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)

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

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_{f_α+1},...,θ_d] ⊆ k[θ₁, θ_{f_α}].