Thuses

Relative tensor products of $\infty$ -categories of local systems

The Lurie tensor product is part of a symmetric monoidal structure on the $\infty$ -category $\mathrm{Pr}^\mathrm{L}$ of presentable $\infty$ -categories.

It enjoys many good formal properties, and is often computable. An example of this computability is that if $C$ is a small $\infty$ -category and $\mathcal{D,E}$ are presentable, then $\mathcal D^C \otimes \mathcal E \simeq (\mathcal D\otimes \mathcal E)^C$ . A special case of this is …

Some families of finite flat group schemes

When people say finite flat group scheme, what exactly do they mean? Sometimes, they just mean finite flat group scheme, presumably over some prefixed base. But often people meant finite flat *commutative* group scheme. It’s confusing. In this blog, I shall always mean the former, and add the adjective “commutative” …

Some examples of algebraic groups

In this post I want to give a few examples of the known “pathological” behavior of algebraic groups defined over general bases. In particular, this post contains examples of the following.

A smooth group scheme over a DVR with generic fiber $\mathbf{G}_m$ and special fiber $\mathbf{G}_a$ ,
An affine smooth group scheme

…

The rank of $y^2 = x^3 - 2$ via Mazur-Tate methods

When I was a young kid, I heard the mathematical fact that the only (positive) integer that is one more than a square and one less than a cube is $26$ . Said differently, the only integer solutions $(x,y)$ to $y^2 = x^3 - 2$ are given by $(3,\pm 5)$ . There are elementary methods to prove this, using …

Isometries of a product of Riemannian manifolds

Theorem. Let $(A, g_A)$ and $(B, g_B)$ be two compact Riemannian manifolds with irreducible holonomy groups. Let $(X, g):= (A \times B, g_A \oplus g_B)$ . Then

$\operatorname{Isom}(X) = \begin{cases} \operatorname{Isom}(A) \times \operatorname{Isom}(B), \ \text{ if } A \text{ is not isometric to } B \\ \left ( \operatorname{Isom}(A) \times \operatorname{Isom}(B) \right ) \rtimes \mathbb{Z}/2\mathbb{Z}, \ \text{ otherwise } \end{cases}$

This result seems to be a folklore, probably well known to the specialists, although it is hard to find it in the literature. The only discussion which I managed to find on Mathoverflow contains …

Finite flat commutative group schemes embed locally into abelian schemes

Let $G$ be a finite flat commutative group scheme over a fixed locally noetherian base scheme $S$ . In this brief note, I want to explain the proof of the following theorem due to Raynaud.

Theorem. There exists, Zariski-locally on $S$ , an abelian scheme $A$ such that $G$ embeds as a closed $S$ -subgroup of …

What Topological Spaces are π₀?

This was a fun question I thought about once. My answer is at the end, in case you’d like to try solving the problem yourself. The question is likely more interesting than my solution.

A well known theorem says that every group occurs as $\pi_1(X)$ for some topological space $X$ . It’s …

A curiosity: “supersmooth” varieties

I want to share a curious condition on varieties for which I have found no use. Let $k$ be a field and let $X$ be a locally finite type $k$ -scheme. Recall that $X$ is said to be smooth if, for every Artin local $k$ -algebra $A$ with a proper ideal $I \subset A$ and every $k$ -morphism $\Spec A/I \to X$ , there …

A proof of a general slice-Bennequin inequality

In this blog post, I’ll provide a slick proof of a form of the slice-Bennequin inequality (as outlined by Kronheimer in a mathoverflow answer.) The main ingredient is the adjunction inequality for surfaces embedded in closed 4-manifolds. To obtain the slice-Bennequin inequality (which is a statement about surfaces embedded …

The Picard number of a Kummer K3 surface

Let $k$ be a separably closed field of characteristic not $2$ , and $A/k$ an abelian surface. Then it is a basic fact (e.g. see Example 1.3 (iii) of Huybrechts’ “K3 Surfaces”) that one can make a K3 surface out of $A$ . The construction is as follows. Consider the involution $\iota : A \to A$ given by $x \mapsto -x.$ The …