Rees decomposition explained

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[''x''₁,...] 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 = o plus _{\alpha} η _{\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 andη_αk[''θ''_{''f''_''α''+1},...,''θ''_''d''] ⊆ k[''θ''₁, ''θ''_{''f''_''α''}].

See also

• Stanley decomposition
• Hironaka decomposition