support of a module について

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support of a module ：ウィキペディア英語版

support of a module
In commutative algebra, the support of a module ''M'' over a commutative ring ''A'' is the set of all prime ideals

\mathfrak

of ''A'' such that

M_\mathfrak \ne 0

.〔EGA 0_I, 1.7.1.〕 It is denoted by

\operatorname(M)

. In particular,

M = 0

if and only if its support is empty.
* Let

0 \to M' \to M \to M'' \to 0

be an exact sequence of ''A''-modules. Then
*:

\operatorname(M) = \operatorname(M') \cup \operatorname(M'').

* If

M

is a sum of submodules

M_\lambda

, then

\operatorname(M) = \cup_\lambda \operatorname(M_\lambda).

* If

M

is a finitely generated ''A''-module, then

\operatorname(M)

is the set of all prime ideals containing the annihilator of ''M''. In particular, it is closed.
*If

M, N

are finitely generated ''A''-modules, then
*:

\operatorname(M \otimes_A N) = \operatorname(M) \cap \operatorname(N).

*If

M

is a finitely generated ''A''-module and ''I'' is an ideal of ''A'', then

\operatorname(M/IM)

is the set of all prime ideals containing

I + \operatorname(M).

This is

V(I)\cap \operatorname(M)

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