# what is a "dévissage" argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the inseparable case. In between the two cases he writes "a straightforward dévissage argument reduces one to the case of an inseparable extension.

What is dévissage argument? I saw in wikipedia something with the same name. Is there a connection?

If it matters, the proposition statement is as follows: Let $K$ be a complete field with respect to a discrete valuation $\nu$ with valuation ring $A$, and let $L/K$ be a finite extension. Let $B$ be the integral closure of $A$. Then $B$ is a discrete valuation ring and is a free $A$-module of rank $[L:K]$; also, $L$ is complete in the topology defined by $B$.

(So in this case the reduction really is easy, as one can break the extension into a separable part and an inseparable part. It is clear that if the statement holds for $L/E$ and for $E/K$ then it holds for $L/K$)

• Are there any better tags? Jul 3, 2013 at 13:39
• Hi I added the terminology tag, that seemed the most appropriate. Jul 3, 2013 at 14:33

The term is used mainly in Grothendieck-style algebraic geometry but has a more general connotation.

Most generally, it means reduction of a problem to a simpler case by some sort of standard rigamarole that is used for many similar reductions of similar problems. For example, you could decompose a problem on field extensions into separable/inseparable parts, or algebraic and transcendental extensions of the prime field, or an induction on the transcendence degree. The simpler case could be very difficult and indeed the term is often used in a way that suggests the reduction is the easy step that pares the problem down to a smaller hard core.

Most typically, it is applied to inductive reduction of a problem on $n$ dimensional algebraic varieties (Noetherian schemes) to a $1$ dimensional version.

Most specifically, there are particular theorems such as the ones at the Wikipedia link that are known by the word devissage and are often used to effectuate the reductions.

• To make sure I got it right, is proving some identity of multiplicative functions by reducing to checking it on prime powers a good example of a devissage argument? Jul 3, 2013 at 15:20
• That is probably too simple for people to give a name to the argument, but it is exactly that kind of idea. I edited the answer a bit to better capture the flavor of how the word is used.
– zyx
Jul 3, 2013 at 15:25