Related papers: On relative grothendieck rings and algebraically c…
We introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to…
The structure of classical non-linear $\cw$ algebras closing on rational functions is analyzed both for the ordinary and the supersymmetric case. Such algebras appear as a result of a coset construction. Their relevance to physical…
A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria…
We study the ring of regular functions on the space of planar electrical networks, which we coin the grove algebra. This algebra is an electrical analogue of the Pl\"ucker ring studied classically in invariant theory. We develop the…
We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck's trace map of the surface as a sum of residues. Points at infinity are then incorporated into the…
This paper explores relations between two separate worlds. They are the algebraic geometry of Alexander Grothendieck and the automorphic representation theory of Robert Langlands. The relation between them would be a very broad example of…
The goal of this short note is to point out three observations around the Grothendieck norm and semidefinite programming. The first is that the Grothendieck norm captures the difficulty of relating the off-diagonal entries of a real,…
This work concerns the study of properties of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain…
We show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general…
The theory of Gromov-Hausdorff convergence is applied to sequences of quotient rings of integers. It is shown the existence of limit rings (fields) as the Gromov-Hausdorff limits of sequences of metric quotient rings. The relation of these…
The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator…
Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring $\mathrm{GW}^{\mathrm{s}}(R)$ of $R$ as an abelian group when $R$ has residue field $\mathbb{F}_2$. This completes a recent…
The theory of Lambda-rings, in the sense of Grothendieck's Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce…
We start an analysis of geometric properties of a structure relative to a reduct. In particular, we look at definability of groups and fields in this context. In the relatively one-based case, every definable group is isogenous to a…
We define the weak-normalization and the seminormalization of a real algebraic variety relative to its central locus. The study is related to the properties of the rings of continuous rational functions and hereditarily rational functions…
We study relationships between certain algebraic properties of groups and rings definable in a first order structure or $*$-closed in a compact $G$-space. As a consequence, we obtain a few structural results about $\omega$-categorical rings…
We characterize the space of restrictions of real rational functions to certain algebraic Jordan curves in the plane via the Dirichlet-to-Neumann map associated to the domain in the complex plane bounded by the curve and its Bergman kernel.…
The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as well. A classical example of the latter is…
Let $R$ be a commutative ring. We show that any complete duality pair gives rise to a theory of relative homological algebra, analogous to Gorenstein homological algebra. Indeed Gorenstein homological algebra over a commutative Noetherian…
We introduce a duality for In\"{o}n\"{u}-Wigner contractions attached to real symmetric Lie algebras. Starting from a symmetric pair $(\mathfrak{g},\theta)$, we define a dual real form $\mathfrak{g}^{*}$ inside the complexification of…