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Dmitry Kerner

Publications and source records attributed to Dmitry Kerner.

At least 19 recordsLinked to original sources
arXiv2026

When does a derivation of a ring admit the exponential?

Exponentials of (real/complex) vector fields are classically defined via the vector field integration. Take a k-algebra k[x] \subset R\subset k[[x]], where k\supseteq \Q is a local domain. Suppose a derivation \xi is x-adically nilpotent. Define the exp-operator via the Taylor expansion, e^\xi:=\sum \frac{\xi^j}{j!}. It is a formal automorphism, e^\xi\in Aut_k(k[[x]]). When does e^\xi act on R? When does the formal power series e^\xi x\in k[[x]] belong to R? We address this question for the following rings. i. The algebraic power series, R=k\bl x\br, differentially finite (holonomic) power series, D(k[x]), and their higher versions, Picard-Vessiot extensions D^\bullet(k[x]), Picard-Vessiot closure D^\infty(k[x]), and differentially-algebraic power series D^{alg}(k[x]). ii. Power series over normed fields. In particular, power series with coefficients of controlled growth, e.g. analytic/Denjoy-Carleman/Gevrey classes. iii. Germs of smooth functions C^\infty(\R^n,o)/J, for arbitrary ideal J\subset C^\infty(\R^n,o). In case i. the operator e^\xi is transcendental, and the power series e^\xi x is ``usually" far from being algebraic. We give various criteria on e^\xi x to belong to k\bl x\br, D(k[x]), D(R), or D^{alg}(k[x]). In case ii. the answer is positive (i.e. e^\xi acts on R ) under rather weak assumptions on R. In case iii. the answer is ``totally negative". For any \xi\neq0 the operator e^\xi (defined as before) does not act on the quotients of the ring of germs of smooth functions, C^\infty(\R^n,o)/J.

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arXiv2025

Equivalence of germs (of mappings and sets) over k vs that over K

Consider real-analytic mapping-germs, (R^n,o)-> (R^m,o). They can be equivalent (by coordinate changes) complex-analytically, but not real-analytically. However, if the transformation of complex-equivalence is identity modulo higher order terms, then it implies the real-equivalence. On the other hand, starting from complex-analytic map-germs (C^n,o)->(C^m,o), and taking any field extension, C to K, one has: if two maps are equivalent over K, then they are equivalent over C. These (quite useful) properties seem to be not well known. We prove slightly stronger properties in a more general form: * for Maps(X,Y) where X,Y are (formal/analytic/Nash) scheme-germs, with arbitrary singularities, over a base ring k; * for the classical groups of (right/left-right/contact) equivalence of Singularity Theory; * for faithfully-flat extensions of rings k -> K. In particular, for arbitrary extension of fields, in any characteristic. The case ``k is a ring" is important for the study of deformations/unfoldings. E.g. it implies the statement for fields: if a family of maps {f_t} is trivial over K, then it is also trivial over k. Similar statements for scheme-germs (``isomorphism over K vs isomorphism over k") follow by the standard reduction ``Two maps are contact equivalent iff their zero sets are ambient isomorphic". This study involves the contact equivalence of maps with singular targets, which seems to be not well-established. We write down the relevant part of this theory.

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arXiv2025

Further results on Artin approximation, for group-actions on mapping-germs Maps(X,Y) and for quivers of maps

Consider (analytic, resp. algebraic) map-germs, Maps((k^n,o),(k^m,o)). These germs are traditionally studied up to the right, let-right and contact equivalences. Below G is one of these groups. An important tool in this study is the Artin approximation: any formal G-equivalence of maps is approximated by ordinary (i.e. analytic, resp. algebraic) G-equivalence. We consider maps of (analytic, resp. algebraic) scheme-germs, with arbitrary singularities, Maps(X,Y), and establish stronger versions of this property (for G): the Strong Artin approximation and the P\l oski approximation. As a preliminary step we study the contact equivalence for maps with singular targets. In many cases one works with multi-germs of spaces, and with their ``muti-maps". More generally, ``quivers of map-germs" occur in various applications. The needed tools are the Strong Artin approximation for quivers and the P\l oski version. We establish these for directed rooted trees.

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arXiv2024

Deforming the weighted-homogeneous foliation, and trivializing families of semi-weighted homogeneous ICIS

Let X_o be a weighted-homogeneous complete intersection germ in (R^N,o) or (C^N,o), with arbitrary singularities, possibly non-reduced. Take the foliation of the ambient space by weighted-homogeneous real arcs, \ga_s. Take a deformation of X_o by higher order terms, X_t. Does the foliation \ga_s deform compatibly with X_t? We identify the ``obstruction locus", \Sigma in X_o, outside of which such a deformation does exist, and possesses exceptionally nice properties. Using this deformed foliation we construct a contact trivialization of the family of defining equations by a homeomorphism that is real analytic (resp. Nash) off the origin, differentiable at the origin, whose presentation in weighted-polar coordinates is globally real-analytic (resp. globally Nash), and with controlled Lipschitz/C^1-properties.

