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2008-05-12T16:34:57Z

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Vipul: New page: ==Definition== Let $A$ be a commutative unital ring and $M$ be a module over $A$ . The '''support''' of $M$ is the subset of $Spec(...$

2008-02-27T16:01:14Z

New page: ==Definition== Let <math>A</math> be a commutative unital ring and <math>M</math> be a module over <math>A</math>. The '''support''' of <math>M</math> is the subset of <math>Spec(...

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==Definition==

Let <math>A</math> be a [[commutative unital ring]] and <math>M</math> be a [[module]] over <math>A</math>. The '''support''' of <math>M</math> is the subset of <math>Spec(A)</math> (the [[spectrum]] of <math>A</math>) comprising those [[prime ideal]]s <math>P</math> such that <math>M_P \ne 0</math>

Here, <math>M_P</math> denotes the [[localization of a module at a prime ideal|localization]] of <math>M</math> at the prime ideal <math>P</math>.

==Facts==

* If a prime ideal <math>P</math> is contained in the support of <math>M</math>, then any prime ideal containing <math>P</math> is in the support of <math>M</math>.
* The support of a module is a union of closed subsets. (This follows from the preceding). Conversely any union of closed subsets, arises as the support of a module.
* For a [[finitely generated module]], the support of the module equals the Galois correspondent closed set to the [[annihilator]] of the module (the ideal that annihilates all elements).

Support of a module - Revision history

Vipul: 1 revision

Vipul: New page: ==Definition== Let A be a commutative unital ring and M be a module over A. The '''support''' of M is the subset of Spec(...

Vipul: New page: ==Definition== Let $A$ be a commutative unital ring and $M$ be a module over $A$ . The '''support''' of $M$ is the subset of $Spec(...$