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In abstract algebra, multiplicity theory concerns the multiplicity of a module M at an ideal I
The notion of the multiplicity of a module is a generalization of the degree of a projective variety. By Serre's intersection formula, it is linked to an intersection multiplicity in the intersection theory.
The main focus of the theory is to detect and measure a singular point of an algebraic variety. Because of this aspect, valuation theory, Rees algebras and integral closure are intimately connected to multiplicity theory.
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