arXiv subjects

Holger Brenner

Publications and source records attributed to Holger Brenner.

At least 19 recordsLinked to original sources

arXiv2024

Module schemes in invariant theory

Let G be a finite group acting linearly on the polynomial ring with invariant ring R. If the action is small, then a classical result of Auslander gives in dimension two a correspondence between linear representations of G and maximal Cohen-Macaulay R-modules. We establish a correspondence for all linear actions between representations and objects over the invariant ring by looking at quotient module schemes (up to modification) instead of the modules of covariants.

math.AC↗

arXiv2021

Deciding Stability of Sheaves on Curves

We give an algorithm to determine whether a kernel sheaf over a smooth projective curve over an algebraically closed field is semistable. The algorithm uses symmetric powers to make destabilizing subbundles visible as global sections.

math.AG↗

arXiv2019

Quantifying singularities with differential operators

The $F$-signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong $F$-regularity. However, it is very difficult to compute. Motivated by different aspects of the $F$-signature, we define a numerical invariant for rings of characteristic zero or $p>0$ that exhibits many of the useful properties of the $F$-signature. We also compute many examples of this invariant, including cases where the $F$-signature is not known. We also obtain a number of results on symbolic powers and Bernstein-Sato polynomials.

math.AC↗

arXiv2019

The fundamental group of binoid varieties

Binoid schemes generalise monoid schemes, which in turn enable us to generalise toric varieties. Let $X$ be a binoid scheme. The aim of this paper is to calculate the topological fundamental group of $KX$, where $K=\mathbb{C}$ or $\mathbb{R}$. For the latter, we will give an explicit way of calculating the fundamental group using methods from 2-category theory. Indeed, we will calculate the more general fundamental groupoid. As a specialisation, we will also look at the Stanley Reisner Rings. Our method simplifies in this case, allowing us to describe the fundamental groupoid in terms of the simplicial complex directly.

math.AG↗

arXiv2018

Differential symmetric signature in high dimension

We study the differential symmetric signature, an invariant of rings of finite type over a field, introduced in a previous work by the authors in an attempt to find a characteristic-free analogue of the F-signature. We compute the differential symmetric signature for invariant rings $k[x_1,\dots,x_n]^G$ where $G$ is a finite small subgroup of $\mathrm{Gl}(n,k)$ and for hypersurface rings $k[x_1,\dots,x_n]/(f)$ of dimension $\geq3$ with an isolated singularity. In the first case, we obtain the value $1/|G|$, which coincides with the F-signature and generalizes a previous result of the authors for the two-dimensional case. In the second case, following an argument by Bruns, we obtain the value $0$, providing an example of a ring where differential symmetric signature and F-signature are different.

math.AC↗

arXiv2018

Local Picard Group of Pointed Monoids and Their Algebras

The main goal of this paper is to give an explicit formula for the cohomology of the sheaf of units of the punctured spectrum of a Stanley-Reisner ring. In particular we compute the local Picard group of $K [\triangle]$. To achieve this we study the corresponding purely combinatorial problem on the punctured spectrum of the pointed monoid defined by $\triangle$. The cohomology of the sheaf of units on ${\rm Spec}^ \bullet K[\triangle ]$ is then the direct sum of this combinatorial cohomology, which has a decomposition along the vertices, and another part depending on the field $K$.

math.AC↗

arXiv2017

Tight closure and continuous closure

We show that for ideals primary to a maximal ideal in a normal domain of finite type over the complex numbers, its tight closure is contained inside the continuous closure.

math.AC↗

arXiv2017

Hilbert-Kunz multiplicity of binoids

We prove in a broad combinatorial setting, namely for finitely generated semipositive cancellative reduced binoids, that the Hilbert-Kunz multiplicity is a rational number independent of the characteristic.

math.AC↗

arXiv2017

Normality and Related Properties of Forcing Algebras

We present a sufficient condition for irreducibility of forcing algebras and study the (non)-reducedness phenomenon. Furthermore, we prove a criterion for normality for forcing algebras over a polynomial base ring with coefficients in a perfect field. This gives a geometrical normality criterion for algebraic (forcing) varieties over algebraically closed fields. Besides, we examine in detail an specific (enlightening) example with several forcing equations. Finally, we compute explicitly the normalization of a particular forcing algebra by means of finding explicitly the generators of the ideal defining it as an affine ring.

math.AC↗

arXiv2016

The symmetric signature

We define two related invariants for a $d$-dimensional local ring $(R,\mathfrak{m},k)$ called syzygy and differential symmetric signature by looking at the maximal free splitting of reflexive symmetric powers of two modules: the top dimensional syzygy module $\mathrm{Syz}^d_R(k)$ of the residue field and the module of Kähler differentials $Ω_{R/k}$ of $R$ over $k$. We compute these invariants for two-dimensional ADE singularities obtaining $1/|G|$, where $|G|$ is the order of the acting group, and for cones over elliptic curves obtaining $0$ for the differential symmetric signature. These values coincide with the F-signature of such rings in positive characteristic.

