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Category Archives: Advanced Algebra
Commutative Algebra 63
Serre’s Criterion for Normality Throughout this article, fix an algebraically closed field k. In this section, A denotes a noetherian domain. We will describe Serre’s criterion, which is a necessary and sufficient condtion for A to be normal. In the … Continue reading
Commutative Algebra 62
Irreducible Subsets of Projective Space Throughout this article, k is an algebraically closed field. We wish to consider irreducible closed subsets of . For that we need the following preliminary result. Lemma 1. Let be a graded ring; a proper homogeneous … Continue reading
Commutative Algebra 61
In this article, we will consider algebraic geometry in the projective space. Throughout this article, k denotes an algebraically closed field. Projective Space Definition. Let . On the set , we consider the equivalence relation: The projective nspace is the set … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, closed subsets, cones, graded rings, nullstellensatz, projective varieties, varieties
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Commutative Algebra 59
Prime Composition Series Throughout this article, A is a noetherian ring and all Amodules are finitely generated. Recall (proposition 1 here) that if M is a noetherian and artinian module, we can find a sequence of submodules whose consecutive factors … Continue reading
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 57
Continuing from the previous article, A denotes a noetherian ring and all Amodules are finitely generated. As before all completions are taken to be stable for a fixed ideal . Noetherianness We wish to prove that the adic completion of … Continue reading
Posted in Advanced Algebra
Tagged aadic filtrations, algebraic geometry, analysis, completion, filtrations, hensels lemma, local rings, padic
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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 55
Exactness of Completion Throughout this article, A denotes a filtered ring. Proposition 1. Let be a short exact sequence of Amodules. Suppose M is filtered, inducing filtrations on N and P. Then is also exact as modules. Proof Without loss of … Continue reading
Posted in Advanced Algebra
Tagged aadic filtrations, artinrees lemma, blowup algebras, completions, filtrations, limits, noetherian
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