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Sophie Marques

Publications and source records attributed to Sophie Marques.

At least 19 recordsLinked to original sources
arXiv2026

Direct-product rigidity and factor reconstruc- tion for monoids with zero

We study direct products of monoids with zero and give a criterion under which their factors can be recovered from the multiplicative structure alone. While classical decomposition theory encodes direct products through factor congruences, central elements, and refinement properties, we give a concrete multiplicative reconstruction mechanism. If the factors have no nontrivial complemented central idempotents, then the coordinate idempotents are precisely the atoms and coatoms of the complemented-central-idempotent poset, and their multiplicative stabilizers are precisely the coordinate factors and cofactors. It follows that every isomorphism between such products is monomial, yielding the corresponding wreath-product description of automorphism groups. More generally, every product decomposition is obtained by grouping the original factors; in particular, strict refinement follows. We apply these results to multiplicative monoids of directly indecomposable unital rings, including connected commutative rings, and to arithmetic examples arising from residue-class rings.

math.RA↗
arXiv2026

Generalized action monads and descent

We develop a monadic approach to actions of internal categories. Given an internal category, we construct a monad on a slice whose algebras are the actions of that internal category. Then, we give a complete characterization of the monads that arise in this way; we call them generalized action monads. Finally, we prove that these monads yield a well-behaved notion of generalized descent.

math.CT↗
arXiv2026

Automorphism groups of direct products of multiplicative monoids of certain rings

In this paper, we establish a rigidity result for automorphisms of multiplicative direct products of $D$-rings which are total ring of fraction that have pairwise distinct cardinalities. Under these assumptions, every automorphism acts independently on each factor, so that no interaction between distinct components occurs; in particular, the automorphism group decomposes canonically as the direct product of the automorphism groups of the factors. As a consequence, the automorphism group of the multiplicative monoid of integers modulo $n$ is entirely determined by its $p$-power components.

math.RA↗
arXiv2025

Inner products for strongly regular near-vector spaces and duality for finite dimensional near-vector spaces

In this paper we develop a duality theory for all finite-dimensional near-vector spaces and introduce a notion of inner product tailored to the broad and natural class of strongly regular near-vector spaces. This generalized construction extends the classical inner product beyond the classical framework, yielding rich families of examples on multiplicative near-vector spaces. Within this setting, several familiar norms-such as those that fail to produce Hilbert spaces in the classical sense-emerge naturally as genuine inner-product-type norms. A further contribution is the extension of the theory of generalized (weighted) means to arbitrary complex datasets. This extension unifies and generalizes the classical power and geometric means, carrying them beyond the domain of positive reals.

math.GM↗
arXiv2025

Comparing the face rings of a boolean complex and its barycentric subdivision

We consider the relationship between the Stanley--Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $\Delta$ and that of its barycentric subdivision. These rings share a distinguished parameter subring. S. Murai asked if they are isomorphic, equivariantly with respect to the automorphism group $\operatorname{Aut}(\Delta)$, as modules over this parameter subring. We show that, in general, the answer is no, but for Cohen--Macaulay complexes in characteristic coprime to $|\operatorname{Aut}(\Delta)|$, it is yes, and we give an explicit construction of an isomorphism. To give this construction, we adapt a pair of tools introduced by A. Garsia in 1980. The first one transfers bases from a Stanley--Reisner ring to closely related rings of which it is a Gr\"obner degeneration, and the second identifies bases to transfer.

math.AC↗
arXiv2025

On the automorphism group of the monoid of the integers modulo a prime power

This paper determines the structure of the automorphism group of the unit group \((U_{p^e}, \cdot)\) and the monoid \((\mathbb{Z}/p^e \mathbb{Z}, \cdot)\). For \( e \geq 5 \), we establish that the automorphism group \( \Aut(U_{2^e}, \cdot) \) is the direct product of \( \mathbb{Z}/2\mathbb{Z} \) with the central product of a dihedral group of order 8 and the cyclic group \( \mathbb{Z}/2^{e-3}\mathbb{Z} \). Moreover, we show that the automorphism group \( \Aut(\mathbb{Z}/p^e \mathbb{Z}, \cdot) \) is isomorphic to a canonical semidirect product of \( U_{p^{e-1}} \) and the subgroup of \( \Aut(U_{p^e}, \cdot) \) consisting of automorphisms that induce an automorphism of \( (U_{p^f}, \cdot) \) for any integer \( f \) such that \( 0 \leq f \leq e \).

math.RA↗
arXiv2025

A note on the moduli spaces of free algebras of rank 2

In this paper, we present a formulation of the moduli problem for rank-2 algebras over general base rings in functorial terms, providing presentations as presheaf quotients of affine schemes by group scheme actions.

math.AG↗
arXiv2025

Why is the category of near-vector spaces abelian?

