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Abstract: We discuss Green’s paper  from a new algebraic perspective, and provide applications of its results to level and Gorenstein algebras, concerning their Hilbert functions and the weak Lefschetz property. In particular, we will determine a new infinite class of symmetric -vectors that cannot be Gorenstein -vectors, which was left open in the recent work . This includes the smallest example, previously unknown, . As M. Green’s results depend heavily on the characteristic of the base field, so will ours. The Appendix contains a new argument, kindly provided to us by M. Green, for Theorems 3 and 4 of , since we had found a gap in the original proof of those results during the preparation of this manuscript. [Copyright &y& Elsevier]

Abstract: Three basic properties which standard graded artinian -algebras may or may not enjoy are the Weak and Strong Lefschetz Properties and the Maximal Rank Property (respectively WLP, SLP, and MRP). In this paper we will assume that the base field has characteristic zero. It is known that SLP implies MRP, which in turn implies WLP, but that both implications are strict. However, it surprisingly turned out (see [T. Harima, J. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for artinian -algebras, J. Algebra 262 (2003) 99–126]) that the set of Hilbert functions admitting any algebras with WLP coincides with the corresponding set for SLP (and therefore with that for MRP). In [J. Migliore, F. Zanello, The Hilbert functions which force the weak Lefschetz property, J. Pure Appl. Algebra 210 (2) (2007) 465–471], J. Migliore and the first author, using Green’s theorem and a result of Wiebe, characterized the Hilbert functions forcing all algebras to enjoy WLP. The purpose of this note is to prove the corresponding characterizations for both SLP and MRP. Unsurprisingly (or surprisingly??), the two characterizations coincide, but they define a class of Hilbert functions strictly smaller than that determined for WLP in the above cited reference. Our methods include the Herzog–Popescu theorem on quotients of -algebras modulo a general form, a result of Wiebe, and gins and stable ideals. At the end, we will also discuss the importance of assuming that the characteristic be zero, and we will exhibit a class of codimension 2 monomial complete intersections for which SLP (but not MRP) fails in positive characteristic. [Copyright &y& Elsevier]

Abstract: The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in [A. Wiebe, The Lefschetz property for componentwise linear ideals and Gotzmann ideals, Comm. Algebra 32 (12) (2004) 4601–4611]) whose Gotzmann ideals have the WLP. This implies that, if a Gotzmann ideal has the WLP, then all algebras with the same Hilbert function (and hence lower Betti numbers) have the WLP as well. However, we will answer in the negative, even in the case of level algebras, the most natural question that one might ask after reading the previous sentence: If is an artinian algebra enjoying the WLP, do all artinian algebras with the same Hilbert function as and Betti numbers lower than those of have the WLP as well? Also, as a consequence of our result, we have another (simpler) proof of the fact that all codimension 2 algebras enjoy the WLP (this fact was first proven in [T. Harima, J. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for Artinian -algebras, J. Algebra 262 (2003) 99–126], where it was shown that even the Strong Lefschetz Property holds). [Copyright &y& Elsevier]

Abstract: The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen–Macaulay algebra in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of . All the examples studied so far have lead to conjecture (see [J. Herzog, X. Zheng, Notes on the multiplicity conjecture. Collect. Math. 57 (2006) 211–226] and [J. Migliore, U. Nagel, T. Römer, Extensions of the multiplicity conjecture, Trans. Amer. Math. Soc. (preprint: math.AC/0505229) (in press)]) that, moreover, the bounds of the MC are sharp if and only if has a pure MFR. Therefore, it seems a reasonable–and useful–idea to seek better, if possibly ad hoc, bounds for particular classes of Cohen–Macaulay algebras. In this work we will only consider the codimension 3 case. In the first part we will stick to the bounds of the MC, and show that they hold for those algebras whose -vector is that of a compressed algebra. In the second part, we will (mainly) focus on the level case: we will construct new conjectural upper and lower bounds for the multiplicity of a codimension 3 level algebra , which can be expressed exclusively in terms of the -vector of , and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of is not pure. Even though proving our bounds still appears too difficult a task in general, we are already able to show them for some interesting classes of codimension 3 level algebras : namely, when is compressed, or when its -vector ends with . Also, we will prove our lower bound when begins with , where , and our upper bound when ends with , where . [Copyright &y& Elsevier]