From the introduction: J. Tits [in: Centre Belge Rech. math. Colloque d’algèbre supérieure, Bruxelles du 19 au 22 déc. 1956, 261–289 (1957; Zbl 0084.15902)] wondered if there would exist a “field of one element” \(\mathbb F_1\) such that for a Chevalley group \(G\) one has \(G (\mathbb F_1)=W\), the Weyl group of \(G\). Recall the Weyl group is defined as \(W=N(T)/Z(T)\) where \(T\) is a maximal torus, \(N(T)\) and \(Z(T)\) are the normalizer and the centralizer of \(T\) in \(G\). He then showed that one would be able to define a finite geometry with \(W\) as automorphism group.
In this paper we will extend the approach of N. Kurokawa, H. Ochiai, and M. Wakayama [Doc. Math., J. DMV Extra Vol. 565–584 (2003; Zbl 1101.11325)] to “absolute mathematics” to define schemes over the field of one element. The basic idea of the approach of [loc. cit.] is that objects over \(\mathbb Z\) have a notion of \(\mathbb Z\)-linearity, i.e., additivity, and that the forgetful functor to \(\mathbb F_1\)-objects therefore must forget about additive structures. A ring \(R\) for example is mapped to the multiplicative monoid \((R,\times)\). On the other hand, the theory also requires a “going up” or base extension functor from objects over \(\mathbb F_1\) to objects over \(\mathbb Z\). Using the analogy of the finite extensions \(\mathbb F_{1^n}\) as in C. Soulé [Mosc. Math. J. 4, 217–244 (2004; Zbl 1103.14003)], we are led to define the base extension of a monoid \(A\) as \[ A\otimes_{\mathbb F_1}\mathbb Z:=\mathbb Z[A], \] where \(\mathbb Z [A]\) is the monoidal ring which is defined in the same way as a group ring. Based on these two constructions, here we lay the foundations of a theory of schemes over \(\mathbb F_1\).
For the entire collection see [Zbl 1078.11002].