1 . . .
         1: . 3 . .
         2: . . 3 .
         3: . . . 1

o4 : BettiTally

i5 : poincare b

           2     4    6
o5 = 1 - 3T  + 3T  - T

o5 : ZZ[T]

Ways to use this method:

poincare(ChainComplex) -- assemble degrees of a chain complex into a polynomial

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldChainComplexes/docs/docs.m2:428:0.

poincare(ChainComplex) -- assemble degrees of a chain complex into a polynomial

Description

See also

Ways to use this method: