In the category of varieties, morphisms are determined (at least locally) by $k$-algebra homomorphisms between the rings of regular functions. Another invariant of a variety $X$ is its function field $K(X)$, whose elements are the germs $\langle U, \varphi_{U} \rangle$ with $\varphi_U$ a regular function on the open set $U$. Multiplying germs is then achieved sheaf-theoretically,

Rational maps answer two questions for us:

Over the last few days I've been learning some machinery used to analyse $\mathbb{G}_{m}$-actions in algebraic geometry. One of the results I've needed for this is Lüroth's theorem. In this write-up I will follow the proof given in Hartshorne, which makes the geometric content of the statement clear and establishes theory about morphisms of curves along the way. This is a favourite theme in my recent work - analysis of field extensions usually provides strong information about morphisms of varieties. I especially enjoy results in the characteristic $p$ case; the field theory developed on its own can feel overly complicated.

One of the more frustrating problems in day-to-day ring theory is determining isomorphisms of rings. In algebraic geometry it is often the case that problems involve adding or removing variables subject to equations to polynomial rings (geometric content: going up and down in dimension, which we do for blowups, vector bundles, covers, cones, etc), and there is no silver bullet for finding the simplest description of such rings. To the beginner in algebraic geometry it is difficult to know what options we have for speeding up these calculations, either through existing algorithms or variants we construct ourselves. Statements which appear obvious can turn out to be difficult, even equipped with the usual tricks like the Nullstellensatz.

Today I spent a few hours solving one of these problems, aiming to reinforce my improvisation skills on the problem and develop some new techniques. I found this problem while preparing to study virtual fundamental classes, which requires some familiarity with intersection theory. This lead me to learning about normal cones and (soon) obstruction theory. As a first step to this, I am looking at the normal cone to the projective variety $X = V(xz, yz) \subseteq \mathbb{P}^3_{[x:y:z:w]}$ (below: a picture of $X$ in the affine chart $w \neq 0$).

I would like this note to be a running document on Galois theory in characteristic $p$ and mixed characteristic for algebraic geometry. Hopefully the scope will become apparent in time!

The last few weeks I've slowly prepared to learn the Artin-Schreier Theorem, a project which was originally motivated by wanting to attend lectures on perfectoid spaces and almost rings. As the weeks of linear algebra tutoring went by it became clear to me that I was unhappy attending without more background, so my goal has instead become to understand this theorem and its application in that context. Now that I've finally achieved part of that, I think the goal is to learn some Kummer theory and the Witt vector version of the story, and then I will see how I can course-correct from there.

Yesterday I spent a little time on a Hartshorne exercise which checks that $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$ are group schemes. By now I've thought about this a few times and it's included in my thesis, but I learnt some new ideas by proving this in the category of varieties, which feels like another piece of working with the variety-theoretic definition of regular functions (something I've always been uneasy about, I gravitate to the scheme theory because I can use more algebra).

To an algebraist the simplest geometric spaces are affine varieties, determined by their ring of functions. Unfortunately, students in algebraic geometry discover early that subvarieties are (a) interesting to study since functions and sheaves restrict to them and (b) indistinguishable by their function rings alone.
Using (b) to detect non-affineness, we can show rigorously that the quasiaffine variety $\Bbb{A}^2 \setminus \{(0,0)\}$ is not affine.

Today I would like to discuss the proof that if $R$ is a local ring with residue field $k$, the functors $h_R$ and $h_{\hat{R}}$ are isomorphic, where $h_S : \mathsf{Art}/k \to \mathsf{Set}, h_{S}(A) = \text{Hom}_{k}(S, A)$ is the functor sending an artinian local ring with residue field $k$ to the set of maps into it from $S$. We will see that this boils down to the fact that artinian local rings are complete (I will not prove it this way, but trust that readers can see the underlying idea), but I would also like to talk about identifying this as the correct deformation functor.

If $R$ is a local $k$-algebra with residue field $k$ and finite tangent space $(\mathfrak{m}_{R}/\mathfrak{m}_{R}^2)^\vee$ of dimension $d$, then $R$ is a quotient of a power series ring $S = k[\mkern-3mu [t_{1}, \dots, t_{d}]\mkern-3mu]$ by an ideal $J \subseteq (t_{1}, \dots, t_{d})^2$. Let $T$ be the tangent space of $S$ and write $n = (t_1, \dots, t_{n})$. Then we have the followingTheorem: For every small extension $0 \to M \to B \to A \to 0$ (ie a surjection of Artinian $k$-algebras with kernel $M$ annihilated by $\mathfrak{m}_{B}$), there is a short exact sequence $0 \to T\otimes_{k} M \to \text{Hom}_{k}(R, B) \to \text{Hom}_{k}(R,A) \xrightarrow{ob} (J/nJ)^\vee \otimes_{k} M,$ which is functorial in small extensions.

This week I attended AustMS, but I still had some time on the long tram trips to do a bit of study. I mostly spent this time learning what a Hall algebra is and studying the example of nilpotent representations for the Jordan quiver. Thus far, my best summary would be that the Hall algebra of a suitable abelian category is an associative algebra which contains information about the short exact sequence structure of the category.