We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes, and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont's rational formality result to formality over Z.The data needed in order to state the presentation is fully encoded in the poset of connected components of intersections of the arrangement.

In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of arithmetic matroids, in order to discuss how the Orlik-Solomon model depends on the poset of layers. The analysis of discriminantal toric arrangements permits us to isolate certain conditions under which two toric arrangements have diffeomorphic complements. We also give combinatorial conditions determining whether the cohomology algebra is generated in degree one.

The theory of matroids has been generalized to oriented matroids and, recently, to arithmetic matroids. We want to give a definition of "oriented arithmetic matroid" and prove some properties like the "uniqueness of orientation". The aim of this paper is to relate two different generalizations of matroids: the oriented matroids and the arithmetic matroids.Oriented matroids have a large use in mathematics and science; they are related to the simplex method for linear programming, to the chirality of molecules in theoretical chemistry, and to knot theory. For instance, the Jones polynomial of a link is a specialization of the signed Tutte polynomial (see [Kau89]) of an oriented graphic matroid [Thi87, Jae88]. Another interesting fact is the correspondence between oriented matroids and arrangements of pseudospheres [FL78] that generalizes the correspondence between realizable matroids and central hyperplane arrangements.Arithmetic matroids appear as the combinatorial object for the cohomology module of the complement of a toric arrangement [DP05, Moc12, CDD + 18]. The study of toric arrangements is related to zonotopes, partition functions, box splines, and Dahmen-Micchelli spaces (see [DPV10, DP11, Moc12]). The obvious correspondence between realizable arithmetic matroids and central toric arrangements has not been generalized to the non-realizable cases, so far.With the aim of filling this gap, we define a class of well-behaviour arithmetic matroids which we call orientable arithmetic matroids (see Definition 1.6) hoping that these correspond to "arrangements of pseudo-tori".An r × n matrix with integer coefficients describes at the same time a central toric arrangement, an oriented matroid, and an arithmetic matroid. It comes natural to say that two matrices are equivalent if they describe the same toric arrangement. Geometrically, the group GL r (Z)×(Z/2Z) n acts on the space M(r, n; Z) by left multiplication and sign reverse of the columns. Two realizations (i.e. matrices) of the arithmetic matroid are equivalent if and only if they belong to the same GL r (Z) × (Z/2Z) n -orbit.

We show that the integral cohomology algebra of the complement of a toric arrangement is not determined by the poset of layers. Moreover, the rational cohomology algebra is not determined by the arithmetic matroid (however it is determined by the poset of layers).

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