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Upper bounds for local cohomology for rings with given Hilbert function. (English) Zbl 1097.13516

From the introduction: In the last few years some theorems were proved which point out that the lexicographic ideal, among all the homogeneous ideals with the same Hilbert function, has extremal properties. Bigatti and Hulett proved independently that, if \(I\) is a homogeneous ideal in a polynomial ring \(R\) over a field \(K\) of characteristic 0 and \(I^{\text{lex}}\) is the uniquely determined lexicographic ideal with the same Hilbert function then all of the graded Betti numbers of \(I^{\text{lex}}\) are greater than or equal to those of \(I\). K. Pardue [Ill. J. Math. 40, 564–585 (1996; Zbl 0903.13004)] was able to show how the same result holds true over fields of any characteristic. Analogously, if \(I\) is a squarefree homogeneous ideal there exists a uniquely determined squarefree lexicographic ideal associated to it. A. Aramova, A. J. Herzog and T. Hibi [J. Algebr. Comb. 12, 207–222 (2000; Zbl 0980.13013)] proved the equivalent of the Bigatti-Hulett-Pardue theorem in the squarefree case. Since, given a homogeneous ideal in the polynomial ring, the local cohomology functors are graded and the homogeneous components are finite \(K\)-vector spaces, one can expect in the same spirit of the previous theorems that, for every \(i,j\), \[ \dim_KH^i_{\mathfrak m}(R/I)_j\leq\dim_KH^i_{\mathfrak m} (R/I^{\text{lex}})_j. \] We shall see that this is indeed the case. First we shall reduce the question to the monomial case, and then study the two problems separately. In the squarefree case we shall prove a formula which relates the dimension of the components of the local cohomology functors with certain Betti numbers of the Alexander dual of the Stanley-Reisner ring, in order to apply the result of [loc. cit.]. In the non-squarefree case we refer to the paper of Pardue where he shows, making use of polarization for monomial ideals, a procedure which leads from the ideal to the associated lexicographic ideal. In order to apply his argument we have to he able to control the behaviour of the Hilbert functions passing from the ideal to its polarization. The crucial step will be extending polarization of ideals to an exact functor on the category of finite multi-graded modules. Finally we shall prove that, if \(I^{\text{lex}}\) is generated in one degree, then the Ext groups of \(R/I^{\text{lex}}\) are cyclic and that the Hilbert series of \(H^i_{\mathfrak m}(R/I)\) in terms of \(t\) has the following upper bounds: \[ \text{Hilb}(H^i_{\mathfrak m}(R/I),t)\leq\frac{t^{\sum_{h\in n-i}v_h-i}\sum_{i=1}^{v_{n-i}}t^{-j}}{(1-t^{-1})^i}, \] for every \(i>0\), where the \(v_i\)’s will be described in terms of the Hilbert polynomial of \(R/I\).

MSC:

13D45 Local cohomology and commutative rings
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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