Blogs about: Commutative Algebra
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E, f, and the remainder theorem
So, now to the next topic in introductory algebraic number theory: ramification. This is a measure of how primes “split.” (No, definitely wrong word there…) e and f Fix a Dedekind … more →
Delta Epsilons
Localization and the strong Nullstellensatz
Qiaochu Yuan wrote 1 day ago: A basic idea in topology and analysis is to study a space by restricting attention to arbitrarily sm … more →
MaxSpec is not a functor
Qiaochu Yuan wrote 2 days ago: For commutative unital C*-algebras and for finitely-generated reduced integral -algebras, we have se … more →
Affine varieties and regular maps
Qiaochu Yuan wrote 3 days ago: I have to admit I’ve been using somewhat unconventional definitions. The usual definition of a … more →
Functoriality
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Qiaochu Yuan wrote 6 days ago: I wanted to talk about the geometric interpretation of localization, but before I do so I should tal … more →
The ideal-variety correspondence
Qiaochu Yuan wrote 3 weeks ago: I guess I didn’t plan this very well! Instead of completing one series I ended one and am righ … more →
Irreducible components
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Qiaochu Yuan wrote 3 weeks ago: If it wasn’t clear, in this discussion all rings are assumed commutative. Given a variety like … more →
The Noetherian condition as compactness
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Qiaochu Yuan wrote 3 weeks ago: Let’s think more about what an abstract theory of unique factorization of primes has to look l … more →
The weak Nullstellensatz and affine varieties
Qiaochu Yuan wrote 3 weeks ago: Hilbert’s Nullstellensatz is a basic but foundational theorem in commutative algebra that has … more →
Spectra of rings of continuous functions
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Qiaochu Yuan wrote 1 month ago: An analyst thinks of the ring of polynomials as a useful tool because, on intervals, it is dense in … more →
Primes and ideals
Qiaochu Yuan wrote 1 month ago: Probably the first important result in algebraic number theory is the following. Let be a finite fie … more →
Variety-style look with Max Spec
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shallow33 wrote 2 months ago: Here is an interesting bit on Max Spec that I pulled from the exercises in Atiyah & Macdonald … more →
Boolean rings and Spec
shallow33 wrote 2 months ago: Boolean Ring A Boolean ring is a ring with every element satisfying From this we can deduce a couple … more →
e, f, and the remainder theorem
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Akhil Mathew wrote 3 months ago: So, now to the next topic in introductory algebraic number theory: ramification. This is a measure o … more →
Extensions of discrete valuations
Akhil Mathew wrote 3 months ago: With the school year starting, I can’t keep up with the one-post-a-day frequency anymore. Stil … more →
Hensel's lemma and a classification theorem
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Akhil Mathew wrote 3 months ago: So, I’ll discuss the proof of a classification theorem that DVRs are often power series rings, … more →
Completions of fields
Akhil Mathew wrote 3 months ago: So again, we’re back to completions, though we’re going to go through it quickly. Except … more →
Dedekind domains
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Akhil Mathew wrote 3 months ago: Today’s (quick) topic focuses on Dedekind domains. These come up when you take the ring of int … more →
DVRs II
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Akhil Mathew wrote 3 months ago: Earlier I went over the definition and first properties of a discrete valuation ring. Today, it … more →
Lifting idempotents à la Grothendieck
Akhil Mathew wrote 3 months ago: I am going to get back shortly to discussing algebraic number theory and discrete valuation rings. B … more →
