## An explicit construction of indecomposable vector bundles over an elliptic curve

In the celebrated paper “Vector bundles over an elliptic curve,” M. Atiyah classifies indecomposable vector bundles, namely he provides a bijection between indecomposable bundles of arbitrary rank and degree (denoted by ) and (where ). The latter is described explicitly: there is a distinguished element such that for any other …

## A proper scheme with infinite-dimensional fppf cohomology

In algebraic geometry, very often one encounters theorems of the following flavor:

Theorem: Let be a proper morphism of spaces. Then for every sheaf on that is finite, so is its pushforward .

Notice how I was being deliberately vague in the theorem above. What are and ? What does …

## Demystification of the Willmore integrand

The Willmore energy for a surface in Euclidean 3-space is defined as , where is the mean curvature of and its area form. It’s known to be invariant under the conformal transformations (whereas the mean curvature itself is not). White, and later Bryant noticed that the 2-form , where stands …

## Looking for a better counterexample

This post is doubling my old question asked on Mathoverflow.

Let be a smooth projective complex variety of dimension , and let be an ample irreducible divisor. Recall that a cohomology class is called -primitive, if . Here, is the first Chern class of the line bundle associated to . …

## The étale cohomology of curves over finite fields

When I was a graduate student, Zev Rosengarten (a former student of Brian Conrad) and I used to eat dinner at Stanford’s Arrillaga dining hall a lot. We’d talk about math for hours, but one thing that will forever be ingrained in my mind is how Zev was able to …

## A shortcut in Kapovich’s proof of Haupt’s theorem

The Teichmüller space of genus curves carries the Hodge bundle , the total space of which maps into the first cohomology space via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image in terms of the integral structure on and …

## The Steinberg Representation

In this post I want to describe a remarkable representation associated to finite groups of Lie type. For this, let be a connected reductive group over a finite field with elements, and let be the unipotent radical of some Borel -subgroup of . Steinberg constructed an irreducible representation of of …

## Fun example: Empty colimit does not commute with empty limit

One important property of filtered colimits is that they commute with finite limits in the category of sets.

Theorem: Let be a functor, where is a filtered small category and is a finite category. Then the natural mapping

is an isomorphism.

This statement is used for example to check that …

## A non-Noetherian local ring with finitely generated maximal ideal

Some time ago I found the following interesting lemma on the stacksproject:

Theorem: (tag/05GH) Let be a finitely generated ideal in a ring . Then the -adic completion is Noetherian if is.

Corollary: Let be a complete local ring with a finitely generated maximal ideal . Then is …