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Tag Archives: localization
Commutative Algebra 58
We have already seen two forms of unique factorization. In a UFD, every nonzero element is a unique product of irreducible (also prime) elements. In a Dedekind domain, every nonzero ideal is a unique product of maximal ideals. Here, we … Continue reading
Posted in Advanced Algebra
Tagged annihilators, associated primes, localization, module division, modules, supports
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Commutative Algebra 45
Invertibility is Local In this article, we again let A be an integral domain and K its field of fractions. We continue our discussion of invertible fractional ideals of A. Proposition 1. A fractional ideal M of A is invertible if and … Continue reading
Commutative Algebra 44
Fractional Ideals Throughout this article, let A be an integral domain and K its field of fractions. We do not assume the ring to be noetherian. The objective here is to develop the theory of multiplying and dividing certain classes of nonzero … Continue reading
Commutative Algebra 40
More on Integrality Lemma 1. Let be an integral extension. If is an ideal and , the resulting injection is an integral extension. Proof Any element of can be written as , . Then x satisfies a monic polynomial relation: . … Continue reading
Posted in Advanced Algebra
Tagged closed maps, fibres, finite extensions, going up, integral extensions, krull dimension, localization
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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 29
Distributivity Finally, tensor product is distributive over arbitrary direct sums. Proposition 1. Given any family of modules , we have: Proof Take the map which takes . Note that this is welldefined: since only finitely many are nonzero, only finitely … Continue reading
Posted in Advanced Algebra
Tagged hom functor, induced modules, localization, rightexact, tensor products, yoneda lemma
2 Comments