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This is a monograph which collects basic techniques, major results and interesting applications of Lefschetz properties of Artinian algebras. The origin of the Lefschetz properties of Artinian algebras is the Hard Lefschetz Theorem, which is a major result in algebraic geometry. However, for the last two decades, numerous applications of the Lefschetz properties to other areas of mathematics have been found, as a result of which the theory of the Lefschetz properties is now of great interest in its own right. It also has ties to other areas, including combinatorics, algebraic geometry, algebraic topology, commutative algebra and representation theory. The connections between the Lefschetz property and other areas of mathematics are not only diverse, but sometimes quite surprising, e.g. its ties to the Schur-Weyl duality. This is the first book solely devoted to the Lefschetz properties and is the first attempt to treat those properties systematically.

Geometry, Algebraic --- Artin rings --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Mathematical Theory --- Artinian rings --- Rings, Artin --- Rings, Artinian --- Algebraic geometry --- Mathematics. --- Algebra. --- Algebraic geometry. --- Combinatorics. --- Algebraic Geometry. --- Geometry, Algebraic. --- Artin rings. --- Associative rings --- Commutative rings --- Geometry --- Geometry, algebraic. --- Combinatorics --- Mathematical analysis

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Highlights developments on artinian modules over group rings of generalized nilpotent groups. This work includes traditional topics such as direct decompositions of artinian modules, criteria of complementability, and criteria of semisimplicity of artinian modules. It also allows a generalization of Maschke Theorem in classes of infinite groups.

Artin rings --- Group rings --- Nilpotent groups. --- Anneaux artiniens --- Anneaux de groupes --- Groupes nilpotents --- Artin rings. --- Group rings. --- Nilpotent groups --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Groups, Nilpotent --- Artinian rings --- Rings, Artin --- Rings, Artinian --- Mathematics. --- Algebra. --- Finite groups --- Group theory --- Rings (Algebra) --- Associative rings --- Commutative rings --- Mathematical analysis