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Many advanced areas of research in mathematics, the Riemann hypothesis, the Taniyama–Shimura conjecture, Green–Tao theorem etc. all have interesting consequences that could be stated using only undergraduate mathematics. Since perfectoid spaces are a hot topic but are out of reach for most of us, I am curious if they have been used to prove anything that an advanced undergraduate could understand.

If not, are there any such consequences of the Langlands program? Though I understand this latter question may be too broad.

user34722
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1 Answers1

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There are some applications to commutative algebra, which I think can be understood by an upper-level undergraduate who has taken a course on commutative rings and modules. You can look at this survey by André for a more in-depth description of applications of perfectoid spaces to commutative algebra.

The following was conjectured by Hochster in 1969:

Direct summand conjecture. Let $R$ be a regular ring, and let $R \to S$ be an extension of rings such that $S$ is finitely generated as a module over $R$. Then, the inclusion $R \to S$ splits as a homomorphism of $R$-modules, i.e., $R$ is a direct summand of $S$.

Hochster proved the case when $R$ contains a field in 1973, and Heitmann proved the case when $\dim R \le 3$ in 2002. The general case was recently settled by André using perfectoid spaces, and Bhatt has also given a shorter proof.

One equivalent formulation is the following:

Monomial conjecture. Let $R$ be a local ring of dimension $d$, and let $x_1,x_2,\ldots,x_d \in R$ be a system of parameters. Then, for every positive integer $t$, we have $$x_1^tx_2^t\cdots x_d^t \notin (x_1^{t+1},x_2^{t+1},\ldots,x_d^{t+1}).$$

One way to think of this latter statement is that an analogous statement for $x_1,x_2,\ldots,x_d$ being a regular sequence is not too hard to show (see one of my answers), hence the conjecture is asking whether being a system of parameters is enough.

Takumi Murayama
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  • Wonderful, thank you! So am I correct in inferring from this answer that there aren't any results significantly simpler than this? Like basic statements about integer or rational equations? – user34722 Oct 01 '18 at 00:01
  • @user34722 I suppose I'm not an expert on perfectoid spaces as a whole (especially its applications to number theory), but this is one of the simpler applications. Maybe someone more knowledgeable can say something about applications to finding rational points, for instance. – Takumi Murayama Oct 01 '18 at 00:11
  • @user34722 I'm definitely not the best person to comment, but I'm not actually aware (due to my ignorance) of an application in finding rational points etc. I can give one example of a toy application to finding rational points, such as showing an isomorphism of l-torsion points of two abelian varities living over a field of char p and a field of char 0 resp. (for l not equal to p). I guess there are more direct ways of proving this though. However, many striking applications of perf. spaces are of the flavor of comparing topos theoretic/cohomological statements in p-adic case to p= 0 case. – S.M Dec 29 '18 at 05:53