Category Archives: Advanced Algebra

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

Commutative Algebra 27

Posted on April 14, 2020 by limsup

Free Modules All modules are over a fixed ring A. We already mentioned finite free modules earlier. Here we will consider general free modules. Definition. Let be any set. The free A-module on I is a direct sum of copies of … Continue reading →

Posted in Advanced Algebra | Tagged exact functors, free modules, left-exact, localization, projective modules, splitting lemma | Leave a comment

Commutative Algebra 26

Posted on April 13, 2020 by limsup

Left-Exact Functors We saw (in theorem 1 here) that the localization functor is exact, which gave us a whole slew of nice properties, including preservation of submodules, quotient modules, finite intersection/sum, etc. However, exactness is often too much to ask … Continue reading →

Posted in Advanced Algebra | Tagged additive functors, hom functor, induced modules, left-exact, right-exact | Leave a comment

Commutative Algebra 25

Posted on April 12, 2020 by limsup

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 | Leave a comment