1,180 research outputs found

    A Generalized Macaulay Theorem and Generalized Face Rings

    Get PDF
    We prove that the ff-vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities. We construct a large family of meet semi-lattices belonging to this class, which includes all posets of multicomplexes, as well as meet semi-lattices with the "diamond property", discussed by Wegner, as spacial cases. Specializing the proof to that later family, one obtains the Kruskal-Katona inequalities and their proof as in Wegner's. For geometric meet semi lattices we construct an analogue of the exterior face ring, generalizing the classic construction for simplicial complexes. For a more general class, which include also multicomplexes, we construct an analogue of the Stanley-Reisner ring. These two constructions provide algebraic counterparts (and thus also algebraic proofs) of Kruskal-Katona's and Macaulay's inequalities for these classes, respectively.Comment: Final version: 13 pages, 2 figures. Improved presentation, more detailed proofs, same results. To appear in JCT

    Linkage of modules over Cohen-Macaulay rings

    Get PDF
    Inspired by the works in linkage theory of ideals, the concept of sliding depth of extension modules is defined to prove the Cohen-Macaulyness of linked module if the base ring is merely Cohen-Macaulay. Some relations between this new condition and other module-theory conditions such as G-dimension and sequentially Cohen-Macaulay are established. By the way several already known theorems in linkage theory are improved or recovered by new approaches.Comment: 12 Page

    On the generalized lower bound conjecture for polytopes and spheres

    Full text link
    In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If PP is a simplicial dd-polytope then its hh-vector (h0,h1,...,hd)(h_0,h_1,...,h_d) satisfies h0≀h1≀...≀h⌊d2βŒ‹h_0 \leq h_1 \leq ... \leq h_{\lfloor \frac d 2 \rfloor}. Moreover, if hrβˆ’1=hrh_{r-1}=h_r for some r≀d2r \leq \frac d 2 then PP can be triangulated without introducing simplices of dimension ≀dβˆ’r\leq d-r. The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this property to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.Comment: 14 pages, improved presentatio
    • …
    corecore