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Oliver Lorscheid

Publications and source records attributed to Oliver Lorscheid.

At least 37 records · Page 2Linked to original sources
arXiv2021

Lift theorems for representations of matroids over pastures

Pastures are a class of field-like algebraic objects which include both partial fields hyperfields and have nice categorical properties. We prove several lift theorems for representations of matroids over pastures, including a generalization of Pendavingh and van Zwam's Lift Theorem for partial fields. By embedding the earlier theory into a more general framework, we are able to establish new results even in the case of lifts of partial fields, for example the conjecture of Pendavingh--van Zwam that their lift construction is idempotent. We give numerous applications to matroid representations, e.g. we show that, up to projective equivalence, every pair consisting of a hexagonal representation and an orientation lifts uniquely to a near-regular representation. The proofs are different from the arguments used by Pendavingh and van Zwam, relying instead on a result of Gelfand-Rybnikov-Stone inspired by Tutte's homotopy theorem.

math.CO↗
arXiv2021

The moduli space of matroids

In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set $E$, the functor taking a pasture $F$ to the set of isomorphism classes of rank-$r$ $F$-matroids on $E$ is representable by an ordered blue scheme $Mat(r,E)$, the moduli space of rank-$r$ matroids on $E$. In the third part, we draw conclusions on matroid theory. A classical rank-$r$ matroid $M$ on $E$ corresponds to a $\mathbb{K}$-valued point of $Mat(r,E)$ where $\mathbb{K}$ is the Krasner hyperfield. Such a point defines a residue pasture $k_M$, which we call the universal pasture of $M$. We show that for every pasture $F$, morphisms $k_M\to F$ are canonically in bijection with $F$-matroid structures on $M$. An analogous weak universal pasture $k_M^w$ classifies weak $F$-matroid structures on $M$. The unit group of $k_M^w$ can be canonically identified with the Tutte group of $M$. We call the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios' the foundation of $M$,. It parametrizes rescaling classes of weak $F$-matroid structures on $M$, and its unit group is coincides with the inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if its foundation is the regular partial field, and a non-regular matroid $M$ is binary if and only if its foundation is the field with two elements. This yields a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.

math.AG↗
arXiv2020

Group completion in the K-theory and Grothendieck-Witt theory of proto-exact categories

We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories, with a particular focus on classes of examples of $\mathbb{F}_1$-linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the $K$-theory or Grothendieck-Witt theories of proto-exact categories defined using the (hermitian) $Q$-construction and group completion.

math.KT↗
arXiv2020

Algebraic K-theory and Grothendieck-Witt theory of monoid schemes

We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $\operatorname{Pic}(X)$ and pointed monoid of regular functions $Γ(X, \mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $\operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.

math.KT↗
arXiv2020

Foundations of matroids I: Matroids without large uniform minors

The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and hyperfields. Using deep results due to Tutte, Dress-Wenzel, and Gelfand-Rybnikov-Stone, we give a presentation for the foundation of a matroid in terms of generators and relations. The generators are certain "cross-ratios" generalizing the cross-ratio of four points on a projective line, and the relations encode dependencies between cross-ratios in certain low-rank configurations arising in projective geometry. Although the presentation of the foundation is valid for all matroids, it is simplest to apply in the case of matroids without large uniform minors, i.e. matroids having no minor corresponding to five points on a line or its dual configuration. For such matroids, we obtain a complete classification of all possible foundations. We then give a number of applications of this classification theorem, for example: - We prove the following strengthening of a theorem of Lee and Scobee: every orientation of a matroid without large uniform minors comes from a dyadic representation, which is unique up to rescaling. - For a matroid $M$ without large uniform minors, we establish the following strengthening of a 2017 theorem of Ardila-Rincón-Williams: if $M$ is positively oriented then $M$ is representable over every field with at least three elements. - Two matroids are said to belong to the same representation class if they are representable over precisely the same pastures. We prove that there are precisely 12 possibilities for the representation class of a matroid without large uniform minors, exactly three of which are not representable over any field.

math.CO↗
arXiv2020

Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call `tropical polynomials' and `sign polynomials', respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polyomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.

math.NT↗
arXiv2019

Tropical geometry over the tropical hyperfield

In this text, we merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism into the tropical hyperfield. In turn, ordered blueprints make it possible to consider the base change of a classical variety to the tropical hyperfield. We call this base change the scheme theoretic tropicalization of the classical variety. Our first main result describes the Berkovich analytification and the tropicalization of a classical variety as sets of rational points of scheme theoretic tropicalizations, including a characterization of the respective topologies. Our second main result shows that the Giansiracusa bend relations can be derived by a natural construction from the scheme theoretic tropicalization.

math.AG↗
arXiv2018

$\mathbb{F}_1$ for everyone

This text serves as an introduction to $\mathbb{F}_1$-geometry for the general mathematician. We explain the initial motivations for $\mathbb{F}_1$-geometry in detail, provide an overview of the different approaches to $\mathbb{F}_1$ and describe the main achievements of the field.

math.AG↗
arXiv2017

Plücker relations for quiver Grassmannians

In this text, we exhibit the quiver Plücker relations for a quiver Grassmannian and show that they describe the quiver Grassmannian as a closed subscheme of a product of usual Grassmannians.

math.RT↗
arXiv2017

Quiver Grassmannians of extended Dynkin type D - Part 2: Schubert decompositions and F-polynomials

Extending the main result of Part 1, in the first part of this paper we show that every quiver Grassmannian of a representation of a quiver of extended Dynkin type $D$ has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. $F$-polynomials). Along these lines, we obtain explicit formulae for all cluster variables of cluster algebras coming from quivers of extended Dynkin type $D$.

