Betti numbers of strongly color-stable ideals and squarefree strongly color-stable ideals

Betti numbers of strongly color-stable ideals and squarefree strongly color-stable ideals

In this paper, we will show that the color-squarefree operation does not change the graded Betti numbers of strongly color-stable ideals. In addition, we will give an example of a nonpure balanced complex which shows that colored algebraic shifting, which was introduced by Babson and Novik, does not always preserve the dimension of reduced homology groups of balanced simplicial complexes.

Minimal Graded Betti Numbers and Stable Ideals

Let k be a field, and let R = k[x1, x2, x3]. Given a Hilbert function H for a cyclic module over R, we give an algorithm to produce a stable ideal I such that R/I has Hilbert function H and uniquely minimal graded Betti numbers among all R/J with the same Hilbert function, where J is another stable ideal in R. We also show that such an algorithm is impossible in more variables and disprove a re...

BETTI NUMBERS FOR p STABLE IDEALS

Stanley Filtrations and Strongly Stable Ideals

We give a new short proof of the fact that the CastelnuovoMumford regularity of a strongly stable ideal is the highest degree of a minimal monomial generator. Our proof depends on results due to D. Maclagan and G. Smith on multigraded regularity. More precisely, we construct a Stanley filtration for strongly stable ideals which provides a bound for the Castelnuovo-Mumford regularity.

Strongly stable ideals and Hilbert polynomials

Strongly stable ideals are a key tool in commutative algebra and algebraic geometry. These ideals have nice combinatorial properties that makes them well suited for both theoretical and computational applications. In the case of polynomial rings with coefficients in a field with characteristic zero, the notion of strongly stable ideals coincides with the notion of Borel-fixed ideals. Ideals of ...