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

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

Ways to use this method:


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