Author Archives: limsup

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 →

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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 →

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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 →

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Commutative Algebra 24

Posted on April 11, 2020 by limsup

Quotient vs Localization Taking the quotient and localization are two sides of the same coin when we look at . Quotient removes the “small” prime ideals in – it only keeps the prime ideals containing . Localization removes the “large” … Continue reading →

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Commutative Algebra 23

Posted on April 10, 2020 by limsup

Localization and Spectrum Recall that the ideals of correspond to a subset of the ideals of A. If we restrict ourselves to prime ideals, we get the following nice bijection. Theorem 1. The above gives a bijection between Useful trick If … Continue reading →

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Commutative Algebra 22

Posted on April 9, 2020 by limsup

Localization Recall that given an integral domain, there is a canonical way to construct the “smallest field containing it”, its field of fractions. Here, we will generalize this construction to arbitrary rings. We let A be a fixed ring throughout. Definition. … Continue reading →

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Commutative Algebra 21

Posted on April 8, 2020 by limsup

Exact Sequences When studying homomorphisms of modules over a fixed ring, we often encounter sequences like this: where each is a homomorphism of modules. This sequence may terminate (on either end) or it may continue indefinitely. Note on indices: usually … Continue reading →

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Commutative Algebra 20

Posted on April 7, 2020 by limsup

Yoneda Lemma For an object , define the covariant functor Proposition 1. Any morphism in gives us a natural transformation In summary, the natural transformation is obtained by right-composing with f. Proof Let be a morphism in . We need … Continue reading →

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Commutative Algebra 19

Posted on April 6, 2020 by limsup

Natural Transformations “I didn’t invent categories to study functors; I invented them to study natural transformations.” – Saunders Mac Lane, one of the founders of category theory A natural transformation is, loosely speaking, a homomorphism between functors. Its definition may … Continue reading →

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Commutative Algebra 18

Posted on April 5, 2020 by limsup

Basics of Category Theory As we proceed, we should cover some rudimentary category theory or many of the subsequent constructions would seem unmotivated. The essence of category is in studying algebraic objects and the homomorphisms between them. By now we have … Continue reading →

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