Finite Type Research Articles

Suberesolving codes

‎We show that any right continuing factor code with retract 0 into an irreducible shift of finite type is right eresolving‎, ‎and we give some sufficient conditions for a right eresolving almost everywhere code being right eresolving everywhere‎. ‎Suberesolving codes as a generalization of ersolving codes have been introduced and we determine some shift spaces which preserved by suberesolving codes‎. ‎Also‎, ‎we show that any bi-eresolving (resp‎. ‎bi-suberesolving) code on an irreducible shift of finite type (resp‎. ‎a synchronized system) is open (resp‎. ‎semi-open) and any right suberesolving code on a synchronized system is right continuing almost everywhere‎.

A Simple Homological Characterization of String Algebras of Finite Representation Type

We prove that among the finite-dimensional algebras of finite representation type those that are string algebras are precisely the ones that have the property that the middle term of an arbitrary extension of indecomposable modules has at most two direct factors. On the other hand, we show that non-domestic string algebras are very far from having that property.

Hyperbolic 5-manifolds that fiber over S^1

We exhibit some finite-volume cusped hyperbolic 5-manifolds that fiber over the circle. These include the smallest hyperbolic 5-manifold known, discovered by Ratcliffe and Tschantz. As a consequence, we build a finite type subgroup of a hyperbolic group that is not hyperbolic.

Special values of L-functions on regular arithmetic schemes of dimension 1

We construct a well-behaved Weil-étale complex for a large class of Z-constructible sheaves on a regular irreducible scheme U of finite type over Z and of dimension 1. We then give a formula for the special value at s=0 of the L-function associated to any Z-constructible sheaf on U in terms of Euler characteristics of Weil-étale cohomology; for smooth proper curves, we obtain the formula of [GS20]. We deduce a special value formula for Artin L-functions of integral representations twisted by a singular irreducible scheme X of finite type over Z and of dimension 1. This generalizes and improves all results in [Tra16]; as a special case, we obtain a special value formula for the arithmetic zeta function of X.

On Tate–Shafarevich groups of one-dimensional families of commutative group schemes over number fields

Given a smooth geometrically connected curve C over a field k and a smooth commutative group scheme G of finite type over the function field K of C, we study the Tate–Shafarevich groups given by elements of \(H^1(K,G)\) locally trivial at completions of K associated with closed points of C. When G comes from a k-group scheme and k is a number field (or k is a finitely generated field and C has a k-point), we prove finiteness of generalizing a result of Saïdi and Tamagawa for abelian varieties. We also give examples of nontrivial in the case when G is a torus and prove other related statements.

EXTENSIONAL REALIZABILITY AND CHOICE FOR DEPENDENT TYPES IN INTUITIONISTIC SET THEORY

AbstractIn [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of $\mathsf {CZF}$ (constructive Zermelo–Fraenkel set theory) and $\mathsf {IZF}$ (intuitionistic Zermelo–Fraenkel set theory), that further validate $\mathsf {AC}_{\mathsf {FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding $\mathsf {AC}_{\mathsf {FT}}$ . We then show that adding such choice principles does not change the arithmetic part of either $\mathsf {CZF}$ or $\mathsf {IZF}$ .

Universal K-matrices for quantum Kac-Moody algebras

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra H H endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra U q L s l 2 U_qL\mathfrak {sl}_{2} .

Local Systems on Diamonds and p-Adic Vector Bundles

Abstract We use Scholze’s framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper, we provide a detailed analysis of local systems on diamonds for the étale, pro-étale, and the $v$-topology and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $\mathbb {C}_p$, this enables us to prove a category equivalence between $\mathbb {C}_p$-local systems with integral models, and modules under the $v$-structure sheaf that modulo each $p^n$ can be trivialized on a proper cover. The flexibility of the $v$-topology together with a descent result on integral models of local systems allows us to prove that the trivializability condition in the module category may be checked on any normal proper cover. This result leads to an extension of the parallel transport theory by Deninger and the second author to vector bundles with numerically flat reduction on a proper normal cover. 2020 MSC: 14G45, 14G22, 11G25.

Topological entropy for shifts of finite type over [formula omitted] and trees

We study the topological entropy of hom tree-shifts and show that, although the topological entropy is not a conjugacy invariant for tree-shifts in general, it remains invariant for hom tree higher block shifts. In [16,17], Petersen and Salama demonstrated the existence of topological entropy for tree-shifts and h(TX)≥h(X), where TX is the hom tree-shift derived from X. We characterize a necessary and sufficient condition when the equality holds for the case where X is a shift of finite type. Additionally, two novel phenomena have been revealed for tree-shifts. There is a gap in the set of topological entropy of hom tree-shifts of finite type, making such a set not dense. Last but not least, the topological entropy of a reducible hom tree-shift of finite type can be strictly larger than that of its maximal irreducible component.

On exotic symmetries of finite type models

On exotic symmetries of finite type models

Quantum difference equation for Nakajima varieties

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in $${{\,\mathrm{Pic}\,}}(X)\otimes {\mathbb {C}}$$ . We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples.

Classification of Surfaces of Coordinate Finite Type in the Lorentz–Minkowski 3-Space

In this paper, we define surfaces of revolution without parabolic points in three-dimensional Lorentz–Minkowski space. Then, we classify this class of surfaces under the condition ΔIIIx=Ax, where ΔIII is the Laplace operator regarding the third fundamental form, and A is a real square matrix of order 3. We prove that such surfaces are either catenoids or surfaces of Enneper, or pseudo spheres or hyperbolic spaces centered at the origin.

Classification of singularities of cluster algebras of finite type: the case of trivial coefficients

AbstractWe provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields of arbitrary characteristic. Furthermore, from the same perspective, we study a family of cluster algebras which are not of finite type and which arise from a star shaped quiver.

