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 …

Brieskorn resolutions via algebraic spaces

I’d like to discuss simultaneous resolutions of surfaces from a moduli-theoretic perspective, following Michael Artin’s paper on Brieskorn resolutions.

Artin’s approach to moduli begins with the most desirable aspect of a moduli space, its universal property. That is to say, define a functor the space should represent and then check …

The torsion component of the Picard scheme

This post is a continuation of Sean Cotner’s most recent post [see An example of a non-reduced Picard scheme]. Since writing that post, Bogdan Zavyalov shared some notes of his proving the following strengthened version of the results described there.

Main Theorem. Let \(S\) be a noetherian local ring and …

Chern connections in the context of CR-geometry

This post is a result of several discussions with Rodion Déev.

1. Preliminaries: Chern’s theorem

Fix a complex manifold \(X\) and a complex vector bundle \(E\) over \(X\). Recall that a structure of holomorphic bundle on \(E\) is given by an operator

\[\overline{\partial}_{\mathcal{E}} \colon \Gamma (E) \to \Gamma(E \otimes \Lambda^{0,1}X)\]

which satisfies \(\overline{\partial}\)-Leibniz identity

\[\overline{\partial}_{\mathcal{E}}(fs) = f\overline{\partial}_{\mathcal{E}}(s) + \overline{\partial}(f) \otimes s\]

and the integrability condition

\[\overline{\partial}_{\mathcal{E}}^2 = 0.\]

For the …

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 …