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Nombres de Betti d'idéaux binomiaux

Abstract : Ha Minh Lam et M. Morales introduced a family of binomial ideals that are binomial extensions of square free monomial ideals. Let I be a square free monomial ideal of k[x] and J a sum of scroll ideals in k[z] with some extra conditions, we define the binomial extension of I as B=I+J⊂k[z]. The aim of this thesis is to study the biggest number p such that the syzygies of B are linear until the step p-1. Due to some order conditions given to the facets of the Stanley-Reisner complex of I we get an order ≻ for the variables of the polynomial ring k[z]. By a calculation of the Gröbner basis of the ideal B we obtain that the initial ideal in(B) is a square free monomial ideal. We will prove that B is 2-regular iff I is 2-regular. In the general case, wheter I is not 2-regular we will find a lower bound for the the maximal integer q which satisfies that the first q-1 sizygies of B are linear. On the other hand, wheter J is toric and supposing other conditions, we will find a upper bound for the integer q which satisfies that the first q-1 syzygies of B are linear. By given more conditions we will prove that the twobounds are equal.
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Hernan de Alba Casillas de Alba Casillas. Nombres de Betti d'idéaux binomiaux. Mathématiques générales [math.GM]. Université de Grenoble, 2012. Français. ⟨NNT : 2012GRENM043⟩. ⟨tel-00772901⟩



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