The Log Lady and the Frobenioid of $\mathbb{Z}$

“Sometimes ideas, like men, jump up and say ‘hello’. They introduce themselves, these ideas, with words. Are they words? These ideas speak so strangely.”

“All that we see in this world is based on someone’s ideas. Some ideas are destructive, some are constructive. Some ideas can arrive in the form of a dream. I can say it again: some ideas arrive in the form of a dream.”

Here’s such an idea.

It all started when Norma wanted to compactify her twisted-prime-fruit pies. Norma’s pies are legendary in Twin Peaks, but if you never ate them at Double R Diner, here’s the concept.

Start with a long rectangular strip of pastry and decorate it vertically with ribbons of fruit, one fruit per prime, say cherry for 2, huckleberry for 3, and so on.

For elegance, I argued, the $p$-th ribbon should have width $log(p)$.

“That may very well look natural to you,” she said, “but our Geometer disagrees”. It seems that geometers don’t like logs.

Whatever. I won.

That’s Norma’s basic pie, or the $1$-pie as we call it. Next, she performs $n$ strange twists in one direction and $m$ magical operations in another, to get one of her twisted-pies. In this case we would order it as her $\frac{m}{n}$-pie.

Marketing-wise, these pies are problematic. They are infinite in length, so Norma can serve only a finite portion, limiting the number of fruits you can taste.

That’s why Norma wants to compactify her pies, so that you can hold the entire pastry in your hands, and taste the infinite richness of our local fruits.

“Wait!”, our Geometer warned, “You can never close them up with ordinary scheme-dough, the laws of scheme-pastry prohibit this!” He suggested to use a ribbon of marzipan, instead.

“Fine, then… Margaret, before you start complaining again, how much marzipan should I use?”, Norma asked.

“Well,” I replied, “ideally you’d want it to have zero width, but that wouldn’t close anything. So, I’d go for the next best thing, the log being zero. Take a marzipan-ribbon of width $1$.”

The Geometer took a $1$-pie, closed it with marzipan of width $1$, looked at the pastry from every possible angle, and nodded slowly.

“Yes, that’s a perfectly reasonable trivial bundle, or structure sheaf if you want. I’d sell it as ${\mathcal{O}}_{\overline{\mathbf{Spec}(\mathbb{Z})}}$ if I were you.”

“In your dreams!  I’ll simply call this  a $1$-pastry, and an $\frac{m}{n}$-pie closed with a $1$-ribbon of marzipan can be ordered from now on as an $\frac{m}{n}$-pastry.”

“I’m afraid this will not suffice,” our Geometer objected, ” you will have to allow pastries having an arbitrary marzipan-width.”

“Huh? You want me to compactify an $\frac{m}{n}$-pie  with marzipan of every imaginable width $r$ and produce a whole collection of … what … $(\frac{m}{n},r)$-pastries? What on earth for??”

“Well, take an $\frac{m}{n}$-pastry and try to unravel it.”

Oh, here we go again, I feared.

Whereas Norma’s pies all looked and tasted quite different to most of us, the Geometer claimed they were all the same, or ‘isomorphic’ as he pompously declared.

“Just reverse the operations Norma performed and you’ll end up with a $1$-pie”, he argued.

So Norma took an arbitrary $\frac{m}{n}$-pastry and did perform the reverse operations, which was a lot more difficult that with pies as now the marzipan-bit produced friction. The end-result was a $1$-pie held together with a marzipan-ribbon of width strictly larger or strictly smaller than $1$, but never gave back the $1$-pastry. Strange!

“Besides”, the Geometer added, “if you take two of your pastries, which I prefer to call $\mathcal{L}$ and $\mathcal{M}$, rather than use your numerical system, then their product $\mathcal{L}\otimes \mathcal{M}$ is again a pastry, though with variable marzipan-width.

In the promotional stage it might be nice to give the product for free to anyone ordering two pastries.”

“And how should I produce such a product-pastry?”

“Well, I’m too lazy to compute such things, it must follow trivially from elementary results in Picard-pastry. Surely, our log lady will work out the details in your notation. No doubt it will involve lots of logs…”

And so I did the calculations in my dreams, and I wrote down all formulas in the Double R Diner log-book, for Norma to consult whenever a customer ordered a product, or power of pastries.

A few years ago we had a Japanese tourist visiting Twin Peaks. He set up office in the Double R Diner, consulted my formulas, observed Norma’s pastry production and had endless conversations with our Geometer.

I’m told he categorified Norma’s pastry-bizness, probably to clone the concept to the Japanese market, replacing pastries by sushi-rolls.

When he left, he thanked me for working out the most trivial of examples, that of the Frobenioid of $\mathbb{Z}$…

Added december 2015:

I wrote this little story some time ago.

The last couple of days this blog gets some renewed interest in the aftermath of the IUTT-Mochizuki-Fest in Oxford last week.

I thought it might be fun to include it, if only in order to decrease the bounce rate.

If you are at all interested in the maths, you may want to start with this google+ post, and work your way back using the links curated by David Roberts here.