Category Archives: algebraic-geometry

Beware of the Two Galois Actions

Let $F$ be a field and $E/F$ be a nontrivial Galois extensions with Galois group $\Gamma$. If $V$ is an $F$-scheme then the points $V(E)$ carry a natural action of $\Gamma$ via the action on $\mathrm{Spec}(E)$. Sometimes, however, $V$ might have two Galois actions. How does this arise? Perhaps the most natural setting is when […]


arXiv: Kindler and Rülling's Intro Notes on l-adic Sheaves

One policy of Aleph Zero Categorical is that any lecture notes posted on the arXiv that I manage to see will be announced and advertised here. Today I saw that Lars Kindler and Kay Rülling have posted their notes entited: Introductory course on $\ell$-adic sheaves and their ramification theory on curves I quote from the […]


Calculation of an Orbital Integral

In the Arthur-Selberg trace formula and other formulas, one encounters so-called 'orbital integrals'. These integrals might appear forbidding and abstract at first, but actually they are quite concrete objects. In this post we'll look at an example that should make orbital integrals seem more friendly and approachable. Let $k = \mathbb{F}_q$ be a finite field […]


From Rational Canonical Form to The Kostant Section

Suppose we have a $2\times 2$ matrix $$ M = \begin{pmatrix} x_{11} & x_{12}\\ x_{21} & x_{22} \end{pmatrix} $$ with entries in a field $F$. The characteristic polynomial of this matrix is $p(t) := {\rm det}(tI_2 – M) = t^2 – (x_{11} + x_{22})t + x_{11}x_{22} – x_{21}x_{12}$. One might ask: how can we produce […]


Guess The Algebraic Group

Suppose one day you run into the following algebraic group, defined on $\mathbb{Z}$-algebras $R$ by $$G(R) = \left\{ \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{11} & -a_{14} & -a_{13} \\ a_{13} & -a_{14} & a_{11} & -a_{12} \\ a_{14} & -a_{13} & -a_{12} & a_{11} \end{pmatrix} \in\mathrm{GL}_4(R) \right\}$$ Can you […]


G-Ideals, Maximal Ideals, and The Nullstellensatz

Let $ R$ be an integral domain and $ K$ is fraction field. If $ K$ is finitely generated over $ R$ then we say that $ R$ is a $ G$-domain, named after Oscar Goldman. This innocuous-looking definition is actually an extremely useful device in commutative algebra that pops up all over the place. […]


2 Dimensional Connected Algebraic Groups are Solvable

Conventions: $ G$ is an algebraic group over an algebraically closed field $ k$ and we identify $ G$ with $ G(k)$. Consider the algebraic groups $ \mathbb{G}_a$ and $ \mathbb{G}_m$. They are the only one-dimensional connected groups and they are both solvable. What about two-dimensional connected groups? It turns out that if $ \mathrm{dim} […]


Extensions of Tori by Tori are Tori

Continuing our previous series, $ G$ is an algebraic group over an algebraically closed field $ k$ and we identify $ G$ with $ G(k)$. Here is an interesting fact: Theorem. In a connected solvable group the unipotent part $ G_u$ is a closed connected normal subgroup of $ G$ and contains the commutator subgroup […]


Finite Normal Subgroups Of Connected Groups Are Central

The previous series on algebraic groups is over. Actually, I barely got to the root system and root datum of a reductive group, but I found that the whole slew of material was getting too complex to organise on this blog, which I feel is better for more self-contained posts. Instead, I have begun to […]


Highlights in Linear Algebraic Groups 14: Singular Tori

From Highlights 12 and Highlights 13, we have gained quite a bit of information on connected reductive groups $ G$ of semisimple rank 1. Recall, this means that $ G/R(G)$ has rank 1 where $ R(G)$ is the radical of $ G$, which is in turn the connected component of the unique maximal normal solvable […]