Abstract. Let X be a smooth scheme of finite type over a field K, let E be a locally free O X -bimodule of rank n, and let A be the non-commutative symmetric algebra generated by E. We construct an internal Hom functor, Hom GrA (−, −), on the category of graded right A-modules. When E has rank 2, we prove that A is Gorenstein by computing the right derived functors of Hom GrA (O X , −). When X is a smooth projective variety, we prove a version of Serre Duality for ProjA using the right derived functors of lim n→∞ Hom GrA (A/A ≥n , −).

Abstract. Let k be a perfect field and let K/k be a finite extension of fields. An arithmetic noncommutative projective line is a noncommutative space of the form ProjS K (V ), where V be a k-central two-sided vector space over K of rank two and S K (V ) is the noncommutative symmetric algebra generated by V over K defined by M. Van den Bergh [26]. We study the geometry of these spaces. More precisely, we prove they are integral, we classify vector bundles over them, we classify them up to isomorphism, and we classify isomorphisms between them. Using the classification of isomorphisms, we compute the automorphism group of an arithmetic noncommutative projective line.

We give a method for constructing maps from a non-commutative scheme to a commutative projective curve. With the aid of Artin-Zhang's abstract Hilbert schemes, this is used to construct analogues of the extremal contraction of a K-negative curve with self-intersection zero on a smooth projective surface. This result will hopefully be useful in studying Artin's conjecture on the birational classification of non-commutative surfaces. As a non-trivial example of the theory developed, we look at non-commutative ruled surfaces and, more generally, at non-commutative P 1 -bundles. We show in particular, that non-commutative P 1 -bundles are smooth, have well-behaved Hilbert schemes and we compute its Serre functor. We then show that non-commutative ruled surfaces give examples of the aforementioned non-commutative Mori contractions.Throughout, all objects and maps are assumed to be defined over some algebraically closed base field k. The first author was supported by an ARC Discovery Project grant.This part is primarily concerned with addressing the question, "Which quasi-schemes are the noncommutative analogues of smooth proper varieties?". We start in section 2 with recounting the notion of base change and Hilbert schemes for categories developed in [AZ01]. Hilbert schemes will be a fundamental tool for us. In section 3, we propose a definition of a non-commutative smooth proper dfold based on various geometric conditions which we impose on a quasi-scheme Y . The list of hypotheses is rather long so in section 4, we show that in the projective case of Y = Proj A, these conditions do follow from hypotheses on A that other authors have studied in the past. Background on base change and Hilbert schemesLet R be a commutative k-algebra. Section B of [AZ01] is devoted to the notion of base change from k −→ R for arbitrary categories. In this section, we review their results and generalise their notion of R-objects to X-objects where X is a separated scheme. We will always work over a field k as opposed to more general commutative rings as in [AZ01]. This simplifies the treatment somewhat, since then any Grothendieck category has k-flat generators and the subtleties involved with defining tensor products disappear.Let Y be a quasi-scheme defined over the ground field k as always.and insisting M ⊗ R − is right exact and commutes with direct sums [AZ01, proposition B3.1]. We may thus define M to be R-flat (or just flat) if this functor is also left exact [AZ01, section C.1]. Also, given a morphism of commutative k-algebras R −→ S, there is a base change functor Mod Y R −→ Mod Y S . With this notion of base change, Grothendieck's theory of flat descent holds [AZ01, theorem C8.6].The categories Mod Y R thus form a stack and this allows us to do base change via arbitrary separated schemes. We copy the definition from that of quasi-coherent modules on a stack (see for example [Vistoli, Appendix]). Let X be a separated scheme on which we put the small Zariski site. An X-object M in Mod Y is the data of an R-object M R ∈ Mod Y R f...

Let k be a field. In this paper, we find necessary and sufficient conditions for a noncommutative curve of genus zero over k to be a noncommutative P 1 -bundle. This result can be considered a noncommutative, one-dimensional version of Tsen's theorem. By specializing this theorem, we show that every arithmetic noncommutative projective line is a noncommutative curve, and conversely we characterize exactly those noncommutative curves of genus zero which are arithmetic. We then use this characterization, together with results from [9], to address some problems posed in [4].

We study the structure of two-sided vector spaces over a perfect field K. In particular, we give a complete characterization of isomorphism classes of simple two-sided vector spaces which are left finite-dimensional. Using this description, we compute the Quillen K-theory of the category of left finite-dimensional, two-sided vector spaces over K. We also consider the closely related problem of describing homomorphisms φ : K → Mn(K).