Vector bundles over projective spaces. The case {\mathbb F_1}
TL;DR: In this paper, it was shown that all line bundles are tensor powers of the Hopf bundle and all vector bundles are direct sums of line bundles over n-dimensional projective spaces.
Abstract: Over the field of one element, vector bundles over n-dimensional projective spaces are considered. It is shown that all line bundles are tensor powers of the Hopf bundle and all vector bundles are direct sums of line bundles. This is in complete analogy to the case of the projective line over an arbitrary classical field, but drastically simpler in comparison with projective spaces of higher dimensions.
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TL;DR: In this article, the Picard group of a monoid scheme and the class group of normal monoid schemes were studied and an ideal theory for pointed abelian noetherian monoids, including primary decomposition and discrete valuations, was developed.
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01 Jan 2013
TL;DR: The history of F1-geometry can be traced from the first mention by Tits in 1956 until the present day as mentioned in this paper, and the main ideas around F1, embedded into the historical context, and the connections to other approaches towards F1 like monoidal schemes after Deit- mar, B1-algebras after Lescot, -schemes after Borger, relative schemes after Toen and Vaqui e, log schemes after Kato and congruence schemes after Berkovich and Deitmar.
Abstract: This overview paper has two parts. In the rst part, we review the develop- ment of F1-geometry from the rst mentioning by Jacques Tits in 1956 until the present day. We explain the main ideas aroundF1, embedded into the historical context, and give an impression of the multiple connections of F1-geometry to other areas of mathematics. In the second part, we review (and preview) the geometry of blueprints. Beyond the basic denitions of blueprints, blue schemes and projective geometry, this includes a theory of Chevalley groups over F1 together with their action on buildings over F1; com- putations of the Euler characteristic in terms of F1-rational points, which involve quiver Grassmannians; K-theory of blue schemes that reproduces the formula Ki(F1) = st (S 0 ); models of the compactications of Spec Z and other arithmetic curves; and explanations about the connections to other approaches towards F1 like monoidal schemes after Deit- mar, B1-algebras after Lescot, -schemes after Borger, relative schemes after Toen and Vaqui e, log schemes after Kato and congruence schemes after Berkovich and Deitmar.
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TL;DR: In this paper, the authors studied the cohomology theory of monoid schemes in general and applied it to vector and line bundles, and showed that any vector bundle is a coproduct of line bundles and that the Pic functor respects finite products.
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TL;DR: For each n ≥ 1, the authors considers a one-parameter family of complex-valued measures on the symmetric group S n, depending on a complex parameter z and shows that they have an interesting internal structure having a representation theoretic interpretation.
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TL;DR: In this paper, the authors studied the cohomology theory of monoid schemes in general and applied it to vector and line bundles, and showed that any vector bundle is a coproduct of line bundles.
Abstract: The aim of this paper is to study the cohomology theory of monoid schemes in general and apply it to vector and line bundles. We will prove that over separated monoid schemes, any vector bundle is a coproduct of line bundles and then go on to study the line bundles in more detail. Amongst other things, we prove that over separated monoid schemes, ${\sf Pic}$ respects finite products.
Next we will introduce the notion of $s$-cancellative monoids. They are monoids for which $ax=ay$ implies that $(xy)^nx=(xy)^ny, n\in\mathbb{N}$. This class is important since it is the biggest class of monoids for which $M^*_{\mathfrak{p}}$ maps injectively into its group of fractions for every prime ideal ${\mathfrak{p}}$. As we will see in section 6, this will enable us to embed $\mathcal{O}^*_X$ injectively in a constant sheaf provided $X$ is locally $s$-cancellative. We develop the theory of $s$-divisors and we prove that for an $s$-cancellative monoid scheme $X$, the group ${\sf Pic}(X)$ can be described in terms of $s$-divisors. For cancellative monoid schemes, $s$-divisors agree with the Cartier divisors.
We then introduce the notion of $s$-smooth monoid schemes, which generalise smooth monoids schemes, and prove that for them $H^i(X,\mathcal{O}^*_X)=0$ for all $i\geq 2$. Furthermore we show that it is a local property and respects finite products.
Finally we investigate the relationship between line bundles over a monoid scheme $X$ and over its geometric realisation $X_k$, where $k$ is a commutative ring. We prove that if $k$ is an integral domain (resp. principal ideal domain) and $X$ is a cancellative and torsion free (resp. seminormal and torsion-free) monoid scheme, then the induced map ${\sf Pic}(X)\to {\sf Pic}(X_k)$ is a monomorphism (resp. isomorphism).
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TL;DR: In this paper, the authors use techniques from relative algebraic geometry and homotopical geometry in order to construct several categories of schemes defined under Spec Z, where from a very intuitive point of view N is the semi-ring of natural numbers.
Abstract: We use techniques from relative algebraic geometry and homotopical algebraic geometry in order to construct several categories of schemes defined "under Spec Z". We define this way the categories of N-schemes, F_1-schemes, S-schemes, S_+-schemes, and S_1-schemes, where from a very intuitive point of view N is the semi-ring of natural numbers, F_1 is the field with one element, S is the sphere ring spectrum, S_+ is the semi-ring spectrum of natural numbers and S_1 is the ring spectrum with one element. These categories of schemes are related by several base change functors, and they all possess a base change functor to Z-schemes (in the usual sense). Finally, we show how the linear group Gl_n and toric varieties can be defined as objects in certain of these categories.
124 citations
TL;DR: In this paper, it was shown that toric varieties can be defined over the field with one element and a motivic interpretation of the image of the J-homomorphism defined by Adams was given.
Abstract: We propose a definition of varieties over the field with one element. These have extensions of scalars to the ring of integers which are varieties in the usual sense. We show that toric varieties can be defined over the field with one element. We also discuss zeta functions for such objects. We give a motivic interpretation of the image of the J-homomorphism defined by Adams. ~ ~ ~ ~
112 citations
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TL;DR: In this paper, a completely algebraic approach to Arakelov geometry is proposed, which does not require the variety under consideration to be generically smooth or projective, and it is shown that the existence of algebraic varieties over Q is shown, and general results are applied to such models.
Abstract: This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective In order to construct such an approach we develop a theory of generalized rings and schemes, which include classical rings and schemes together with "exotic" objects such as F_1 ("field with one element"), Z_\infty ("real integers"), T (tropical numbers) etc, thus providing a systematic way of studying such objects
This theory of generalized rings and schemes is developed up to construction of algebraic K-theory, intersection theory and Chern classes Then existence of Arakelov models of algebraic varieties over Q is shown, and our general results are applied to such models
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TL;DR: In this paper, it was shown that the algebra and the endomotive of the quantum statistical mechanical system of Bost-Connes naturally arise by extension of scalars from the "field with one element" to rational numbers.
57 citations