math.RT↗
arXiv2017

Representation type via Euler characteristics and singularities of quiver Grassmannians

In this text, we characterize the representation type of an acyclic quiver by the properties of its associated quiver Grassmannians. This characterization utilizes and extends known results about singular quiver Grassmannians and cell decompositions into affine spaces. While all quiver Grassmannians for indecomposable representations of quivers of finite representation types $A$ and $D$ are smooth and admit cell decompositions, it turns out that all quiver Grassmannians for indecomposable representations of quivers of tame types $A$ and $D$ admit cell decompositions, but some of these quiver Grassmannians are singular (even as varieties). A quiver is wild if and only if there exists a quiver Grassmannian with negative Euler characteristic.

math.RT↗
arXiv2016

Blue schemes, semiring schemes, and relative schemes after Toën and Vaquié

It is a classical insight that the Yoneda embedding defines an equivalence of schemes as locally ringed spaces with schemes as sheaves on the big Zariski site. Similarly, the Yoneda embedding identifies monoid schemes (or $\mathbb{F}_1$-schemes in the sense of Deitmar) with schemes relative to sets (in the sense of Toën and Vaquié). In this paper, we investigate the generalization to blue schemes and to semiring schemes. We establish Yoneda functors for both schemes theories. These functors fail, however, to be equivalences in both situations. The reason for this failure is a divergence in the Grothendieck pretopologies coming from schemes as topological spaces and schemes as sheaves. Restricted to blue schemes that are locally of finite type over a blue field, we construct an inverse to the Yoneda functor, which establishes an equivalence for this subclass of blue schemes. Moreover, we verify the compatibility of the Yoneda functors with the base extension from blue schemes to semiring schemes and with the base extension from semiring schemes to usual schemes.

math.AG↗
arXiv2016

On the relation between hyperrings and fuzzy rings

We construct a full embedding of the category of hyperfields into Dress's category of fuzzy rings and explicitly characterize the essential image --- it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker's theory of matroids over hyperfields with Dress's theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker's partial demifields.

math.CO↗
arXiv2016

Quasicoherent sheaves on projective schemes over F_1

Given a graded monoid A with 1, one can construct a projective monoid scheme MProj(A) analogous to Proj(R) of a graded ring R. This paper is concerned with the study of quasicoherent sheaves (of pointed sets) on MProj(A), and we prove several basic results regarding these. We show that: 1.) Every quasicoherent sheaf F on MProj(A) can be constructed from a graded A--set in analogy with the construction of quasicoherent sheaves on Proj(R) from graded R--modules. 2.) High enough twists of coherent sheaves are generated by finitely many global sections, hence that every coherent sheaf is a quotient of a locally free sheaf. 3.) Coherent sheaves have finite spaces of global sections. The last part of the paper is devoted to classifying coherent sheaves on P^1 in terms of certain directed graphs and gluing data. The classification of these over F_1 is shown to be much richer and combinatorially interesting than in the case of ordinary P^1, and several new phenomena emerge.

math.AG↗
arXiv2015

Cech cohomology over $\mathbb{F}_{1^2}$

In this text, we generalize Cech cohomology to sheaves $\mathcal F$ with values in blue $B$-modules where $B$ is a blueprint with $-1$. If $X$ is an object of the underlying site, then the cohomology sets $H^l(X,\mathcal F)$ turn out to be blue $B$-modules. For locally free $\mathcal O_X$-module $\mathcal F$ on a monoidal scheme $X$, we prove that $H^l(X,\mathcal F)^+=H^l(X^+,\mathcal F^+)$ where $X^+$ is the scheme associated with $X$ and $\mathcal F^+$ is the locally free $\mathcal O_{X^+}$-module associated with $\mathcal F$. In an appendix, we show that the naive generalization of cohomology as a right derived functor is infinite-dimensional for the projective line over $\mathbb F_1$.

math.AG↗
arXiv2015

Schemes as functors on topological rings

Let $X$ be a scheme. In this text, we extend the known definitions of a topology on the set $X(R)$ of $R$-rational points from topological fields, local rings and adèle rings to any ring $R$ with a topology. This definition is functorial in both $X$ and $R$, and it does not rely on any restriction on $X$ like separability or finiteness conditions. We characterize properties of $R$, such as being a topological Hausdorff ring, a local ring or having $R^\times$ as an open subset for which inversion is continuous, in terms of functorial properties of the topology of $X(R)$. Particular instances of this general approach yield a new characterization of adelic topologies, and a definition of topologies for higher local fields.

math.AG↗
arXiv2015

Quiver Grassmannians of type extended Dynkin type D - Part 1: Schubert systems and decompositions into affine spaces

Let $Q$ be a quiver of extended Dynkin type $D$. In this first of two papers, we show that the quiver Grassmannian $Gr_e(M)$ has a decomposition into affine spaces for every dimension vector $e$ and every indecomposable representation $M$ of defect $-1$ and defect $0$, with exception of the non-Schurian representations in homogeneous tubes. We characterize the affine spaces in terms of the combinatorics of a fixed coefficient quiver for $M$. The method of proof is to exhibit explicit equations for the Schubert cells of $Gr_e(M)$ and to solve this system of equations successively in linear terms. This leads to an intricate combinatorial problem, for whose solution we develop the theory of Schubert systems. In the sequel to this paper, we extend the result of this paper to all indecomposable representations $M$ of $Q$ and determine explicit formulae for their $F$-polynomials.

math.RT↗