239 research outputs found

    Algebraic shifting of strongly edge decomposable spheres

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    Recently, Nevo introduced the notion of strongly edge decomposable spheres. In this paper, we characterize the algebraic shifted complex of those spheres. Algebraically, this result yields the characterization of the generic initial ideal of the Stanley--Reisner ideal of Gorenstein* complexes having the strong Lefschetz property in characteristic 0.Comment: 19 pages. Add a few examples in the Introduction. To appear in J. Combin. Theory Ser.

    Spheres arising from multicomplexes

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    In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex Δ\Delta on the vertex set VV with Δ≠2V\Delta \ne 2^V, the deleted join of Δ\Delta with its Alexander dual Δ∨\Delta^\vee is a combinatorial sphere. In this paper, we extend Bier's construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable, which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that these spheres are related to polarizations and Alexander duality for monomial ideals which appear in commutative algebra theory.Comment: 20 pages. Improve presentation. To appear in Journal of Combinatorial Theory, Series

    Lefschetz Elements for Stanley-Reisner Rings and Annihilator Numbers

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    I Introduction 1 1 Basic algebraic definitions and constructions 3 1.1 Some homological algebra .......................... 3 1.1.1 Free resolutions............................ 3 1.1.2 Cochain complexes and injective resolutions . . . . . . . . . . . . 6 1.1.3 Tor- and Ext-groups ......................... 7 1.1.4 The Eliahou-Kervaire resolution ................... 8 1.1.5 The Cartan complex ......................... 10 1.2 The generic initial ideal ............................ 12 1.2.1 The basic construction ........................ 12 1.2.2 Main properties............................ 13 1.2.3 Algebraic invariants of the generic initial ideal with respect to the reverse lexicographic order...................... 14 1.2.4 The generic initial ideal over the exterior algebra . . . . . . . . . . 15 2 Simplicial complexes 17 2.1 Simplicial complexes – the basic definition . . . . . . . . . . . . . . . . . 17 2.2 Classes of simplicial complexes ....................... 20 2.2.1 Cohen-Macaulay complexes ..................... 20 2.2.2 Shellable complexes ......................... 22 2.3 Operations and constructions on simplicial complexes . . . . . . . . . . . . 24 2.3.1 Several standard operations ..................... 24 2.3.2 The barycentric subdivision ..................... 24 2.3.3 Algebraic shifting: The exterior shifting of a simplicial complex . . 26 II Lefschetz Properties for Classes of Simplicial Complexes 29 3 The Lefschetz property: classical and more recent results 31 3.1 The classical g-theorem and the g-conjecture ................ 31 3.2 More recent results .............................. 34 3.2.1 The strong Lefschetz property for matroid complexes . . . . . . . . 37 3.2.2 The strong Lefschetz property for simplicial complexes admitting a convex ear decomposition ...................... 37 3.2.3 The behavior of Lefschetz properties under join, union and connected sum .............................. 38 3.2.4 The behavior of Lefschetz properties under stellar subdivisions of simplicial complexes ......................... 39 3.2.5 Lefschetz properties for strongly edge decomposable complexes . . 40 3.2.6 The non-negativity of the cd-index. . . . . . . . . . . . . . . . . . 41 3.3 Algebraic methods .............................. 42 4 The Lefschetz property for barycentric subdivisions of simplicial complexes 47 4.1 The motivation for studying barycentric subdivisions of simplicial complexes 48 4.2 The almost strong Lefschetz property for shellable complexes . . . . . . . 50 4.3 Numerical consequences for the h-vector................... 57 4.4 Inequalities for a special refinement of the Eulerian numbers . . . . . . . . 58 4.5 Open problems and conjectures........................ 64 III Notion of Depth and Annihilator Numbers 67 5 Exterior depth and generic annihilator numbers 69 5.1 The exterior depth............................... 70 5.2 Annihilator numbers ............................. 76 5.2.1 Symmetric annihilator numbers ................... 77 5.2.2 Exterior annihilator numbers ..................... 79 5.2.3 An application of almost regular sequences and generic annihilator numbers................................ 84 5.3 A counterexample to a minimality conjecture of Herzog . . . . . . . . . . . 85 5.4 The exterior depth and exterior annihilator numbers for Stanley-Reisner rings 91 5.4.1 The exterior depth for Stanley-Reisner rings of simplicial complexes 91 5.4.2 Annihilator numbers for Stanley-Reisner rings of simplicial complexes ................................. 96 References 10

    Gorenstein rings through face rings of manifolds

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    The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the sphere gg-conjecture implies all enumerative consequences of its far reaching generalization (due to Kalai) to manifolds. A special case of Kalai's manifold gg-conjecture is established for homology manifolds that have a codimension-two face whose link contains many vertices
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