In this paper we present a unified proof of the fact that the category of modules over a ring and the category of near-vector spaces in the sense of J. André, over an appropriate scalar system (a 'scalar group'), are both abelian categories. The unification is possible by viewing each of these categories as subcategories of the (abelian) category of modules over a multiplicative monoid $M$. Although in the case of near-vector spaces all elements of $M$ except one (the 'zero' element) are invertible, we show that this requirement is not necessary for the corresponding category to be abelian in analogy to the well-known fact that modules over a ring form an abelian category even if the ring is not a field (i.e., modules over it are not vector spaces).

math.RA↗
arXiv2025

A study of a recursive sequence of polynomials revealing weighted Catalan Numbers

This paper examines the recursive sequence of polynomials $p_n(x)$, defined by $p_0(x) = x^2 - 2$ and $p_n(x) = p_{n-1}(x)^2 - 2$ for $n \geq 1$. It describes the field-theoretic motivations behind this sequence, derives a recursive formula for its coefficients, and identifies invariants that uncover combinatorial connections, including links to weighted Catalan numbers.

math.CO↗
arXiv2024

A characterization of ramification groups via Taylor morphism

In this paper, we present a functorial method to define ramification groups, identifying them as inertia groups of an induced action on composite jet algebras. This framework lays the foundation for defining higher ramification groups for actions involving group schemes. To achieve this, we introduce Taylor maps within the category of commutative unitary rings at prime ideals of an R-algebra and compute their kernels for algebras of finite type over a field with separably generated residue fields.

math.AG↗
arXiv2024

A note on quadratic cyclotomic extensions

This paper provides two characterizations of the primitive roots of unity in quadratic cyclotomic extensions over arbitrary fields. Firstly, we introduce a mapping from $\mathbb{N}$ to $\mathbb{N}$ crucial for describing these roots, closely tied to their order over the field. Secondly, for any prime $p$, we determine the maximal natural number $n$ such that $ζ_{p^n}$ defines a quadratic cyclotomic extension over the field $F$. This characterization is uniform across different fields, regardless of their characteristic, and applies to both odd and even primes.

math.NT↗
arXiv2024

When is a 2-Power Cyclotomic Extension cyclic?

This paper characterizes the cyclicity property of $2$-power cyclotomic extensions through various means: the structure of the Galois groups, the nature of their subextensions, tower decompositions, and, most importantly, specific conditions on the base field.

math.NT↗
arXiv2024

Distributive decomposition of near-vector spaces

This paper provides two characterizations of regularity for near-vector spaces: first, by expressing them as a direct sum of vector spaces over division rings formed by distributive elements; second, by expressing their dimension in term of the dimension of these summands. These results offer new insights into the structure and properties of near-vector spaces.

math.RA↗
arXiv2024

Gluing Data categories and gluing data functors

We present a novel approach to the concept of gluing in mathematics by introducing the notions of a gluing data category and a gluing data functor. Our work provides a formal categorical characterization of the notion of gluing in algebraic geometry. By using this characterization, we are able to describe gluing in a unified way that applies to a wide range of mathematical structures, including topological spaces, presheaves, sheaves, ringed topological spaces, locally ringed topological spaces, and schemes. Our results provide a fresh perspective on gluing that is both abstract and formal, offering a deeper understanding of this fundamental concept in mathematics.

math.CT↗
arXiv2024

A Categorical Perspective on Gluing

This paper introduces the concept of gluing in a general category, enabling us to define categories that admit glued-up objects. To achieve this, we introduce the notion of a gluing index category. Subsequently, we provide an entirely abstract definition of a gluing data functor requiring only the given category to admit pushouts. We explore various characterizations of cones and limits over these functors. We introduce the concept of refined gluing, which in turn enables us to combine different gluing data effectively. Furthermore, we demonstrate that several categories of topological spaces admit glued-up objects. This, in turn, allows us to establish a concept of gluing covering and to prove that the collection of those coverings forms a Grothendieck topology.

math.CT↗
arXiv2023

Near-linear algebra

In this paper, we prove that the world of near-vector spaces allows us to work with non-linear problems and yet, gives access to most of the tools linear algebra has to offer. We establish some fundamental results for near-vector spaces toward extending classical linear algebra to near-linear algebra. In the present paper, we finalize the algebraic proof that any non-empty $F$-subspace stable under addition and scalar multiplication is an $F$-subspace. We demonstrate that any quotient of a near-vector space by an $F$-subspace is a near-vector space and the First Isomorphism Theorem for near-vector spaces. In doing this, we obtain fundamental descriptions of the span. Defining linear independence outside the quasi-kernel, we prove that near-vector spaces are characterized in terms of the existence of a scalar basis, and we obtain a new important notion of basis.

math.RA↗
arXiv2022

Near-field structures on a given scalar group

With this paper, we gain a better understanding of the set of near-field structures on a fixed scalar group. If we were able to describe all near-field structures on a fixed scalar group, we could describe all near-vector spaces. The near-field structures induced by isomorphisms of canonical near-vector spaces differ by quasi-multiplicative bijections while those induced by isomorphisms of near-fields differ by multiplicative bijections. This reveals one of the fundamental differences between linear algebra and near-linear algebra. We find an explicit description of all the elementary near-vector spaces. Significantly, we construct an addition $\boxplus$ on $\mathbb{Q}$ such that $(\mathbb{Q},\boxplus, \cdot)$ is isomorphic to $(\mathbb{Q}(\sqrt{-19}),+, \cdot)$. We also describe explicitly sufficient conditions for such an isomorphism to exist for more general extensions of $\mathbb{Q}$. Moreover, under extra conditions, we still describe those structures on $(\mathbb{R}, \cdot)$, and $(\mathbb{C}, \cdot)$.

math.RA↗