In this paper, we introduce ordered blueprints and ordered blue schemes, which serve as a common language for the different approaches to tropicalizations and which enhances tropical varieties with a schematic structure. As an abstract concept, we consider a tropicalization as a moduli problem about extensions of a given valuation v : k → T between ordered blueprints k and T . If T is idempotent, then we show that a generalization of the Giansiracusa bend relation leads to a representing object for the tropicalization, and that it has yet another interpretation in terms of a base change along v. We call such a representing object a scheme theoretic tropicalization.This theory recovers and improves other approaches to tropicalizations as we explain with care in the second part of this text.The Berkovich analytification and the Kajiwara-Payne tropicalization appear as rational point sets of a scheme theoretic tropicalization. The same holds true for its generalization by Foster and Ranganathan to higher rank valuations.The scheme theoretic Giansiracusa tropicalization can be recovered from the scheme theoretic tropicalizations in our sense. We obtain an improvement due to the resulting blueprint structure, which is sufficient to remember the Maclagan-Rincón weights.The Macpherson analytification has an interpretation in terms of a scheme theoretic tropicalization, and we give an alternative approach to Macpherson's construction of tropicalizations.The Thuillier analytification and the Ulirsch tropicalization are rational point sets of a scheme theoretic tropicalization. Our approach yields a generalization to any, possibly nontrivial, valuation v : k → T with idempotent T and enhances the tropicalization with a schematic structure.Contents commutes. We denote by Val v (B, S) the set of valuations w : B → S over v.

ABSTRACT. Let Q be a quiver, M a representation of Q with an ordered basis B and e a dimension vector for Q. In this note we extend the methods of [7] to establish Schubert decompositions of quiver Grassmannians Gr e (M) into affine spaces to the ramified case, i.e. the canonical morphism F : T → Q from the coefficient quiver T of M w.r.t. B is not necessarily unramified.In particular, we determine the Euler characteristic of Gr e (M) as the number of extremal successor closed subsets of T 0 , which extends the results of Cerulli Irelli ([4]) and Haupt ([6]) (under certain additional assumptions on B).

No abstract

We classify graph C * -algebras, namely, Cuntz-Krieger algebras associated to the Bass-Hashimoto edge incidence operator of a finite graph, up to strict isomorphism. This is done by a purely graph theoretical calculation of the K-theory of the C * -algebras and the method also provides an independent proof of the classification up to Morita equivalence and stable equivalence of such algebras, without using the boundary operator algebra. A direct relation is given between the K 1 -group of the algebra and the cycle space of the graph.

In this paper, we introduce Schubert decompositions for quiver Grassmannians and investigate example classes of quiver Grassmannians with a Schubert decomposition into affine spaces. The main theorem puts the cells of a Schubert decomposition into relation to the cells of a certain simpler quiver Grassmannian. This allows us to extend known examples

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.

In this paper we provide a first realization of an idea of Jacques Tits from a 1956 paper, which first mentioned that there should be a field of charactéristique une, which is now called F 1 , the field with one element. This idea was that every split reductive group scheme over Z should descend to F 1 , and its group of F 1 -rational points should be its Weyl group. We connect the notion of a torified scheme to the notion of F 1 -schemes as introduced by Connes and Consani. This yields models of toric varieties, Schubert varieties and split reductive group schemes as F 1 -schemes. We endow the class of F 1 -schemes with two classes of morphisms, one leading to a satisfying notion of F 1 -rational points, the other leading to the notion of an algebraic group over F 1 such that every split reductive group is defined as an algebraic group over F 1 . Furthermore, we show that certain combinatorics that are expected from parabolic subgroups of GL(n) and Grassmann varieties are realized in this theory.

The space of toroidal automorphic forms was introduced by Zagier in 1979. Let F be a global field. An automorphic form on GL(2) is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems from the fact (amongst others) that an Eisenstein series of weight s is toroidal if s is a non-trivial zero of the zeta function, and thus a connection with the Riemann hypothesis is established.In this paper, we concentrate on the function field case. We show the following results. The (n − 1)-th derivative of a non-trivial Eisenstein series of weight s and Hecke character χ is toroidal if and only if L(χ, s + 1/2) vanishes in s to order at least n (for the "only if"-part we assume that the characteristic of F is odd). There are no non-trivial toroidal residues of Eisenstein series. The dimension of the space of derivatives of unramified Eisenstein series equals h(g − 1) + 1 if the characterisitc is not 2; in characteristic 2, the dimension is bounded from below by this number. Here g is the genus and h is the class number of F . The space of toroidal automorphic forms is an admissible representation and every irreducible subquotient is tempered. CONTENTS