arXiv subjects

A. Nyman

Publications and source records attributed to A. Nyman.

12 recordsLinked to original sources

arXiv2020

Morphisms to noncommutative projective lines

Let $k$ be a field, let ${\sf C}$ be a $k$-linear abelian category, let $\underline{\mathcal{L}}:=\{\mathcal{L}_{i}\}_{i \in \mathbb{Z}}$ be a sequence of objects in ${\sf C}$, and let $B_{\underline{\mathcal{L}}}$ be the associated orbit algebra. We describe sufficient conditions on $\underline{\mathcal{L}}$ such that there is a canonical morphism from the noncommutative space ${\sf Proj }B_{\underline{\mathcal{L}}}$ to a noncommutative projective line in the sense of \cite{abstractp1}, generalizing the usual construction of a map from a scheme $X$ to $\mathbb{P}^{1}$ defined by an invertible sheaf $\mathcal{L}$ generated by two global sections. We then apply our results to construct, for every natural number $d>2$, a degree two cover of Piontkovski's $d$th noncommutative projective line by a noncommutative elliptic curve in the sense of Polishchuk.

math.AG↗

arXiv2017

An abstract characterization of noncommutative projective lines

Let $k$ be a field. We describe necessary and sufficient conditions for a $k$-linear abelian category to be a noncommutative projective line, i.e. a noncommutative $\mathbb{P}^{1}$-bundle over a pair of division rings over $k$. As an application, we prove that $\mathbb{P}^{1}_{n}$, Piontkovski's $n$th noncommutative projective line, is the noncommutative projectivization of an $n$-dimensional vector space.

math.AG↗

arXiv2015

Species and non-commutative P^1's over non-algebraic bimodules

We study non-commutative projective lines over not necessarily algebraic bimodules. In particular, we give a complete description of their categories of coherent sheaves and show they are derived equivalent to certain bimodule species. This allows us to classify modules over these species and thus generalize, and give a geometric interpretation for, results of C. Ringel.

math.RT↗

arXiv2015

Witt's theorem for noncommutative conics

Let k be a field. We show that all homogeneous noncommutative curves of genus zero over k are noncommutative P^1-bundles over a (possibly) noncommutative base. Using this result, we compute complete isomorphism invariants of homogeneous noncommutative curves of genus zero, allowing us to generalize a theorem of Witt.

math.AG↗

arXiv2015

Noncommutative Tsen's theorem in dimension one

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 regarding arithmetic noncommutative projective lines, to address some problems posed by D. Kussin.

math.AG↗

arXiv2015

Serre duality for non-commutative P^1-bundles

Let E be a locally free, rank n bimodule over a smooth projective scheme X, and let A be the non-commutative symmetric algebra generated by E. We construct an internal Hom functor 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 the internal Hom functor. When X is a smooth projective variety, we use the Gorensteinness of A to prove a version of Serre duality on Proj A, the non-commutative P^1 bundle defined by A.

math.RA↗

arXiv2011

Duals of simple two-sided vector spaces

Let $K$ be a perfect field and let $k \subset K$ be a subfield. In previous work of the second author and C. Pappacena, left finite dimensional simple two-sided $k$-central vector spaces over $K$ were classified by arithmetic data associated to the extension $K/k$. In this paper, we continue to study the relationship between simple two-sided vector spaces and their associated arithmetic data. In particular, we determine which arithmetic data corresponds to simple two-sided vector spaces with the same left and right dimension, and we determine the arithmetic data associated to the left and right dual of a simple two-sided vector space. As an immediate application, we prove the existence of the non-commutative symmetric algebra of any $k$-central two-sided vector space over $K$ which has the same left and right dimension.

math.RA↗

arXiv2009

Points on quantum projectivations

The use of geometric invariants has recently played an important role in the solution of classification problems in non-commutative ring theory. We construct geometric invariants of non-commutative projectivizations, a significant class of examples in non-commutative algebraic geometry.

math.RA↗

arXiv2009

Two-sided vector spaces

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 $ϕ:K\to M_n(K)$.

math.KT↗

arXiv2009

Serre finiteness and Serre vanishing for non-commutative P^1-bundles

Suppose $X$ is a smooth projective scheme of finite type over a field $K$, $\mathcal{E}$ is a locally free ${\mathcal{O}}_{X}$-bimodule of rank 2, $\mathcal{A}$ is the non-commutative symmetric algebra generated by $\mathcal{E}$ and ${\sf Proj}\A$ is the corresponding non-commutative $\mathbb{P}^{1}$-bundle. We use the properties of the internal $\operatorname{Hom}$ functor $\HU(-,-)$ to prove versions of Serre finiteness and Serre vanishing for ${\sf Proj}\A$. As a corollary to Serre finiteness, we prove that ${\sf Proj}\A$ is Ext-finite. This fact is used in \cite{izu} to prove that if $X$ is a smooth curve over $\operatorname{Spec}K$, ${\sf Proj }\A$ has a Riemann-Roch theorem and an adjunction formula.

math.RA↗

arXiv2009

Grassmannians of two-sided vector spaces

Let $k \subset K$ be an extension of fields, and let $A \subset M_{n}(K)$ be a $k$-algebra. We study parameter spaces of $m$-dimensional subspaces of $K^{n}$ which are invariant under $A$. The space $\mathbb{F}_{A}(m,n)$, whose $R$-rational points are $A$-invariant, free rank $m$ summands of $R^{n}$, is well known. We construct a distinct parameter space, $\mathbb{G}_{A}(m,n)$, which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection $\mathbb{F}_{A}(m,n) \cap \mathbb{G}_{A}(m,n)$, which we denote by $\mathbb{H}_{A}(m,n)$. Under suitable hypotheses on $A$, we construct affine open subschemes of $\mathbb{F}_{A}(m,n)$ and $\mathbb{H}_{A}(m,n)$ which cover their $K$-rational points. We conclude by using $\mathbb{F}_{A}(m,n)$, $\mathbb{G}_{A}(m,n)$, and $\mathbb{H}_{A}(m,n)$ to construct parameter spaces of two-sided subspaces of two-sided vector spaces.

math.AG↗

arXiv2008

A generalization of Watts's Theorem: Right exact functors on module categories

Watts's Theorem says that a right exact functor F:Mod R-->Mod S that commutes with direct sums is isomorphic to -\otimes_R B where B is the R-S-bimodule FR. The main result in this paper is the following: if A is a cocomplete abelian category and F:Mod R --> A is a right exact functor commuting with direct sums, then F is isomorphic to - \otimes_R B where B is a suitable R-module in A, i.e., a pair (B,f) consisting of an object B in A and a ring homomorphism f:R --> Hom_A(B,B). Part of the point is to give meaning to the notation -\otimes_R B. That is done in the paper by Artin and Zhang on Abstract Hilbert Schemes. The present paper is a natural extension of some of the ideas in the first part of their paper.

math.RA↗