Monthly Archives: March 2020

Commutative Algebra 13

Posted on March 31, 2020 by limsup

Zariski Topology for Rings In this article, we generalize earlier results in algebraic geometry to apply to general rings. Recall that points on an affine variety V correspond to maximal ideals . For general rings, we have to switch to … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , , | Leave a comment

Commutative Algebra 12

Posted on March 30, 2020 by limsup

Some Results on Posets In this article we have two goals in mind: to introduce the idea of noetherian posets, and to state Zorn’s lemma and give some examples. The latter is of utmost importance in diverse areas of mathematics. … Continue reading →

Posted in Advanced Algebra | Tagged , , , , | Leave a comment

Commutative Algebra 11

Posted on March 29, 2020 by limsup

Coordinate Rings as k-algebras Let k be an algebraically closed field. Recall that a closed subset is identified by its coordinate ring k[V], which is a finitely generated k-algebra since Definition. An affine k-variety is a finitely generated k-algebra A which is … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , | Leave a comment

Commutative Algebra 10

Posted on March 28, 2020 by limsup

Algebras Over a Ring Let A be any ring; we would like to look at A-modules with a compatible ring structure. Definition. An –algebra is an -module , together with a multiplication operator such that becomes a commutative ring (with 1); multiplication … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , , | 5 Comments

Commutative Algebra 9

Posted on March 27, 2020 by limsup

Direct Sums and Direct Products Recall that for a ring A, a sequence of A-modules gives the A-module where the operations are defined component-wise. In this article, we will generalize the construction to an infinite collection of modules. Throughout this article, let denote … Continue reading →

Posted in Advanced Algebra | Tagged , , , , | 5 Comments

Commutative Algebra 8

Posted on March 26, 2020 by limsup

Generated Submodule Since the intersection of an arbitrary family of submodules of M is a submodule, we have the concept of a submodule generated by a subset. Definition. Given any subset , let denote the set of all submodules of M containing … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , , | Leave a comment

Commutative Algebra 7

Posted on March 25, 2020 by limsup

Modules Having dipped our toes into algebraic geometry, we are back in commutative algebra. Next we would like to introduce “linear algebra” over a ring A. Most of the proofs should pose no difficulty to the reader so we will … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , | Leave a comment

Commutative Algebra 6

Posted on March 24, 2020 by limsup

Injective and Surjective Maps Proposition 1. Let be a morphism of closed sets, with corresponding . is injective if and only if is dense. is surjective if and only if is an embedding of V as a closed subspace of … Continue reading →

Posted in Advanced Algebra | Tagged , , , , | 4 Comments

Commutative Algebra 5

Posted on March 23, 2020 by limsup

Morphisms in Algebraic Geometry Next we study the “nice” functions between closed subspaces of . Definition. Suppose and are closed subsets. A morphism is a function which can be expressed as: for some polynomials . We also say f is a regular … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , | 2 Comments

Commutative Algebra 4

Posted on March 22, 2020 by limsup

More Concepts in Algebraic Geometry As before, k denotes an algebraically closed field. Recall that we have a bijection between radical ideals of and closed subsets of . The bijection reverses the inclusion so if and only if . Not too … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , | 7 Comments