Rees decomposition

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

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 ηα is a homogeneous element and the d elements θi are a homogeneous system of parameters for R and ηαk[θf α+1 ,...,θd] ⊆ k[θ1, θf α ].