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 \(\alpha_{(r, d)}\) between indecomposable bundles of arbitrary rank \(r\) and degree \(d\) (denoted by \(\mathcal{E}_{(r, d)}\)) and \(\mathcal{E}_{(h, 0)}\) (where \(h = \text{gcd}(d, r)\)). The latter is described explicitly: there is a distinguished element \(F_r \in \mathcal{E}_{(r, 0)}\) 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 \(f : X \to S\) be a proper morphism of spaces. Then for every sheaf \(\mathcal{F}\) on \(X\) that is finite, so is its pushforward \(Rf_\ast \mathcal{F}\).

Notice how I was being deliberately vague in the theorem above. What are \(X\) and \(Y\)? What does …

Simple Proof of Tokuyama’s Formula

Tokuyama’s Formula is a combinatorial result that expresses the product of the deformed Weyl denominator and the Schur polynomial as a sum over strict Gelfand-Tsetlin patterns. This result implies Gelfand’s parametrization of the Schur polynomial, Weyl’s Character Formula, and Stanley’s formula on Hall-Littlewood polynomials — all for \(\text{GL}_n\); …

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

The Teichmüller space \(T_g\) of genus \(g\) curves carries the Hodge bundle \(\Omega T_g\), the total space of which maps into the first cohomology space \(V = H^1(S_g,\mathbb{C})\) via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image \(per(\Omega T_g) \subset V\) in terms of the integral structure on \(V = H^1(S, \mathbb{Z}) \otimes \mathbb{C}\) and …

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 \(F \colon \mathcal{C}\times \mathcal{D} \to \mathbf{Sets}\) be a functor, where \(\mathcal{C}\) is a filtered small category and \(\mathcal{D}\) is a finite category. Then the natural mapping

\[\mathrm{colim}_{\mathcal{C}} \lim_{\mathcal{D}} F (c, d) \to \lim_{\mathcal{D}} \mathrm{colim}_{\mathcal{C}} F(c, d)\]

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 \(I\) be a finitely generated ideal in a ring \(R\). Then the \(I\)-adic completion \(\widehat{R}\) is Noetherian if \(R/I\) is.

Corollary: Let \(R\) be a complete local ring with a finitely generated maximal ideal \(\mathfrak{m}\). Then \(R\) is …