Arguments on translation of affine space.

Question

Let $k$ be an algebraically closed field. I have found in some algebraic geometry books arguments such as "by translation we may suppose that every maximal ideal $\mathcal{m} = \langle x_1-a_1, \ldots, x_n-a_n \rangle$ is of the form $\mathcal{m} = \langle x_1, \ldots, x_n \rangle$". My question is, what are all the transformations that we usually can use over $\mathbb{A}^n$ to simplify arguments? What does it mean formally to have such an invariance? Is there any book where this arguments are treated?

Answer 1

This is an aspect of what you might call "the yoga of isomorphisms": the point is that problems can always be transformed into isomorphic problems, and here the isomorphism is translation. I'm not aware of any text where this sort of thing is formally discussed; it's a basic meta-pattern you pick up on as you go along.

Slightly more formally, you have some problem, and your problem concerns the behavior of some objects, in this case the maximal ideals $m$ of $k[x_1, \dots x_n]$. You can ask yourself, "which of these objects are isomorphic" (in the sense appropriate to your problem)? In this case two maximal ideals will behave in the same way if one is the image of the other under some automorphism of $k[x_1, \dots x_n]$ as a $k$-algebra, and the translations are the simplest such automorphisms. Mathematicians are implicitly doing this kind of thing all the time without commenting too explicitly on it; the phrase to look for is "without loss of generality" or "WLOG".

In this case the action of the automorphisms is transitive so to answer a question for every maximal ideal it suffices to answer it for any maximal ideal. In general the action of the automorphism group has some orbits and it suffices to answer the question for an element of each orbit.

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A more general version of this strategy is what you might call "the yoga of reductions," where you ask for something weaker than isomorphisms, as follows. Suppose you want to prove some property for all objects of a certain class, and you can prove "reductions," meaning you can find pairs $(P, Q)$ such that if $P$ has the property then so does $Q$. In this case we say that the problem for $Q$ has been reduced to the problem for $P$ (note that reduction goes the opposite direction from the proof of the implication!). Then it suffices to prove the property for a collection of objects $P_i$ such that every object $Q$ can be reduced by some sequence of reductions to some $P_i$.

(Isomorphisms are the simplest reductions.)

This more general strategy is also very common in algebraic geometry (and other fields), where we often reduce to the case of local rings, or finitely generated modules, etc. Note that as stated this idea is so general that it includes the principle of mathematical induction, where we reduce a problem for $n+1$ to the same problem for $n$ and so we can inductively reduce all the way to the base case $1$.