Hard implications that become easy with the right intermediate step

Question

I’m interested in examples of theorems of the form “If $P$ then $Q$“ that were either unsolved or thought to require difficult arguments until someone came up with an $X$ for which “If $P$ then $X$” and “If $X$ then $Q$” are significantly easier to prove.

Answer 1

This is an example of the special case: an inclusion $P \subseteq Q$ which seems difficult or impossible to prove, until someone comes up with an $X$, with the property that $P \subseteq X$ and $X \subseteq Q$ are relatively easy to prove.

Let $Z$ be a finite set of points in projective space $\mathbb{P}^n$, let $I$ be the homogenous ideal of $Z$, and let $m$ be a positive integer. Let $P$ be the ideal of homogeneous forms that vanish to order $mn$ at each point of $Z$. This is the $mn$’th symbolic power of $I$, denoted $I^{(mn)}$. And let $Q$ be the $m$’th ordinary power, $I^m$. Then $P \subseteq Q$, or $I^{(mn)} \subseteq I^m$.

This is not obvious. Say $F$ vanishes to order $4$ at each point of $Z$. Then $F$ can be written as a sum of products $G_i H_i$, where each $G_i,H_i$ vanishes at each point of $Z$. Why? It’s not easy to see, and this is just the $m=2$ case.

This was proved by Ein-Lazarsfeld-Smith [1] using $X=$ an asymptotic multiplier ideal, which was introduced in this paper (they introduce asymptotic multiplier ideals and give several applications, one of which is the theorem $P \subseteq Q$), and simultaneously proved by Hochster-Huneke [2] using $X=$ a tight closure ideal. The theorems are more general than I’ve stated, e.g., for ELS, $Z$ can be a reduced scheme on a smooth variety; HH work in an arbitrary reduced Noetherian ring. Yet, I don’t believe there is, to this day, any “elementary” proof known, even in the case of points in the projective plane $\mathbb{P}^2$, in characteristic zero. Bocci-Harbourne proved it by “elementary” means when $Z$ is a general set of points in $\mathbb{P}^2$, but not for arbitrary $Z$.

I might be wrong. Since ELS and HH in around 2001 there has been a ton of work on this topic, under the heading of “containment problems” or more specifically, the problem of containments of symbolic powers. Lots of generalizations, strengthenings, special cases, etc. But if anyone found a proof of the original, basic result $P \subseteq Q$, without going through some “non-elementary” $X$ like an asymptotic multiplier ideal or test ideal (only using “classical” plane geometry), I missed the news.

Answer 2

Three examples come to mind, all with $P=ZF$ or some other base theory.

1) Fermat / Heath-Brown / Zagier:

• $Q$ = "every prime of the form $4n+1$ is the sum of two squares";
• $X$ = "the map $$(x,y,z)\mapsto \begin{cases} (x+2z,~z,~y-x-z),\quad \textrm{if}\,\,\, x < y-z \\ (2y-x,~y,~x-y+z),\quad \textrm{if}\,\,\, y-z < x < 2y\\ (x-2y,~x-y+z,~y),\quad \textrm{if}\,\,\, x > 2y \end{cases} $$ is an involution with an odd number of fixed points on the set $\{(x,y,z)\in\mathbb{N}^3:x^2+4yz=p\}$.

2) Hadamard / de la Vallee Poussin / Newman / Zagier:

• $Q$ = the prime number theorem;
• $X_1,\ldots,X_7$ = Zagier's statements $I-VI$ and the Analytic Theorem.

3) Lebesgue / Caratheodory:

• $Q$ = the Lebesgue-measurable sets are closed under countable unions;
• $X$ = the Caratheodory-measurable sets are closed under countable unions.

I think giving the statements $X$ and asking for proofs could be a reasonable homework or group project for a graduate class in Number Theory, Complex Analysis, or Real Analysis, which is saying something given the significance of the $Q$.