https://jagr2808.github.io/blog/ Recent content in Jacob's math blog on Jacob's blog Hugo -- gohugo.io en-us Mon, 14 Sep 2020 12:00:04 +0200 https://jagr2808.github.io/blog/posts/ar-duality/ Mon, 14 Sep 2020 12:00:04 +0200 https://jagr2808.github.io/blog/posts/ar-duality/ \( \require{AMScd} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\proj}{proj} \def\L{\Lambda} \def\mod{\operatorname{mod}} \) In this blogpost I want to talk about the concept of AR duality. In short AR duality says that in suitably nice abelian categories we have natural isomorphisms $$\underline{\Hom}(\tau^- Y, X) \cong D\Ext^1(X, Y) \cong \overline{\Hom}(Y, \tau X)$$ Where $\underline{\Hom}$ is the stable homfunctor, $\overline{\Hom}$ is the costable homfunctor and $\tau$ is the Auslander-Reiten translate. In this blogpost I will define what these things mean and try to give an outline of the proof. https://jagr2808.github.io/blog/posts/cogroups/ Sat, 01 Aug 2020 11:08:13 +0200 https://jagr2808.github.io/blog/posts/cogroups/ \( \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\C}{\mathbb{C}} \DeclareMathOperator{\Z}{\mathbb{Z}} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Mod}{Mod} \) Introduction I was watching Richard E. Borchards lecture series on schemes where he talked about groups in the category of schemes. Thinking about this I stumbled upon the idea of cogroups. I was already vaguely familiar with the idea of coalgebras, but I hadn’t really looked at cogroups before (or so I thought). So this blogpost will just be my ramblings of what I found out. https://jagr2808.github.io/blog/posts/geometry-of-polynomial-rings/ Tue, 14 Jul 2020 10:19:23 +0200 https://jagr2808.github.io/blog/posts/geometry-of-polynomial-rings/ \( \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\C}{\mathbb{C}} \) Inspired by Ravi Vakil’s summer project about algebraic geometry I wanted to make this blog post about the spectrum of polynomial rings and how to conceptualize them geometrically when your field isn’t algebraically closed. The spectrum of a ring $R$, denoted $\Spec R$, is the set of all prime ideals of $R$ (with some additional structure). We think of a $\Spec R$ as a geometric space where the points are the maximal ideals, and the prime ideals contained in a maximal ideal represent geometric objects containing the point. https://jagr2808.github.io/blog/posts/kan-extensions/ Sat, 30 May 2020 21:09:44 +0200 https://jagr2808.github.io/blog/posts/kan-extensions/ \( \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\C}{\mathbb{C}} \DeclareMathOperator{\Z}{\mathbb{Z}} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Mod}{Mod} \) Recently I learned about Kan extension which is a generalization of (co)limits in category theory. So I thought it would be fun to write a little blog post introducing limits and Kan extensions. The idea of a (co)limit is to capture information about a diagram in just one object. The idea of Kan extensions is similar, but instead of trying to capture information in a single object we do it in some other diagram.