Étale algebras


One interesting thing we could do at this point is work out the much more interesting structure theory of separable algebras if we relax some of the standing assumptions such as commutativity. Instead, we’re going to start working in a more general situation, but the idea is to put stronger conditions into our definition to keep the structure close to the same.

Note that since we were working over a field all of our algebras were automatically flat. For this reason, they were all something called étale algebras. Geometrically speaking this means they are smooth and the fibers all have dimension {0}. Another equivalent way to say this is that the structure map is flat and unramified. This condition of an algebra being unramified is what we’ll discuss today.

If you look in Milne’s book Étale Cohomology, you’ll find that there’s a mind-boggling large number of equivalent ways to check a map is étale. The way that algebraic geometers (at least those working with functors) tend to check this condition is almost entirely passed over in the book. There is just one quick mention that this method can be done, so today we’ll write down this condition and next time we’ll work out a neat example using it.

The idea of being unramified can maybe be stated more easily in terms of spaces. A variety {X/k} is formally unramified if for any {Y\rightarrow X} and any infinitesimal thickening, say {\overline{Y}} of {Y}, there is at most one way to extend {Y\rightarrow X} to the thickened scheme {\overline{Y}\rightarrow X}. This is intentionally vague, but now if we think about everything being affine, by the equivalence of categories with algebras we could work out what the exact condition should be. One should be careful about the word “formally” existing everywhere, but since we’ll assume our algebras are again unital, associative, commutative, and finite dimensional this formal condition is completely equivalent to being unramified/étale in the usual sense.

An {A}-algebra {B} is called (formally) unramified if for any sequence of {A}-algebras {0\rightarrow I\rightarrow C'\rightarrow C\rightarrow 0} with {I} nilpotent we have that any {A}-map {B\rightarrow C} has at most one extension through {C'}, i.e. there is at most one composition {B\rightarrow C'\rightarrow C} which yields {B\rightarrow C}. If in addition one can always find such an extension, {B} is called an étale {A}-algebra. One might have learned in Hartshorne that this latter condition is called the infinitesimal lifting criterion and implies the map is smooth. We already talked about this here.

To wrap up today I’ll point out why this is the condition that comes up most for me. Suppose you are given the points of some scheme {h_X: \text{k-alg} \rightarrow Set}, but you don’t know much else about it. Maybe you want to check that {X} is étale. If you try to use one of the standard methods, you may get stuck since those other methods tend to assume you know more about {X} already. Merely from knowing the functor of points you only need to check that it is formally unramified by checking that given the situation above {0\rightarrow I\rightarrow C'\rightarrow C\rightarrow 0} the natural map coming from the functor {h_X(C')\rightarrow h_X(C)} is injective.

Next time we’ll work out one common place where this happens, and then see that checking this map is injective (which will be easy in the example) implies some incredibly strong things about the variety {X}.

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