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m Minor edit but crucial. The direct sum is a colimit while the product is a limit, and tensors commute with colimits, not limits. Because of this, the product of flat modules is not necessarily flat, while the sum is. →Localization at primes |
m continuation of the last edit -- the direct sum is not a ring →Localization at primes |
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Properties of a ring that can be characterized on its local rings are called ''local properties'', and are often the algebraic counterpart of geometric [[local property|local properties]] of [[algebraic varieties]], which are properties that can be studied by restriction to a small neighborhood of each point of the variety. (There is another concept of local property that refers to localization to Zariski open sets; see {{slink||Localization to Zariski open sets}}, below.)
Many local properties are a consequence of the fact that the
:<math>\bigoplus_\mathfrak p R_\mathfrak p</math>
is a [[faithfully flat module]] when the direct sum is taken over all prime ideals (or over all [[maximal ideal]]s of {{mvar|R}}). See also [[Faithfully flat descent]].
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