Schemes over $F_1$

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Using the approach of Kurokawa, Ochiai, and Wakayama to 'absolute mathematics' we define a corresponding notion of schemes.

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Algebraic Geometry Over Hyperrings (2015, background): “This motivated Connes and Consani to search possible other candidates rather than monoids and the study of algebraic geometry over more general algebraic objects by many others (cf. [3], [13], [28], [35], [37]).”
Lambda-rings and the field with one element (2009, unknown): “For these approaches see the following references: Baez [3], Connes–Consani [13], Connes–Consani–Marcoli [14], Deitmar [19][18][20], Diers [26], Durov [27], Ha-ran [36] [35], Kapranov [38], Kapranov–Smirnov [39], Kurokawa [42][43][44], Kurokawa– Ochiai–Wakayama [45], Smirnov [55], Tate–Voloch [59],…”