SAD

Motivations

Hello and welcome to SAD, short for Small AG (Algebraic Geometry) Discussion. The goal of this project is to describe the foundations of algebraic geometry from a purely functorial point of view. You can access the GitHub for this project here. SAD is primarily inspired by this course taught by Sam Raskin and this blog post made by David Ben-Zvi. Much of this project might seem cosmetic and to some extent this is true once the theory is really off the ground. That being said, there are major reasons to do things using the factor-of-points rather than, say, locally ringed spaces. I will organize these roughly by my personal order of importance.

  • Many objects in algebraic and arithmetic geometry arise naturally as solutions to moduli problems. Projective space and Grassmannians are both good examples. Basically by definition, effectively describing moduli problems relies on the functor-of-points.
  • Algebraic groups are best understood as functors. This is especially the case if one wants to work with underlying objects like Hopf algebras.
  • Working with the functor-of-points from the get-go makes it much more intuitively natural to introduce general sites and topoi, which have proved indispensable to modern algebraic and arithmetic geometry. Students of these subjects already have so much to learn so we should be mindful about them also having to “unlearn” things in the process.
  • Learning algebraic geometry in this way makes it much easier to learn and read derived algebraic geometry (DAG) and spectral algebraic geometry (SAG). It is my personal belief that both of these fields will grow immensely in importance in the coming decades so algebraic and arithmetic geometers will need to be at least passingly acquainted. This is currently difficult in large part because many texts on DAG and SAG tacitly assume their readers have already thoroughly categorified the foundations.
  • Doing things in a largely non-topological way actually helps us understand the role that topology plays! This is certainly true in algebraic geometry (e.g., for notions like genericity) but where this really matters is in arithmetic geometry. The work of Scholze using adic spaces has shown us that viewing things in terms of locally (topologically) ringed spaces is absolutely essential and so it is important to understand why this is the case.
  • Even if something in algebraic geometry can be described topologically, it is often best understood or constructed in a more functorial way. Such is the case for, say, closed immersions of schemes — checking we have a topological closed immersion on the level of underlying topological spaces is often difficult in practice and doesn’t actually shed much light on the situation. This is even more true for things encoded by valuative criteria like formal smoothness.
  • Building off of the previous point, topology can sometimes fail to capture the appropriate geometry. In topology, open and closed sets exist on somewhat equal footing in the sense that the complement of one type is always of the other type and this process is involutary. However, in algebraic geometry there is reason to want the complement of a closed immersion to be an open immersion while the complement of an open immersion need not be a closed immersion.
  • As another illustration of the above point, the topological operation of closure is sometimes difficult to “geometrize.” In nice situations (e.g., working with a quasicompact open immersion), scheme-theoretic closure agrees with topological closure. However, in more general situations this need not be the case. This presents a problem since it is really the scheme-theoretic closure we care about, which won’t behave as expected if we approach the matter from a naïve topological viewpoint.

Challenges

Here are some things that need to be figured out and/or implemented.

  • Valuative criteria for universal closedness and properness (reference)
  • Dimension theory “as geometric as possible”
  • Limits of topological closure
  • Dealing with (set-theoretic) surjective morphisms of schemes
  • “De-topologizing” genericity theorems
  • Intersection theory (working with cycle classes in the appropriate manner)
  • Comparison of topological and functorial perspectives on formal geometry
  • Reasoning about connected components, especially the connected component of the identity $G^0$ for $G$ a group scheme

Teaching

My friend Ehsan Shahoseini has kindly volunteered to work through project ideas with me. Below you can see our progress.

  • Worked Examples, November 18, 2021
  • Pushforward and Pullback for Sheaves, November 7, 2021
  • Schemes, October 21, 2021
  • Zariski Sheaves, October 7, 2021
  • Relative Spaces, September 23, 2021
  • More on Open Coverings, September 9, 2021
  • Open Coverings of Affine Schemes, September 2, 2021
  • Algebraic Geometry: Broad Perspectives, August 26, 2021
  • Affine Schemes, August 19, 2021