# Category Archives: Advanced Algebra

## Commutative Algebra 64

Segre Embedding Throughout this article, k is a fixed algebraically closed field. We wish to construct the product in the category of quasi-projective varieties. For our first example, let be the projective variety defined by the homogeneous equation . We define … Continue reading

## 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 n-space is the set … Continue reading

## Commutative Algebra 60

Primary Decomposition of Ideals Definition. Let be a proper ideal. A primary decomposition of is its primary decomposition as an A-submodule of A: where each is -primary for some prime , i.e. . Here is a quick way to determine if … Continue reading

## Commutative Algebra 59

Prime Composition Series Throughout this article, A is a noetherian ring and all A-modules 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 non-zero element is a unique product of irreducible (also prime) elements. In a Dedekind domain, every non-zero ideal is a unique product of maximal ideals. Here, we … Continue reading

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## 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

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## Commutative Algebra 56

Throughout this article, A denotes a noetherian ring and is a fixed ideal. All A-modules are finitely generated. Consequences of Artin-Rees Lemma Suppose we have an exact sequence of finitely generated A-modules Let M be given the 𝔞-adic 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 A-modules. Suppose M is filtered, inducing filtrations on N and P. Then is also exact as -modules. Proof Without loss of … Continue reading