arXiv subjects

Diego Sulca

Publications and source records attributed to Diego Sulca.

7 recordsLinked to original sources

arXiv2026

Counting subalgebras of $\mathfrak{o}^n$

Let $\mathfrak{o}$ be a compact discrete valuation ring and $n\geq 2$. We introduce a method to study the cotype zeta function of subalgebras of $\mathfrak{o}^n$. This multivariable series encodes the number of finite-index subalgebras $\Lambda$ of the $\mathfrak{o}$-algebra $\mathfrak{o}^n$ of a given elementary divisor type. We express this zeta function as a finite sum of $\mathfrak{o}$-adic integrals and compute these integrals in many cases. As a first application, we recover known results in a natural way from our approach. For instance, we obtain a lower bound for the abscissa of convergence of the subalgebra zeta function of $\mathfrak{o}^n$ by exhibiting an explicit pole. We also determine the number of irreducible subrings of $\mathfrak{o}^n$ of small index. As a second application, we give an explicit formula for the cotype zeta function of subalgebras of $\mathfrak{o}^4$.

math.NT↗

arXiv2024

Purely Inseparable ring extensions

We revisit the concept of special algebras, also known as \textit{purely inseparable ring extensions}. This concept extends the notion of purely inseparable field extensions to the more general context of extensions of commutative rings. We use differential operators methods to provide a characterization for a ring extension to be purely inseparable in terms of a condition on certain modules of differential operators associated to the ring extension. This approach is also used to recover an already known characterization involving the modules of principal parts. Next, given a purely inseparable ring extension $A\subset C$, we aim to understand which intermediate rings $A\subset B\subset C$ satisfy the property that both $A\subset B$ and $B\subset C$ are both flat extensions by considering only the subalgebra $\operatorname{End}_B(C)$ of $\operatorname{End}_A(C)$. To achieve this, we prove a generalization of the Jacobson-Bourbaki theorem on Galois correspondence for field extensions to the setting of commutative ring extensions with homeomorphic spectra. Finally, given a tower of ring extensions $A\subset B\subset C$, we consider the question of whether the fact that two of the three extensions $A\subset C$, $A\subset B$, and $B\subset C$ are purely inseparable implies that the third one is also purely inseparable.

math.AC↗

arXiv2022

Differentiably simple rings and ring extensions defined by $p$-basis

We review the concept of differentiably simple ring and we give a new proof of Harper's Theorem on the characterization of Noetherian differentiably simple rings in positive characteristic. We then study flat families of differentiably simple rings, or equivalently, finite flat extensions of rings which locally admit $p$-basis. These extensions are called "Galois extensions of exponent one". For such an extension $A\subset C$, we introduce an $A$-scheme, called the "Yuan scheme", which parametrizes subextensions $A\subset B\subset C$ such that $B\subset C$ is Galois of a fixed rank. So, roughly, the Yuan scheme can be thought of as a kind of Grassmannian of Galois subextensions. We finally prove that the Yuan scheme is smooth and compute the dimension of the fibers.

math.AC↗

arXiv2022

On the degree of polynomial subgroup growth of nilpotent groups

Let $N$ be a finitely generated nilpotent group. The subgroup zeta function $ζ_N^{\leq}(s)$ and the normal zeta function $ζ_N^\lhd(s)$ of $N$ are Dirichlet series enumerating the finite index subgroups or the finite index normal subgroups of $N$. We present results about their abscissae of convergence $α_N^\leq$ and $α_N^\lhd$, also known as the degrees of polynomial subgroup growth and polynomial normal subgroup growth of $N$, respectively. We first prove some upper bounds for the functions $N\mapsto α_N^\leq$ and $N\mapstoα_N^\lhd$ when restricted to the class of torsion-free nilpotent groups of a fixed Hirsch length. We then show that if two finitely generated nilpotent groups have isomorphic $\mathbb{C}$-Mal'cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. This follows, via the Mal'cev correspondence, from a similar result that we establish for zeta functions of rings. This result is obtained by proving that the abscissa of convergence of an Euler product of certain Igusa-type local zeta functions introduced by du Sautoy and Grunewald remains invariant under base change. We also apply this methodology to formulate and prove a version of our result about nilpotent groups for virtually nilpotent groups. As a side application of our result about zeta functions of rings, we present a result concerning the distribution of orders in number fields.

math.GR↗

arXiv2021

Zeta functions of the 3-dimensional almost-Bieberbach groups

The subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group can be expressed as finite sums of Dirichlet series admitting Euler product factorization. We compute these series except for a finite number of local factors when the group is virtually nilpotent of Hirsch length 3. We deduce that they can be meromorphically continued to the whole complex plane and that they satisfy local functional equations. The complete computation (with no exception of local factors) is presented for those groups that are also torsion-free, that is, for the 3-dimensional almost-Bieberbach groups.

math.GR↗

arXiv2021

Zeta functions of virtually nilpotent groups

We prove that the subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group are finite sums of Euler products of cone integrals over $\mathbb{Q}$ and we deduce from this that they have rational abscissa of convergence and some meromorphic continuation. We also define Mal'cev completions of a finitely generated virtually nilpotent group and we prove that the subgroup growth and the normal subgroup growth of the latter are invariants of its $\mathbb{Q}$-Mal'cev completion.

math.GR↗

arXiv2021

Multiplicity along points of a radicial covering of a regular variety

We study the maximal multiplicity locus of a variety $X$ over a field of characteristic $p>0$ that is provided with a finite surjective radical morphism $δ:X\rightarrow V$, where $V$ is regular, for example, when $X\subset\mathbb{A}^{n+1}$ is a hypersurface defined by an equation of the form $T^{q}-f(x_{1},\ldots,x_{n})=0$ and $δ$ is the projection onto $V:=\operatorname{Spec}(k[x_{1},\ldots,x_{n}])$. The multiplicity along points of $X$ is bounded by the degree, say $d$, of the field extension $K(V)\subset K(X)$. We denote by $F_{d}(X)\subset X$ the set of points of multiplicity $d$. Our guiding line is the search for invariants of singularities $x\in F_{d}(X)$ with a good behavior property under blowups $X'\rightarrow X$ along regular centers included in $F_{d}(X)$, which we call \emph{invariants with the pointwise inequality property}. A finite radicial morphism $δ:X\to V$ as above will be expressed in terms of an $\mathcal{O}_{V}^{q}$-submodule $\mathscr{M}\subseteq\mathcal{O}_{V}$. A blowup $X'\to X$ along a regular equimultiple center included in $F_{d}(X)$ induces a blowup $V'\to V$ along a regular center and a finite morphism $δ':X'\to V'$. A notion of transform of the $\mathcal{O}_{V}^{q}$-module $\mathscr{M}\subset\mathcal{O}_{V}$ to an $\mathcal{O}_{V'}^{q}$-module $\mathscr{M}'\subset\mathcal{O}_{V'}$ will be defined in such a way that $δ':X'\to V'$ is the radicial morphism defined by $\mathscr{M}'$. Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of $V$ on $\mathcal{O}_{V}^{q}$-submodules $\mathscr{M}\subset\mathcal{O}_{V}$.

math.AG↗