HilbertSamuel

HilbertSamuel refers to the Hilbert-Samuel theory in commutative algebra, a framework developed by David Hilbert and Pierre Samuel to study the growth of lengths of modules over Noetherian local rings with respect to powers of the maximal ideal. It provides a local analogue of the classical Hilbert function for graded algebras and plays a central role in measuring size and complexity of modules in algebraic geometry and singularity theory.

For a Noetherian local ring (R, m) and a finitely generated R-module M, the Hilbert-Samuel function is defined by H_M(n) = length_R(M / m^{n+1}M). When M is of dimension d, there exists a polynomial P_M(n) of degree d such that H_M(n) = P_M(n) for all sufficiently large n. This polynomial is called the Hilbert-Samuel polynomial. The leading coefficient of P_M(n) is e(M)/d!, where e(M) is the Hilbert-Samuel multiplicity of M and is an important invariant in singularity theory and intersection theory.

The theory connects to the associated graded ring gr_m(M) = ⊕_{n≥0} m^n M / m^{n+1}M, since P_M(n) equals the Hilbert polynomial of gr_m(M). Hilbert-Samuel multiplicity e(M) reflects geometric and algebraic properties such as dimension, regularity, and depth, and has applications in studying reductions, equisingularity, and the behavior of families of algebraic varieties.

Historically, Hilbert introduced the growth of dimensions in graded settings, and Samuel extended these ideas to local rings, developing a foundational framework for multiplicities and local intersection theory.