A Betti tally is a special type of Tally that is printed as a display of graded Betti numbers. The class was created so the function betti could return something that both prints nicely and from which information can be extracted. The keys are triples (i,d,h) encoding:
Only i and h are used in printing, and the weight covector can be modified by specifying the betti(...,Weights=>...) option to betti(BettiTally).
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For convenience, the operations of direct sum (++), tensor product (**), codim, degree, dual, pdim, poincare, regularity, and degree shifting (numbers in brackets or parentheses), have been implemented for Betti tallies. These operations mimic the corresponding operations on chain complexes.
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If the Betti tally represents the Betti numbers of a resolution of a module $M$ on a polynomial ring $R = K[x_0,...,x_n]$, then while the data does not uniquely determine $M$, it suffices to compute the Hilbert polynomial and Hilbert series of $M$.
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A Betti tally can be multiplied by an integer or by a rational number, and the values can be lifted to integers, when possible.
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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.
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Hans-Christian von Bothmer donated the last feature.
The object BettiTally is a type, with ancestor classes VirtualTally < HashTable < Thing.