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Tag Archives: exact functors
Commutative Algebra 56
Throughout this article, A denotes a noetherian ring and is a fixed ideal. All Amodules are finitely generated. Consequences of ArtinRees Lemma Suppose we have an exact sequence of finitely generated Amodules Let M be given the 𝔞adic filtration; the induced filtration on … Continue reading
Commutative Algebra 31
Flat Modules Recall from proposition 3 here: for an Amodule M, is a rightexact functor. Definition. We say M is flat over A (or Aflat) if is an exact functor, equivalently, if A flat Aalgebra is an Aalgebra which is flat as … Continue reading
Commutative Algebra 27
Free Modules All modules are over a fixed ring A. We already mentioned finite free modules earlier. Here we will consider general free modules. Definition. Let be any set. The free Amodule on I is a direct sum of copies of … Continue reading
Posted in Advanced Algebra
Tagged exact functors, free modules, leftexact, localization, projective modules, splitting lemma
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Commutative Algebra 24
Quotient vs Localization Taking the quotient and localization are two sides of the same coin when we look at . Quotient removes the “small” prime ideals in – it only keeps the prime ideals containing . Localization removes the “large” … Continue reading
Posted in Advanced Algebra
Tagged algebras, exact functors, induced modules, localization, universal properties
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