Resolutions and Castelnuovo-Mumford Regularity

Abstract : In this thesis, we study square-free monomial ideals of the polynomial ring S which have a linear resolution. By remarkable result of Bayer and Stilman [BS] and the technique of polarization, classification of ideals with linear resolution is equivalent to classification of square-free monomial ideals with linear resolution. For this reason, we consider only square-free monomial ideals in S. However, classification of square-free monomial ideals with linear resolution seems to be so difficult because by Eagon-Reiner Theorem [ER], this is equivalent to classification of Cohen-Macaulay ideals. It is worth to note that, square-free monomial ideals in S are in one-to-one correspondence to Stanley-Reisener ideals of simplicial complexes on one hand and the circuit ideal of clutters from another hand. This correspondence motivated mathematicians to use the combinatorial and geometrical properties of these objects in order to get the desired algebraic results. Classification of square-free monomial ideals with 2-linear resolution, was successfully done by Froberg [Fr] in 1990. Froberg observed that the circuit ideal of a graph G has a 2-linear resolution if and only if G is chordal, that is, G does not have an induced cycle of length > 3. In [Em, ThVt, VtV, W] the authors have partially generalized the Fr¨oberg's theorem for degree greater than 2. They have introduced several definitions of chordal clutters and proved that, their corresponding circuit ideals have linear resolutions. Viewing cycles as geometrical objects (triangulation of closed curves), in this thesis we try to generalize the concept of cycles to triangulation of pseudo-manifolds and get a partial generalization of Froberg's theorem for higher dimensional hypergraphs. All the results in Chapters 4 and 5 and some results in Chapter 3 are devoted to be the original results.
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Ali Akbar Yazdan Pour. Resolutions and Castelnuovo-Mumford Regularity. Differential Geometry [math.DG]. Université de Grenoble, 2012. English. ⟨NNT : 2012GRENM083⟩. ⟨tel-01547825⟩

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