October 21st : Dear Diary,
today’s seminar was fun, though a bit unconventional. My goal was to explain faithfully flat descent, but at the last moment i had this urge to let students discover the main idea themselves (in the easiest of examples) by means of a thought experiment :
“I am an alien (laughter…), and a very stubborn alien at that. There’s just one field, the complex numbers $\mathbb{C}$, and all rings are $\mathbb{C}$-algebras. I’ve heard strange rumours that you humans believe in a geometry ‘hidden under $\mathbb{C}$’, something called real manifolds. What then is an algebra over this obviously virtual ‘real’ field?”
Their first hurdle was to convey the concept of complex conjugation as the alien(me) was unwilling to decompose a complex number $c$ into two ‘ghost components’ $a+bi$. Still i had to concede that i knew about addition and multiplication, i had a $1$ and a square root of $-1$, which for some reason they preferred to call $i$.
‘Oh, but then you know about $\mathbb{Z}[i]$! You just add a number of times $1$’s with $i$’s.’
‘Why are you humans so obsessed with counting? We do not count! We can’t! We have neither fingers nor toes!’
Admittedly a fairly drastic intervention, but i had to keep them on the path leading to Galois descent… After a while we agreed on a map (they called it conjugation) sending sums to sums, products to products and taking a root of unity to its inverse.
Next, they asked me to be a bit flexible and allow for ‘generalized’ fields such as the one consisting of all elements fixed under conjugation! Clearly, the alien refused : ‘We’re not going on that slippery road called generalization, we’ve seen the havock caused by it in human-mathematics.’
It took them a while to realize they would never be able to sell me an $\mathbb{R}$-algebra $A$, but perhaps they could try to sell me the complex algebra $B= A \otimes_{\mathbb{R}} \mathbb{C}$?
Alien : ‘But, how do i recognize one of your algebras among mine? Is there a test to detect them?’
Humans : ‘Yes, they have a map (which we know to be the map $a \otimes c \mapsto a \otimes \overline{c}$, but you cannot see it) sending sums to sums, products to products that extends the conjugation on $\mathbb{C}$.’
Alien : ‘But if i take a basis for any of my algebras and apply conjugation to all its coordinates, then surely all my algebras have this property, not?’
Humans : ‘No, such maps are good for sums, but not always for products. For example, take $\mathbb{C}[x]/(x^2-c)$ for $c$ a complex-number not fixed under conjugation.’
Alien : ‘Point taken. But then, your algebras are just a subclass of my algebras, right?’
Humans : ‘No! An algebra can have several of such additional maps. For example, take $B = \mathbb{C} \times \mathbb{C}$ then there is one sending $(a,b)$ to $(\overline{a},\overline{b})$ and another sending it to $(\overline{b},\overline{a})$. (because we know there are two distinct real algebras $\mathbb{R} \times \mathbb{R}$ and $\mathbb{C}$ of dimension two, tensoring both to $\mathbb{C} \times \mathbb{C}$.)’
By now, the alien and humans agreed on a dictionary : what to humans is the $\mathbb{R}$-algebra $A$ is to the alien the complex algebra $B=A \otimes \mathbb{C}$ together with a map $\gamma_B : B \rightarrow B$ sending sums to sums, products to products and extending conjugation on $\mathbb{C}$ (this extra structure, the map $\gamma_B$, is called the ‘descent data’).
A human-observed $\mathbb{R}$-algebra morphism $\phi : A \rightarrow A’$ is to the alien the $\mathbb{C}$-algebra morphism $\Phi = \phi \otimes id_{\mathbb{C}} : B \rightarrow B’$ which commutes with the extra structures, that is, $\Phi \circ \gamma_B = \gamma_{B’} \circ \Phi$.
Phrased differently (the alien didn’t want to hear any of this) : there is an equivalence of categories between the category $\mathbb{R}-\mathsf{algebras}$ of commutative $\mathbb{R}$-algebras and the category $\gamma-\mathsf{algebras}$ consisting of complex commutative algebras $B$ together with a ringmorphism $\gamma_B$ extending complex conjugation and with morphisms $\mathbb{C}$-algebra morphisms compatible with the $\gamma$-structure.
Further, what to humans is the base-extension (or tensor) functor
$- \otimes_{\mathbb{R}} \mathbb{C}~:~\mathbb{R}-\mathsf{algebras} \rightarrow \mathbb{C}-\mathsf{algebras}$
is (modulo the above equivalence) to the alien merely the forgetful functor
$\mathsf{Forget}~:~\gamma-\mathsf{algebras} \rightarrow \mathbb{C}-\mathsf{algebras}$
stripping off the descent-data.
After the break (yes, it took us that long to get here) we used this idea to properly define obviously non-existing rings living ‘under $\mathbb{Z}$’, or if you like silly terminology, algebras over the field with one element $\mathbb{F}_1$.
Alien : ‘Ha-ha-ha, a field with one element? Surely you’re joking Mr. Human’
Note to self : Dare to waste time like this in a seminar.
