arXiv subjects
Publications and source records attributed to Qimh Richey Xantcha.
Hall's binomial rings, rings with binomial coefficients, are given an axiomatisation and proved identical to the numerical rings studied by Ekedahl. The Binomial Transfer Principle is established, enabling combinatorial proofs of algebraical identities. The finitely generated binomial rings are completely classified. An application to modules over binomial rings is given.
We introduce the notion of numerical functors to generalise Eilenberg & MacLane's polynomial functors to modules over a binomial base ring. After shewing how these functors are encoded by modules over a certain ring, we record a precise criterion for a numerical (or polynomial) functor to admit a strict polynomial structure in the sense of Friedlander & Suslin. We also provide several characterisations of analytic functors.
We develop the theory of 2-quivers and quiver 2-categories to run in parallel with the classical theory of quiver algebras. A quiver 2-category is always finitary, and, conversely, every finitary 2-category will be bi-equivalent with a quiver 2-category for a unique underlying reduced 2-quiver. As an application, we produce an example of a fiat 2-category, one of whose Duflo involutions is not self-adjoint.
A classification is provided of functors, in particular polynomial ones, from a category with a zero object in which every object is a finite sum of copies of a generating object, into an abelian category. This classification is extended to include functors from a category with sums and a zero object, carrying a small, regular projective generator.
The article focuses on three different notions of polynomiality for maps of modules. In addition to the polynomial maps studied by Eilenberg and Mac Lane and the strict polynomial maps ("lois polynomes") considered by Roby, we introduce numerical maps and investigate their properties. Even though our notion require the existence of binomial co-efficients in the base ring, we argue that it constitutes the correct way to extend Eilenberg and Mac Lane's polynomial maps of abelian groups to incorporate modules over more general rings. The main theorem propounds that our maps admit a description corresponding, word by word, to Roby's definition of strict polynomial maps.
A partly (anti-)commutative quiver algebra is a quiver algebra bound by an (anti-)commutativity ideal, that is, a quadratic ideal generated by monomials and (anti-)commutativity relations. We give a combinatorial description of the ideals and the associated generator graphs, from which one can quickly determine if the ideal is admissible or not. We describe the center of a partly (anti-)commutative quiver algebra and state necessary and sufficient conditions for the center to be finitely generated as a K-algebra. As an application, necessary and sufficient conditions for finite generation of the Hochschild cohomology ring modulo nilpotent elements for a partly (anti-)commutative Koszul quiver algebra are given.
A formula for calculating Extensions of (mainly integral) Polynomial Functors is established, based upon projective resolutions. Sample computations are performed, which, in particular, exhibit a surprising non-trivial extension of Divided Cubes by Symmetric Cubes. An explicit description of the latter is given.
We propose a new description of Endofunctors of Module Categories, based upon a combinatorial category comprising finite sets and so-called mazes. Polynomial and numerical functors both find a natural interpretation in this frame-work. Since strict polynomial functors, according to the work of Salomonsson, are encoded by multi-sets, the two strains of functors may be compared and contrasted through juxtaposing the respective combinatorial structures, leading to the Polynomial Functor Theorem, giving an effective criterion for when a numerical (polynomial) functor is strict polynomial.