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Tag Archives: local rings
Commutative Algebra 57
Continuing from the previous article, A denotes a noetherian ring and all A-modules 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 a-adic filtrations, algebraic geometry, analysis, completion, filtrations, hensels lemma, local rings, p-adic
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Commutative Algebra 34
Nakayama’s Lemma The following is a short statement which has far-reaching applications. Since its main applications are for local rings, we will state the result in this context. Throughout this section, is a fixed local ring. Theorem (Nakayama’s Lemma). Let … Continue reading
Commutative Algebra 25
Arbitrary Collection of Modules Finally, we consider the case where we have potentially infinitely many modules. Proposition 1. For a collection of A-modules , we have Proof First claim: we will show that the LHS satisfies the universal property for … Continue reading
Posted in Advanced Algebra
Tagged direct products, direct sums, exact sequences, local properties, local rings, localization
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Commutative Algebra 23
Localization and Spectrum Recall that the ideals of correspond to a subset of the ideals of A. If we restrict ourselves to prime ideals, we get the following nice bijection. Theorem 1. The above gives a bijection between Useful trick If … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, local rings, localization, prime ideals, rational functions, spectrum, zariski topology
12 Comments
Krull-Schmidt Theorem
Here, we will prove that the process of decomposing is unique, given that M is noetherian and artinian. Again, R is a ring, possibly non-commutative. Definition. A decomposition of an R-module M is an expression for non-zero modules An R-module M is said … Continue reading
Posted in Notes
Tagged indecomposable modules, krull-schmidt, local rings, matrix rings, splitting lemma, unique factorisation
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Local Rings
Mathematicians are generally more familiar with the case of local commutative rings, so we’ll begin from there. Definition. A commutative ring R is said to be local if it has a unique maximal ideal. Note that every non-zero commutative ring has … Continue reading
Posted in Notes
Tagged algebra, associative algebra, indecomposable modules, local rings, units
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Mathematics and Such 