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# Ring ^ BettiTally -- construct a chain complex with prescribed Betti table

- Operator: ^
- Usage:
    `R^t`
- Inputs:
  - R, a ring,
  - t, a Betti tally,
- Outputs:
  - C, a chain complex, whose Betti table matches t;

## Description

Given a ring $R$, a chain complex with zero maps over $R$ that has a prescribed Betti table can be constructed. Negative entries are ignored and rational entries produce an error. Multigraded rings work only if the Betti tally contains degrees of the correct degree length.

    i1 : R = QQ[x,y]

    o1 = R

    o1 : PolynomialRing

    i2 : t = new BettiTally from { (0,{0},0) => 1, (1,{1},1) => 2, (2,{3},3) => 3, (2,{4},4) => 4 }

                0 1 2
    o2 = total: 1 2 7
             0: 1 2 .
             1: . . 3
             2: . . 4

    o2 : BettiTally

    i3 : C = R^t

          1      2      7
    o3 = R  <-- R  <-- R

         0      1      2

    o3 : ChainComplex

    i4 : betti C

                0 1 2
    o4 = total: 1 2 7
             0: 1 2 .
             1: . . 3
             2: . . 4

    o4 : BettiTally

    i5 : C.dd

              1         2
    o5 = 0 : R  <----- R  : 1
                   0

              2         7
         1 : R  <----- R  : 2
                   0

    o5 : ChainComplexMap

## Contributors

Hans-Christian von Bothmer implemented this feature.

## See also

- betti -- display or modify a Betti diagram
- BettiTally -- the class of all Betti tallies

## Ways to use this method:

- Ring ^ BettiTally -- construct a chain complex with prescribed Betti table

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldChainComplexes/docs/betti-doc.m2:108:0.