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 }}]}](../I/m/9d628bd1a873bd519c797c79a1de2d16274a1707.svg)
where each ηα is a homogeneous element and the d elements θi are a homogeneous system of parameters for R and ηαk[θfα+1,...,θd] ⊆ k[θ1, θfα].
See also
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, MR 1122013, doi:10.1007/BF01205079
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