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# pureCharFree -- first Betti number of specific exact complex

- Usage:
    `pureCharFree`
- Inputs:
  - a list, a strictly increasing sequence of degrees
- Outputs:
  - an integer, The zero-th Betti number of the corresponding pure resolution construction

## Description

pureCharFree corresponds to the construction in math.AC/0712.1843v2, "Betti Numbers of Graded Modules and Cohomology of Vector Bundles", Section 5.

    i1 : L = {0,2,3,9}

    o1 = {0, 2, 3, 9}

    o1 : List

    i2 : B = pureBettiDiagram L

                0  1  2 3
    o2 = total: 7 27 21 1
             0: 7  .  . .
             1: . 27 21 .
             2: .  .  . .
             3: .  .  . .
             4: .  .  . .
             5: .  .  . .
             6: .  .  . 1

    o2 : BettiTally

    i3 : pureCharFree L

    o3 = 56

    i4 : L1 = {0,3,4,6}

    o4 = {0, 3, 4, 6}

    o4 : List

    i5 : B1 = pureBettiDiagram L1

                0 1 2 3
    o5 = total: 1 8 9 2
             0: 1 . . .
             1: . . . .
             2: . 8 9 .
             3: . . . 2

    o5 : BettiTally

    i6 : pureCharFree L1

    o6 = 5

Thus, for large enough multiples m, m*B occurs as the Betti diagram of a module from the pureCharFree construction

However, we can find B itself as the Betti diagram of a module:

    i7 : betti res randomSocleModule(L,1)

                0  1  2 3
    o7 = total: 7 27 21 1
             0: 7  .  . .
             1: . 27 21 .
             2: .  .  . .
             3: .  .  . .
             4: .  .  . .
             5: .  .  . .
             6: .  .  . 1

    o7 : BettiTally

    i8 : betti res randomModule(L,1)

                0  1  2  3
    o8 = total: 7 27 35 15
             0: 7  .  .  .
             1: . 27 11  .
             2: .  . 24 15

    o8 : BettiTally

    i9 : betti res randomModule({0,6,7,9},1)

                0  1  2 3
    o9 = total: 1 21 27 7
             0: 1  .  . .
             1: .  .  . .
             2: .  .  . .
             3: .  .  . .
             4: .  .  . .
             5: . 21 27 .
             6: .  .  . 7

    o9 : BettiTally

    i10 : betti res randomSocleModule(L1,1)

                 0 1 2 3
    o10 = total: 1 8 9 2
              0: 1 . . .
              1: . . . .
              2: . 8 9 .
              3: . . . 2

    o10 : BettiTally

    i11 : betti res randomModule(L1,1)

                 0 1 2 3
    o11 = total: 1 8 9 2
              0: 1 . . .
              1: . . . .
              2: . 8 9 .
              3: . . . 2

    o11 : BettiTally

    i12 : betti res randomModule({0,2,3,6},1)

                 0 1  2 3
    o12 = total: 2 9 10 3
              0: 2 .  . .
              1: . 9  7 .
              2: . .  3 3

    o12 : BettiTally

    i13 : betti res randomSocleModule({0,2,3,6},1)

                 0 1 2 3
    o13 = total: 2 9 8 1
              0: 2 . . .
              1: . 9 8 .
              2: . . . .
              3: . . . 1

    o13 : BettiTally

## See also

- pureAll -- Vector of first Betti number of our three specific exact complexes
- pureWeyman -- first Betti number of specific exact complex
- pureTwoInvariant -- first Betti number of specific exact complex

## Ways to use pureCharFree:

- `pureCharFree(List)`

## For the programmer

The object pureCharFree is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/BoijSoederberg.m2:1965:0.