The Hilbert–Samuel function is an important concept in commutative algebra and algebraic geometry, particularly in the study of the structure of space defined by ideals in rings and the geometry of schemes. It provides a way to measure the growth of the dimensions of the graded components of the quotient of a Noetherian ring by an ideal.
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For an ideal of definition of a Noetherian local ring and a finitely generated module , the Hilbert–Samuel function isFor sufficiently large , it equals the Hilbert-Samuel polynomial.