Finitely star regular domains

Let R be an integral domain, S t a r ( R ) the set of all star operations on R and S t a r F C ( R ) the set of all star operations of finite type on R . Then R is said to be star regular if | S t a r ( T ) | ≤ | S t a r ( R ) | for every overring T of R . In this paper we introduce the notion of finitely star regular domain as an integral domain R such that | S t a r F C ( T ) | ≤ | S t a r F C ( R ) | for each overring T of R . First, we show that the notions of star regular and finitely star regular domains are completely different and do not imply each other. Next, we extend/generalize well-known results on star regularity in Noetherian and Prüfer contexts to finitely star regularity. Also we handle the finite star regular domains issued from classical pullback constructions to construct finitely star regular domains that are not star regular.

Computability of topological pressure on compact shift spaces beyond finite type* *Burr was partially supported by grants from the National Science Foundation (CCF-1527193 and DMS-1913119). Das was partially supported by ONR MURI Grant N00014-19-1-242 and ONR YIP Grant N00014-16-1-264. Wolf was partially supported by grants from the Simons Foundation (#637594 to Christian Wolf) and PSC-CUNY (TRADB-51-63715 to Christian Wolf). Yang

We investigate the computability (in the sense of computable analysis) of the topological pressure P top(ϕ) on compact shift spaces X for continuous potentials . This question has recently been studied for subshifts of finite type (SFTs) and their factors (sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalised gap shifts, and particular beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalised pressure function (X, ϕ) ↦ P top(X, ϕ| X ) is not computable for a large set of shift spaces X and potentials ϕ. In particular, the entropy map X ↦ h top(X) is computable at a shift space X if and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.

BOHR’S FORMULA FOR ONE-DIMENSIONAL SCHRÖDINGER OPERATORS DEFINED BY SELF-SIMILAR MEASURES WITH OVERLAPS

We study Bohr’s formula for fractal Schrödinger operators on the half line. These fractal Schrödinger operators are defined by fractal measures with overlaps and a locally bounded potential that tends to infinity. We first derive an analog of Bohr’s formula for these Schrödinger operators under some suitable conditions. Then we demonstrate how this result can be applied to self-similar measures with overlaps, including the infinite Bernoulli convolution associated with the golden ratio, m-fold convolution of Cantor-type measures, and a family of graph-directed self-similar measures that are essentially of finite type.

Relative pressure functions and their equilibrium states

AbstractFor a subshift $(X, \sigma _{X})$ and a subadditive sequence ${\mathcal F}=\{\log f_{n}\}_{n=1}^{\infty }$ on X, we study equivalent conditions for the existence of $h\in C(X)$ such that $\lim _{n\rightarrow \infty }(1/{n})\int \log f_{n}\, d\kern-1pt\mu =\int h \,d\kern-1pt\mu $ for every invariant measure $\mu $ on X. For this purpose, we first we study necessary and sufficient conditions for ${\mathcal F}$ to be an asymptotically additive sequence in terms of certain properties for periodic points. For a factor map $\pi : X\rightarrow Y$ , where $(X, \sigma _{X})$ is an irreducible shift of finite type and $(Y, \sigma _{Y})$ is a subshift, applying our results and the results obtained by Cuneo [Additive, almost additive and asymptotically additive potential sequences are equivalent. Comm. Math. Phys.37 (3) (2020), 2579–2595] on asymptotically additive sequences, we study the existence of h with regard to a subadditive sequence associated to a relative pressure function. This leads to a characterization of the existence of a certain type of continuous compensation function for a factor map between subshifts. As an application, we study the projection $\pi \mu $ of an invariant weak Gibbs measure $\mu $ for a continuous function on an irreducible shift of finite type.

On dimension theory of supermodules, super-rings, and superschemes

We introduce the notion of Krull super-dimension of supermodules over certain super-commutative Noetherian super-rings. We investigate how this notion relates to the notion of odd regular sequence due T. Schmitt and how it behaves with respect to the transition to the graded and bigraded super-modules and super-rings associated with the original ones. We also apply these results to the super-dimension theory of superschemes of finite type and their morphisms.

Torsion and torsion-free classes from objects of finite type in Grothendieck categories

In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of FPn-injective objects to be a torsion class. By doing so, we propose a notion of n-hereditary categories. We also define and study the class of FPn-flat objects in Grothendieck categories with a generating set of small projective objects, and provide several equivalent conditions for this class to be torsion-free. At the end, we present several applications and examples of n-hereditary categories in the contexts modules over a ring, chain complexes of modules and categories of additive functors from an additive category to the category of abelian groups. Concerning the latter setting, we find a characterization of when these functor categories are n-hereditary in terms of the domain additive category.

Density of rational points on a family of del Pezzo surfaces of degree one

Let k be an infinite field of characteristic 0, and X a del Pezzo surface of degree d with at least one k-rational point. Various methods from algebraic geometry and arithmetic statistics have shown the Zariski density of the set X(k) of k-rational points in X for d≥2 (under an extra condition for d=2), but fail to work in generality when the degree of X is 1, leaving a large class of del Pezzo surfaces for which the question of density of rational points is still open. In this paper, we prove the Zariski density of X(k) when X has degree 1 and is represented in the weighted projective space P(2,3,1,1) with coordinates x,y,z,w by an equation of the form y2=x3+az6+bz3w3+cw6 for a,b,c∈k with a,c non-zero, under the condition that the elliptic surface obtained by blowing up the base point of the anticanonical linear system |−KX| contains a smooth fiber above a point in P1∖{(1:0),(0:1)} with positive rank over k. When k is of finite type over Q, this condition is sufficient and necessary.