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| In algebra, the '''support''' of a [[module (mathematics)|module]] ''M'' over a commutative ring ''A'' is the set of all [[prime ideal]]s <math>\mathfrak{p}</math> of ''A'' such that <math>M_\mathfrak{p} \ne 0</math>.<ref>EGA 0<sub>I</sub>, 1.7.1.</ref> It is denoted by <math>\operatorname{Supp}(M)</math>. In particular, <math>M = 0</math> if and only if its support is empty.
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| * If <math>0 \to M' \to M \to M'' \to 0</math> be an exact sequence of ''A''-modules. Then
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| *:<math>\operatorname{Supp}(M) = \operatorname{Supp}(M') \cup \operatorname{Supp}(M'').</math>
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| * If <math>M</math> is a sum of submodules <math>M_\lambda</math>, then <math>\operatorname{Supp}(M) = \bigcup_\lambda \operatorname{Supp}(M_\lambda).</math>
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| * If <math>M</math> is a finitely generated ''A''-module, then <math>\operatorname{Supp}(M)</math> is the set of all prime ideals containing the [[Annihilator (ring theory)|annihilator]] of ''M''. In particular, it is closed.
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| *If <math>M, N</math> are finitely generated ''A''-modules, then
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| *:<math>\operatorname{Supp}(M \otimes_A N) = \operatorname{Supp}(M) \cap \operatorname{Supp}(N).</math>
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| *If <math>M</math> is a finitely generated ''A''-module and ''I'' is an ideal of ''A'', then <math>\operatorname{Supp}(M/IM)</math> is the set of all prime ideals containing <math>I + \operatorname{Ann}(M).</math> This is <math>V(I)\cap \operatorname{Supp}(M)</math>.
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| ==See also==
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| *[[Associated prime]]
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| == References ==
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| {{reflist}}
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| *{{EGA|book=I}}
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| {{algebra-stub}}
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| [[Category:Module theory]]
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Revision as of 19:06, 11 February 2014
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