The Weak Lefschetz Property For Artinian Quadratic Monomial Algebras

Sammanfattning: In this thesis we aim to study the Lefschetz properties ofmonomial algebras. First, we present the necessary concepts and resultsfrom commutative algebra, in particular we build up to the Hilbert-Serre theorem regarding the rationality of Hilbert series. We then reviewsome important results from the literature on the Lefschetz properties(whereof many provide drastic computational shortcuts under certainconditions) and provide some examples of these. The second half isdevoted to the study of Artinian quadratic monomial algebras of theform A(Δ) = K[x1, . . . , xn]/JΔ, where JΔ = (x21, . . . , x2n) + IΔ and IΔis the ideal obtained from some (abstract) simplicial complex Δ viathe Stanley-Reisner correspondence. In particular, we review a recentarticle [2] by H. Dao and R. Nair, provide examples and refine some ofthe formulations and results.