Are you dreading the social responsibilities that come with this time of the year? You can spice up the conversations with your relatives and friends using the following fun facts in algebraic geometry. Whenever a conversational lull comes up, drop one of these and things will turn fun fun fun again!

Absolute flatification

Assuming your friends know what flatness is, you can amaze them using the absolute flatification of a ring. The reference for all this is Greg Stevenson's paper Derived categories of absolutely flat rings, published in Homology Homotopy Applications.

We say a ring $A$ is absolutely flat (or von Neumann regular) if for every $a\in A$ there exists an $x\in A$ such that $a=a^2x$. This element $x$ is unique with the property that $x=x^2a$, and deserves to be called the weak inverse.

You can now easily prove that for such a beast the Zariski spectrum has the following properties:

  • it is zero-dimensional
  • it is Hausdorff
  • it is totally disconnected
  • the closed sets are precisely the quasicompact subspaces
  • the open sets are precisely the Thomason subsets
  • it has infinitely many points
  • it is not noetherian

And now comes the baffling fact. To any ring you can associate its absolute flatification, which forms the left adjoint to the inclusion of the full subcategory of absolutely flat rings. The unit of the adjunction proves us with a morphism $\mathop{\rm Spec} A^{\rm abs}\to\mathop{\rm Spec} A$, that:

  • is a bijection
  • gives a homeomorphism between the Zariski topology on $\mathop{\rm Spec}A^{\rm abs}$ (recall its weird properties stated above) and the constructible topology on $\mathop{\rm Spec}A$!

Also, the absolute flatification of a ring is easily constructed by formally adding all these weak inverses (as is common for constructing left adjoints).

Also, fear not if you think that the Stacks project can't give you more facts about absolutely flat rings! It's just that a different term is used and different results are proven: a morphism of rings is absolutely flat or weakly étale if the map and the diagonal are both flat. And this notion is very important when developing the tools of the pro-étale topology on schemes, but that is something that I might discuss later on.