Author Archives: limsup

Commutative Algebra 37

Posted on April 25, 2020 by limsup

Artinian Modules Instead of the ascending chain condition, we can take its reverse. Definition. Let M be an A-module. Consider the set of submodules of M, ordered by inclusion, i.e. if and only if . We say M is artinian … Continue reading →

Posted in Advanced Algebra | Tagged artinian, composition factors, composition series, length of module, noetherian, simple modules | Leave a comment

Commutative Algebra 36

Posted on April 24, 2020 by limsup

In this article, we will study the topology of Spec A when A is noetherian. For starters, let us consider irreducible topological spaces in greater detail. Irreducible Spaces Recall that an irreducible topological space is a non-empty space X satisfying any of the … Continue reading →

Posted in Advanced Algebra | Tagged irreducible components, irreducible spaces, noetherian, noetherian spaces, zariski topology | Leave a comment

Commutative Algebra 35

Posted on April 23, 2020 by limsup

Noetherian Modules Through this article, A is a fixed ring. For the first two sections, all modules are over A. Recall that a submodule of a finitely generated module is not finitely generated in general. This will not happen if we constrain … Continue reading →

Posted in Advanced Algebra | Tagged flat modules, hilbert basis theorem, ideals, modules, noetherian, projective modules | Leave a comment

Commutative Algebra 34

Posted on April 22, 2020 by limsup

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 →

Posted in Advanced Algebra | Tagged cotangent spaces, finitely presented, flat modules, local rings, locally free, nakayamas lemma, projective modules | Leave a comment

Commutative Algebra 33

Posted on April 21, 2020 by limsup

Snake Lemma Let us introduce a useful tool for computing kernels and cokernels in a complicated diagram of modules. Although it is only marginally useful for now, it will become a major tool in homological algebra. Snake Lemma. Suppose we … Continue reading →

Posted in Advanced Algebra | Tagged finitely presented, hom functor, induced modules, local properties, projective modules, snake lemma | Leave a comment

Commutative Algebra 32

Posted on April 20, 2020 by limsup

Torsion and Flatness Definition. Let A be a ring and M an A-module; let . If satisfies , we call it an –torsion element. If is an -torsion for some non-zero-divisor we call it a torsion element. M is said … Continue reading →

Posted in Advanced Algebra | Tagged algebraic geometry, flat modules, torsion, varieties, zero-divisors | Leave a comment

Commutative Algebra 31

Posted on April 19, 2020 by limsup

Flat Modules Recall from proposition 3 here: for an A-module M, is a right-exact functor. Definition. We say M is flat over A (or A-flat) if is an exact functor, equivalently, if A flat A-algebra is an A-algebra which is flat as … Continue reading →

Posted in Advanced Algebra | Tagged direct sums, exact functors, flat modules, local properties, localization, projective modules, tensor products | Leave a comment

Commutative Algebra 30

Posted on April 17, 2020 by limsup

Tensor Product of A-Algebras Proposition 1. Let B, C be A-algebras. Their tensor product has a natural structure of an A-algebra which satisfies . Proof Fix . The map is A-bilinear so it induces an A-linear map Now varying (b, c) gives … Continue reading →

Posted in Advanced Algebra | Tagged algebraic geometry, algebras, coproducts, fibres, tensor product, varieties | 2 Comments

Commutative Algebra 29

Posted on April 16, 2020 by limsup

Distributivity Finally, tensor product is distributive over arbitrary direct sums. Proposition 1. Given any family of modules , we have: Proof Take the map which takes . Note that this is well-defined: since only finitely many are non-zero, only finitely … Continue reading →

Posted in Advanced Algebra | Tagged hom functor, induced modules, localization, right-exact, tensor products, yoneda lemma | 2 Comments

Commutative Algebra 28

Posted on April 15, 2020 by limsup

Tensor Products In this article (and the next few), we will discuss tensor products of modules over a ring. Here is a motivating example of tensor products. Example If and are real vector spaces, then is the vector space with … Continue reading →

Posted in Advanced Algebra | Tagged bilinear maps, distributive property, modules, tensor product, universal properties | 2 Comments