| Reducing flat groupoids to smooth groupoids (pdf)
This note is written for the stacks project. It contains a streamlined exposition of Artin's theorem that flat groupoids in schemes admit smooth presentations under mild separation conditions.
| Étale morphisms of schemes (pdf)
This note was written for what has become the stacks project. It contains an exposition of basic properties of étale, unramified, and smooth morphisms, especially the functorial characterisations. It has now been incorporated into the stacks project, and the most current (and heavily modified) version can be found here.
| Formal glueing of module categories (pdf)
This note is written for the stacks project. It contains an exposition of Artin's theorem that coherent sheaves on a (reasonable) scheme can be constructed formal locally around a closed subscheme provided the value on the complement has been specified.
| Counting points on images (pdf)
This note discusses a proof of the rationality of the generating function counting the size of the image of the set of rational points under a morphism of varieties over a finite field.
| What is a ... perfectoid space? (pdf)
This note is written for the AMS Notices, and attempts to give an introduction to Scholze's perfectoid spaces.
| A non-algebraic Hom-stack (pdf)
This note is written for the stacks project. It gives an example of a non-representable Hom-stack.
| Flat maps are not limits of finitely presented flat maps (pdf)
This note is written for the stacks project, following a query of B. Conrad and M. Emerton. It gives an example of a flat map of rings that cannot be written as a direct limit of finitely presented flat maps; the base ring is as nice as possible.
| An imperfect ring with a trivial cotangent complex (pdf)
This note records an example (suggested by Gabber) of a non-reduced ring in characteristic p which has a trivial cotangent complex.
| Torsion derived completions are small (pdf)
This note records an application to the Banach Open Mapping Theorem to algebra: we show that derived I-complete modules that are acyclic outside the vanishing locus of I are actually killed by a power of I.
| Refined alterations (with Andrew Snowden) (pdf)
This note records an refinement of de Jong's alteration theorems; the main improvement is that we are able to exercise some control on the etale locus of the alterations.
| Non-liftability of vector bundles to the Witt vectors (pdf)
We give an example of a vector bundle on a smooth projective variety in characteristic p that does not extend to the Witt vector thickenings of the variety.