Category Archives: Advanced Algebra

Commutative Algebra 64

Posted on May 30, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged cones, krull dimension, product, projective varieties, quasi-projective varieties, segre embedding | Leave a comment

Commutative Algebra 63

Posted on May 28, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged discrete valuation rings, hartogs theorem, irreducible components, krull dimension, normal domains, primary decompositions, serres criterion | Leave a comment

Commutative Algebra 62

Posted on May 25, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged algebraic geometry, mobius transformations, projective varieties, quasi-projective varieties | Leave a comment

Commutative Algebra 61

Posted on May 22, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged algebraic geometry, closed subsets, cones, graded rings, nullstellensatz, projective varieties, varieties | Leave a comment

Commutative Algebra 60

Posted on May 18, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged associated primes, embedded primes, localization, minimal primes, primary decompositions, primary ideals | Leave a comment

Commutative Algebra 59

Posted on May 17, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged associated primes, composition series, minimal primes, primary decompositions, supports | Leave a comment

Commutative Algebra 58

Posted on May 16, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged annihilators, associated primes, localization, module division, modules, supports | 4 Comments

Commutative Algebra 57

Posted on May 15, 2020 by limsup

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 | 1 Comment

Commutative Algebra 56

Posted on May 14, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged a-adic filtrations, artin-rees lemma, completion, exact functors, flat modules, induced modules, jacobson radical, krulls intersection theorem | 3 Comments

Commutative Algebra 55

Posted on May 13, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged a-adic filtrations, artin-rees lemma, blowup algebras, completions, filtrations, limits, noetherian | Leave a comment