# Challenge Accepted

Remember this challenge? Probably not. But wait, don’t click! Namely, I will do something more general in this post. Suppose we have a ring A and a contravariant functor F on (Sch/A) with the following properties: F satisfies the sheaf property … Continue reading

# Another challenge

Let A —> B be a finitely presented ring map. Let M be a finitely presented B-module flat over A. Then the projective dimension of M as an A-module is at most 1. This follows from a result of Jensen … Continue reading

# Challenge

Let f : X —> B be a morphism of algebraic spaces. Let u : F —> G be a map of quasi-coherent O_X-modules. Consider the functor F : (Sch/B) —> (Sets), T |—> singleton if u_T is zero and … Continue reading

# A challenge

Here is a challenge to an commutative algebraist out there. Give a direct algebraic proof of the following statement (see Lemma Tag 05U9): Let A —> B be a local ring homomorphism which is essentially of finite type. Let N … Continue reading

# Minimal Cech coverings

Please don’t read this unless you want to be distracted in these difficult times and you enjoy thinking about elementary problems in etale cohomology. I am going to pose a challenge computing etale cohomology by Cech coverings. The setting. We … Continue reading

# Comments and the Stacks project

As discussed here the new Stacks project website allows you to leave comments. In this blog post I’d like to advertise this as a fun activity and as well discuss what kind of comments I’d like to encourage. Before we … Continue reading

# Purity

Let f : X —> S be a morphism of finite type. The relative assassin Ass(X/S) of X/S is the set of points x of X which are embedded points of their fibres. So if f has reduced fibres or … Continue reading

# Splitting chapters

So I was wondering if it would maybe help to split the chapters in the stacks project into shorter chapters. To do this I wrote a python script that can automatically fix tags and references that point to the wrong … Continue reading