\( \require{AMScd} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\proj}{proj} \def\L{\Lambda} \def\mod{\operatorname{mod}} \) 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.

\( \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\C}{\mathbb{C}} \DeclareMathOperator{\Z}{\mathbb{Z}} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Mod}{Mod} \) 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.

\( \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\C}{\mathbb{C}} \) 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.