math.AC↗

arXiv2016

Generalized Hilbert-Kunz function in graded dimension two

We prove that the generalized Hilbert-Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$-domain over an algebraically closed field $K$ of prime characteristic $p$ has the form $gHK(M,q)=e_{gHK}(M)q^{2}+γ(q)$, with rational generalized Hilbert-Kunz multiplicity $e_{gHK}(M)$ and a bounded function $γ(q)$. Moreover we prove that if $R$ is a $\mathbb{Z}$-algebra, the limit for $p\rightarrow+\infty$ of the generalized Hilbert-Kunz multiplicity $e_{gHK}^{R_p}(M_p)$ over the fibers $R_p$ exists and it is a rational number.

math.AC↗

arXiv2015

On a question of Mehta and Pauly

In this short note we provide explicit examples in characteristic $p$ on certain smooth projective curves where for a given semistable vector bundle $\mathcal{E}$ the length of the Harder-Narasimhan filtration of $F^\ast \mathcal{E}$ is longer than $p$. This answers a question of Mehta and Pauly raised in arXiv:math/0607565.

math.AG↗

arXiv2013

Irrational Hilbert-Kunz multiplicities

We interpret Hilbert-Kunz theory of a graded ring of positive characteristic in terms of Frobenius asymptotic of cohomology of vector bundles on projective varieties. With this method we show that for almost all prime numbers there exist three-dimensional quartic hypersurface domains and modules of finite length with irrational Hilbert-Kunz multiplicity. From this we deduce that also the Hilbert-Kunz multiplicity of a local noetherian domain might be an irrational number.

math.AG↗

arXiv2012

An explicit example of Frobenius periodicity

In this note we show that the restriction of the cotangent bundle $Ω_{\mathbb P}^2$ of the projective plane to a Fermat curve $C$ of degree $d$ in characteristic $p \equiv -1 \mod 2d$ is, up to tensoration with a certain line bundle, isomorphic to its Frobenius pull-back. This leads to a Frobenius periodicity $F^*({\mathcal E}) \cong {\mathcal E} $ on the Fermat curve of degree 2d, where ${\mathcal E}= {\rm Syz}(U^2,V^2,W^2)(3)$.

math.AG↗

arXiv2012

On the connectedness of the spectrum of forcing algebras

We study the connectedness property of the spectrum of forcing algebras over a noetherian ring. In particular we present for an integral base ring a geometric criterion for connectedness in terms of horizontal and vertical components of the forcing algebra. This criterion allows further simplifications when the base ring is local, or one-dimensional, or factorial. Besides, we discuss whether the connectedness is a local property. Finally, we present a characterization of the integral closure of an ideal by means of the universal connectedness of the forcing algebra

math.AC↗

arXiv2012

A Direct Limit for Limit Hilbert-Kunz Multiplicity for Smooth Projective Curves

This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert-Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an affirmative answer for one of the main cases for which the limit Hilbert-Kunz multiplicity is even known to exist, namely that of graded ideals in the homogeneous coordinate ring of smooth projective curves. The proof involves more careful estimates of bounds found independently by Brenner and Trivedi on the dimensions of the cohomologies of twists of the syzygy bundle as the characteristic p goes to infinity and uses asymptotic results of Trivedi on the slopes of Harder-Narasimham filtrations of Frobenius pullbacks of bundles. In view of unpublished results of Gessel and Monsky, the case of maximal ideals in diagonal hypersurfaces is also discussed in depth.

math.AC↗

arXiv2012

On the behaviour of strong semistability in geometric deformations

Let $Y \to B$ be a relative smooth projective curve over an affine integral base scheme $B$ of positive characteristic. We provide for all prime characteristics example classes of vector bundles $\mathcal{S}$ over $Y$ such that $\mathcal{S}$ is generically strongly semistable and semistable but not strongly semistable for some special fibre. This also provides new examples of the behaviour of Hilbert-Kunz multiplicities in geometric families.

math.AG↗

arXiv2012

Dagger closure in regular rings containing a field

We prove that dagger closure is trivial in regular domains containing a field and that graded dagger closure is trivial in polynomial rings over a field. We also prove that Heitmann's full rank one closure coincides with tight closure in positive characteristic under some mild finiteness conditions. Furthermore, we prove that dagger closure is always contained in solid closure and that the forcing algebra for an element contained in dagger closure is parasolid.

math.AC↗