The combinatorial-motivic nature of \(\mathbb F_1\)-schemes (Q2827346)

The author describes several attempts to define schemes ``defined over \(\mathbb{F}_1\)'' in eight sections. Section one addresses the quetion whether there exists a large site containing Grothendieck schemes, in which one can define ``absolute Descartes powers'' \(\mathrm{Spec}\mathbb{Z}\times_{\Upsilon}\times\cdots\times_{\Upsilon}\mathrm{Spec}\mathbb{Z}\) over some deeper base ``\(\Upsilon\)'' than \(\mathbb{Z}\) so that \(\mathrm{Spec}\mathbb{Z}\times_{\Upsilon}\mathrm{Spec}\mathbb{Z}\) becomes a surface.NEWLINENEWLINEIn Section two the author explains Deitmar's ``monoidal scheme theory'' in some detail, and in Section three he describes some fundamental examples, like affine and projective spaces. In Section four he introduces the notion of a loose graph and shows how one can construct projective completion from the loose graphs. Section five presents his own theory of \(\Upsilon\)-schemes, which is another approach to \(\mathbb{F}_1\) schemes. In Section six and seven he reviews several attempts to define absolute motives and their absolute zeta functions. In the final section eight he describes the approach of \textit{A. Connes} and \textit{C. Consani} [J. Number Theory 131, No. 2, 159--194 (2011; Zbl 1221.14002)] to understand the adèle class space through hyperring extension theory, in which a connection is revealed with certain group actions on projective spaces.NEWLINENEWLINEFor the entire collection see [Zbl 1343.14002].