As we saw in the last chapter, we can determine if a set of points X in (Formula Presented) is ACM directly from a combinatorial description of the points. In this chapter we show that this combinatorial description, in particular, the tuples αX and βX, also allows us to determine the bigraded Betti numbers in the bigraded minimal free resolution of I(X) when X is ACM. Consequently, the Hilbert function of X when X is ACM can also be computed directly from αX and βX or from the set SX. We conclude this chapter by answering the interpolation question introduced in Chapter 1 Specifically, we classify what functions H: ℕ 2→ ℕ are the Hilbert functions of ACM reduced sets of points in (Formula Presented).

Homological invariants

Guardo E.;Van Tuyl A.
2015-01-01

Abstract

As we saw in the last chapter, we can determine if a set of points X in (Formula Presented) is ACM directly from a combinatorial description of the points. In this chapter we show that this combinatorial description, in particular, the tuples αX and βX, also allows us to determine the bigraded Betti numbers in the bigraded minimal free resolution of I(X) when X is ACM. Consequently, the Hilbert function of X when X is ACM can also be computed directly from αX and βX or from the set SX. We conclude this chapter by answering the interpolation question introduced in Chapter 1 Specifically, we classify what functions H: ℕ 2→ ℕ are the Hilbert functions of ACM reduced sets of points in (Formula Presented).
2015
978-3-319-24164-7
978-3-319-24166-1
Betti Number
Complete Intersection
Difference Function
Induction Hypothesis
Short Exact Sequence
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11769/522364
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