Stanley decomposition

In commutative algebra, a Stanley decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by Richard Stanley (1982).

Definition

Suppose that a ring R is a quotient of a polynomial ring k[x₁,...] over a field by some ideal. A Stanley decomposition of R is a representation of R as a direct sum (of vector spaces)

$R=\bigoplus _{\alpha }x_{\alpha }k(X_{\alpha })$

where each x_α is a monomial and each X_α is a finite subset of the generators.

See also

• Rees decomposition
• Hironaka decomposition

References

• Stanley, Richard P. (1982), "Linear Diophantine equations and local cohomology", Invent. Math., 68 (2): 175–193, doi:10.1007/bf01394054, MR 0666158
• Sturmfels, Bernd; White, Neil (1991), "Computing combinatorial decompositions of rings", Combinatorica, 11 (3): 275–293, doi:10.1007/BF01205079, MR 1122013