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Summary:
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 101

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