A Geometric Reason to Care About Discrete Valuation Rings

Math
Perhaps those that have taken an introductory commutative algebra class have come across the profoundly opaque definition of a discrete valuation ring (DVR), thinking to themselves "why have the math gods condemned me?" From a purely algebraic perspective, it is a remarkable that a ring $A$ is a valuation ring if and only if it is a domain so that for any element in the fraction field for $A$, $x$ or $x^{-1}$ is contained in  $A$. Even further, if a valuation ring is Noetherian (and not a field) it is a DVR. Again, algebraically, there are a lot of reasons to care about these rings, but today isn't about algebra. While there are a huge number of equivalences we can speak of, but instead we will make use of the following definition:   Definition:…
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Pullbacks of maximal ideals in a finitely generated k-algebra are again maximal

Math
A few days ago, some classmates and I were thinking about about whether or not morphisms in $\mathrm{maxSpec}(A)$ were well defined. In particular, given a map of $k$-algebras $f:A \to B$, we consider the induced morphism $f^*:\mathrm{maxSpec} A \to \mathrm{maxSpec B}$ given by $X \mapsto f^{-1}(X)$. We all knew that pullbacks of maximal ideals need not be maximal for ring morphisms, which is precisely what led to a great deal of confusion-- thankfully this was all fixed as soon as we were reminded that we were considering morphisms of $k$-algebras. While this was funny, I think the proof for this fact is both an elementary and interesting check up in commutative algebra: Theorem 1: let $A,B$ be finitely generated $k$-algebras. Then the pullback of any maximal ideal in a $k$-algebra homomorphism  is again…
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Tensors are interesting in Representation Theory

Math
Initially, I was baffled by the emphasis on tensor products in Representation Theory. I thought of tensors as a way to construct new from the old, in the sense that we can easily obtain new and potentially interesting representations of a group just by "tensoring." This was definitely important in representations of finite groups, since it gave an easy way to deduce an exhaustive list of irreducible representations for some group $G$. Yet there was still some motivation missing. This post is more or less how I remedied my uneasiness. The not-so-well hidden secret is that we can actually discuss representation theory with the language of modules. In particular, there is this really nice object known as the group ring, ( but for our purposes the group algebra over $\mathbb…
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A Non-Algebraic Proof for the Hairy Ball Theorem.

Math
Here, I want to share a peculiar analytic proof of the Hairy Ball Theorem, which states colloquially that "you cannot perfectly comb a coconut." It may not be clear that this can never occur, but if you start going around the $2$-sphere, there is no way to "smooth out" a bunch of vectors without creating either a swirl or some "clash." There is an absolutely standard proof using a notion of degree, which I will include at the end of this post just for completeness. Yet, there's a relatively unknown proof  due to John Milnor, and we'll borrow from his paper  liberally. I think it is very nice that we need virtually no machinery beyond multi-variable calculus and general topology. Somehow when different theories can establish the same theorem, the result becomes more…
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A funny Proof For Sunzi’s Remainder Theorem

Math
This post will assume familiarity with some basic notions from commutative algebra such as localization for modules and exact sequences. Here we will mean "ring" to mean a commutative ring with unity. This post is about to the  "Chinese Remainder Theorem.) however, I prefer a renaming— while Sunzi's identity is unknown, I do not think "Sunzi's remainder theorem" will cause any great confusion. Anyhow, the theorem is best remembered by the informal statement "you can simultaneously solve a system of modular conditions, given that the moduli are coprime." We'll consider a mild generalization to arbitrary rings, granted that we can make sense of "coprime" outside of principal ideal domains, leading us to the following definition: Definition 1 Two ideals $I,J$ of a ring $R$ are said to be coprime if…
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