1,180 research outputs found
A Generalized Macaulay Theorem and Generalized Face Rings
We prove that the -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
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
In 1971, McMullen and Walkup posed the following conjecture, which is called
the generalized lower bound conjecture: If is a simplicial -polytope
then its -vector satisfies . Moreover, if for some then can be triangulated without introducing simplices of
dimension .
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
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