\(\mathbb{F}_1\)-geometry, schemes and \(\mathbb{T}\)-modules

Matthew Taylor

Frank Adams 2,

Sometimes quite elementary objects can fit into a fascinating framework; so it is with \(\mathbb{T}\)-modules, which I've spent most of my last year working with.
 
In 1957, Jacques Tits conjectured the existence of a "field with one element", \(\mathbb{F}_1\). There's a problem here - fields need to have at least two elements. Two years ago, Jeff and Noah Giansiracusa were trying to figure out a way to simplify scheme theory by using tropical maths. In the end, the way they did it was to provide a general construction of an \(\mathbb{F}_1\)-scheme which also worked for semirings.
 
So - what do we mean by \(\mathbb{F}_1\) if it can't exist? What's an affine scheme over the tropical semiring? How do \(\mathbb{T}\)-modules fit into all this? I'll attempt to answer these questions - and more - from as close to first principles as I can.
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