Rees decomposition

In commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by David Rees (1956).

Definition

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

$R=\bigoplus _{\alpha }\eta _{\alpha }k[\theta _{1},\ldots ,\theta _{f_{\alpha }}]$

where each η_α is a homogeneous element and the d elements θ_i are a homogeneous system of parameters for R and η_αk[θ_fα+1,...,θ_d] ⊆ k[θ₁, θ_fα].

See also

• Stanley decomposition
• Hironaka decomposition

References

• Rees, D. (1956), "A basis theorem for polynomial modules", Proc. Cambridge Philos. Soc., 52: 12–16, MR 0074372
• Sturmfels, Bernd; White, Neil (1991), "Computing combinatorial decompositions of rings", Combinatorica, 11 (3): 275–293, doi:10.1007/BF01205079, MR 1122013