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)

[math]\displaystyle{ R = \bigoplus_\alpha \eta_\alpha k[\theta_1,\ldots,\theta_{f_\alpha}] }[/math]

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 
• Sturmfels, Bernd; White, Neil (1991), "Computing combinatorial decompositions of rings", Combinatorica 11 (3): 275–293, doi:10.1007/BF01205079