By Koen Thas

It's been recognized for a while that geometries over finite fields, their automorphism teams and sure counting formulae concerning those geometries have fascinating guises whilst one we could the scale of the sector visit 1. nevertheless, the nonexistent box with one aspect, F1

, provides itself as a ghost candidate for an absolute foundation in Algebraic Geometry to accomplish the Deninger–Manin application, which goals at fixing the classical Riemann Hypothesis.

This publication, that is the 1st of its variety within the F1

-world, covers a number of components in F1

-theory, and is split into 4 major elements – Combinatorial idea, Homological Algebra, Algebraic Geometry and Absolute Arithmetic.

Topics handled comprise the combinatorial thought and geometry in the back of F1

, specific foundations, the mix of other scheme theories over F1

which are almost immediately on hand, factors and zeta features, the Habiro topology, Witt vectors and overall positivity, moduli operads, and on the finish, even a few arithmetic.

Each bankruptcy is thoroughly written by way of specialists, and along with elaborating on recognized effects, fresh effects, open difficulties and conjectures also are met alongside the way.

The variety of the contents, including the secret surrounding the sector with one aspect, should still allure any mathematician, despite speciality.

Keywords: the sector with one aspect, F1

-geometry, combinatorial F1-geometry, non-additive type, Deitmar scheme, graph, monoid, rationale, zeta functionality, automorphism crew, blueprint, Euler attribute, K-theory, Grassmannian, Witt ring, noncommutative geometry, Witt vector, overall positivity, moduli area of curves, operad, torificiation, Absolute mathematics, counting functionality, Weil conjectures, Riemann speculation

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**Extra info for Absolute Arithmetic and F1-geometry**

**Example text**

3 Base change . . . . . . . . . . . . . 4 General sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 . 39 . 40 . 48 . 53 . 54 . 62 . 65 . 67 . . . . . . 70 . 70 . 72 . 75 . 76 References . . . . . . . . . . . . . . . . . . . 77 Index . . . . . . . . . . . . . . . . . . . . 79 1. 1. Introduction. The ultimate goal of F1 -geometry is to extend the classical correspondence between function fields and number fields so as to allow transfer of algebro-geometric methods to the number field case and thus make it possible to attack deep number theoretical problems.

1. Introduction. The ultimate goal of F1 -geometry is to extend the classical correspondence between function fields and number fields so as to allow transfer of algebro-geometric methods to the number field case and thus make it possible to attack deep number theoretical problems. These methods include cohomology theories of various flavors. There are many different approaches to F1 -geometry, see [1], [2], [4], [5], [10], [13], [18], [19], [21], but their common core seems to be the “non-additive geometry” as defined in [5], [6], [7] or [22], which is a version of algebraic geometry not based on rings, but on monoids.

A1 We work up to point-line duality: this is why we are allowed to ask, without loss of generality, that lines have at most two points. We do not ask that they have precisely two points, one motivation being for instance (combinatorial) affine spaces over F1 , in which any line has precisely one point. Similarly, their scheme-theoretic versions have precisely one closed point: we refer the reader to later chapters for a formal definition. CL Referring to the preceding remark, note that, later on, F1 -geometries with precisely 2 points per line will correspond to closed subschemes (in the setting of [17]) of the appropriate ambient projective F1 -space, seen as a scheme.