In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
, a Rees decomposition is a way of writing a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
by some homogeneous ideal. A Rees decomposition of ''R'' is a representation of ''R'' as a direct sum (of vector spaces)
:
where each ''η''
''α'' is a homogeneous element and the ''d'' elements ''θ''
''i'' are a homogeneous system of parameters for ''R'' and
''η''
''α''''k''
'θ''''f''''α''+1,...,''θ''''d''⊆ ''k''
'θ''1, ''θ''''f''''α''
See also
*
Stanley decomposition In commutative algebra, a Stanley 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 id ...
*
Hironaka decomposition In mathematics, a Hironaka decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions are named after Heisuke Hironaka, who used this in ...
References
*
*
Commutative algebra
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