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

Synopsis

Function: poincare
Usage:

poincare C
Inputs:
- C, a complex,
Outputs:
- a ring element, in the ring of degrees of the underlying ring of $C$

Description

This method encodes information about the degrees of the generators of each term in a chain complex. It returns an element in the degrees ring, which is the ring of Laurent polynomials whose monomials correspond to the degrees of the monomials in the underlying ring of $C$. When the $i$-th term $C_i$ of the complex $C$ is generated by elements of degree $d_{i,1}, d_{i,2}, \dotsc$, this Laurent polynomial is $\sum_i (-1)^i \sum_j T^{d_{i,j}}$, where we use multi-index notation $T^d = T_0^{d_0} T_1^{d_1} \dotsb T_r^{d_r}$.

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

i2 : K = koszulComplex vars R

      1      6      15      20      15      6      1
o2 = R  <-- R  <-- R   <-- R   <-- R   <-- R  <-- R
                                                   
     0      1      2       3       4       5      6

o2 : Complex

i3 : betti K

            0 1  2  3  4 5 6
o3 = total: 1 6 15 20 15 6 1
         0: 1 6 15 20 15 6 1

o3 : BettiTally

i4 : p = poincare K

                 2      3      4     5    6
o4 = 1 - 6T + 15T  - 20T  + 15T  - 6T  + T

o4 : ZZ[T]

i5 : factor p

            6
o5 = (1 - T)

o5 : Expression of class Product

i6 : C = freeResolution ideal(a*b, b*c*d, c*d*e^2)

      1      3      2
o6 = R  <-- R  <-- R
                    
     0      1      2

o6 : Complex

i7 : betti C

            0 1 2
o7 = total: 1 3 2
         0: 1 . .
         1: . 1 .
         2: . 1 1
         3: . 1 1

o7 : BettiTally

i8 : poincare C

          2    3    5
o8 = 1 - T  - T  + T

o8 : ZZ[T]

Since the number of variables in the degrees ring is equal to the rank of the grading group, the Poincare polynomial may have more than one variable.

i9 : S = ZZ/101[x,y,z, DegreeRank => 3];

i10 : L = koszulComplex vars S

       1      3      3      1
o10 = S  <-- S  <-- S  <-- S
                            
      0      1      2      3

o10 : Complex

i11 : poincare L

o11 = 1 - T  - T  - T  + T T  + T T  + T T  - T T T
           0    1    2    0 1    0 2    1 2    0 1 2

o11 : ZZ[T ..T ]
          0   2

If the terms in the complex are not free, then this method uses just the degrees of the generators of each term.

i12 : D = C ** coker vars R

o12 = cokernel | a b c d e f | <-- cokernel {2} | a b c d e f 0 0 0 0 0 0 0 0 0 0 0 0 | <-- cokernel {4} | a b c d e f 0 0 0 0 0 0 |
                                            {3} | 0 0 0 0 0 0 a b c d e f 0 0 0 0 0 0 |              {5} | 0 0 0 0 0 0 a b c d e f |
      0                                     {4} | 0 0 0 0 0 0 0 0 0 0 0 0 a b c d e f |      
                                                                                            2
                                   1

o12 : Complex

i13 : poincare D

           2    3    5
o13 = 1 - T  - T  + T

o13 : ZZ[T]

i14 : betti D

             0 1 2
o14 = total: 1 3 2
          0: 1 . .
          1: . 1 .
          2: . 1 1
          3: . 1 1

o14 : BettiTally

This is equivalent to computing the Poincare polynomial of the Betti table.

i15 : poincare betti D

           2    3    5
o15 = 1 - T  - T  + T

o15 : ZZ[T]

Ways to use this method: