The n-Category Café

First: I’ve learned a lot about separable algebras over fields, but I need to understand them over commutative rings — especially the integers.

A separable algebra $A$ over the commutative ring $k$ is one such that $A$ is projective as an $A \otimes_k A^{op}$ module. There are also some equivalent characterizations in the nLab article. But I want to know what finite separable $k$-algebras are actually like when $k$ is the integers. Of course an algebra over the integers is usually called a ring, so I might as well talk about “separable rings”.

I know I should read this:

• F. DeMeyer and E. Ingraham, Separable Algebras over Commutative Rings, Lecture Notes in Mathematics 181, Springer, Berlin, 1971.

… but I’m having trouble getting ahold of it right now. And I feel like chatting with you. So, I’ll just throw out this question:

Question: is a finite separable commutative ring the same as a finite product of finite fields?

Second, Lieven le Bruyn said that “in categorical terms, studying the monoidal category of commutative separable $k$-algebras is the same as studying the étale site of $k$”.

I would like to understand this comment better. I know I should read this:

• Milne, Étale Cohomology, Sections 1.3 and 2.1 (esp. Thm. 1.9), Princeton U. Press, Princeton, 1980.

… but again I’m having trouble getting ahold of it, and I think it might be more to just chat a bit about these ideas.

Here’s my vague guess. When we’re working over a commutative ring $k$, a commutative separable $k$-algebra acts like ‘the algebra of functions on a finite 0-dimensional space’. For example, when $k$ is the complex numbers, any commutative separable $k$-algebra is just the algebra of complex functions on a finite set.

In topology, when we have a finite covering space of a space, the fibers are finite sets. So, when we go over to algebraic geometry, commutative separable $k$-algebras play the role of ‘fibers of covering spaces’. But instead of saying ‘covering space’, we say ‘étale map of schemes’.

So, there’s a relation between the étale topology and commutative separable $k$-algebras. And I’m hoping that in the case where $k$ is the integers, these algebras are just finite products of finite fields. After all, a finite field acts like a ‘point’, so a finite product of finite fields should act like a ‘finite 0-dimensional space’.