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arXiv2024

Detecting fast vanishing loops in complex-analytic germs (and detecting germs that are inner metrically conical)

Let X be a reduced complex-analytic germ of pure dimension n\ge2, with arbitrary singularities (not necessarily normal or complete intersection). Various homology cycles on Link_\ep[X] vanish at different speeds when \ep\to0. We give a condition ensuring fast vanishing loops on X. The condition is in terms of the discriminant and the covering data for "convenient" coverings X\to (C^n,o). No resolution of singularities is involved. For surface germs (n=2) this condition becomes necessary and sufficient. A corollary for surface germs that are strictly complete intersections detects fast loops via singularities of the projectivized tangent cone of X. Fast loops are the simplest obstructions for X to be inner metrically conical. Hence we get simple necessary conditions to the IMC property. For normal surface germs these conditions are also sufficient. We give numerous classes of non-IMC germs and IMC germs.

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arXiv2023

Fast vanishing cycles on perturbations of complex weighted-homogeneous complete intersection germs

Take a complex-analytic germ X in (C^N,o) with arbitrary singularity. In many cases Link[X] contains cycles that vanish faster than linearly, when Link[X] shrinks to the origin. These ``fast cycles" capture the crucial metric/Lipschitz properties of X. We consider germs (of arbitrary dimensions and codimensions) that are perturbations of weighted-homogeneous complete intersections. For such germs we determine the fast cycles (i.e. their homotopy type, tangent cone, vanishing rates) via the weights. This gives a vast zoo of fast cycles with prescribed properties. As an immediate application we get countable families of (distinct) exotic Lipschitz structures on germs of topological manifolds, for each fixed dimension, codimension, and multiplicity. Another application is the obstruction for germs to be inner metrically conical, e.g.: * (with certain assumptions) If X is IMC then the n lowest weights coincide. * Let the surface germ X in (C^3,o) be Newton-non-degenerate and IMC. Then for some of the faces of the Newton diagram the two lowest weights coincide.

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arXiv2023

Results on left-right approximation for algebraic morphisms and for analytic morphisms of weakly finite singularity type

The classical Artin approximation (AP) reads: any formal solution of a system of (analytic, resp. algebraic) equations of implicit function type is approximated by ``ordinary" solutions (i.e. analytic, resp. algebraic). Morphisms of scheme-germs, e.g. Maps((k^n,o),(k^m,o)) are usually studied up to the left-right equivalence. The natural question is the left-right version of Artin approximation: when is the formal left-right equivalence of morphisms approximated by the ``ordinary" (i.e. analytic, resp. algebraic) equivalence? In this case the standard Artin approximation is not directly applicable, as the involved (functional) equations are not of implicit function type. Moreover, the na\"ive extension does not hold in the analytic case, because of Osgood-Gabrielov-Shiota examples. The left-right version of Artin approximation (LRAP) was established by M. Shiota for morphisms that are either Nash or [real-analytic and of finite singularity type]. We establish LRAP and its stronger version of P\l oski (LRAPP) for Maps(X,Y) where X,Y are analytic/algebraic germs of schemes of any characteristic. More precisely: * LRAP, LRAPP, the inverse Artin approximation (and its P\l oski's version) hold for algebraic morphisms and for finite analytic morphisms. * LRAP holds for analytic morphisms of weakly-finite singularity type. (For char>0 we impose certain integrability condition.) This latter class of morphisms of ``weakly-finite singularity type" (which we introduce) is of separate importance. It extends naturally the traditional class of morphisms of ``finite singularity type", while preserving their non-pathological behavior. The definition goes via the higher critical loci and higher discriminants of morphisms with singular targets. We establish basic properties of these critical loci. In particular: any map is finitely (right) determined by its higher critical loci.

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arXiv2022

Unfoldings of maps, the first results on stable maps, and results of Mather-Yau/Gaffney-Hauser type in arbitrary characteristic

Consider the (formal/analytic/algebraic) map-germs Maps(X,(k^p,o)). Let G be the group of right/contact/left-right transformations. I extend the following (classical) results from the real/complex-analytic case to the case of arbitrary field k. * A separable unfolding is locally trivial iff it is infinitesimally trivial. * An unfolding is locally versal iff it is infinitesimally versal. * The criterion of factorization of map-germs in zero characteristic. * Criteria of trivialization of unfoldings over affine base. * Fibration of K-orbits into A-orbits. * A map is locally stable iff it is infinitesimally stable. * Stable maps are unfodings of their genotypes. * Stable maps are determined by their local algebras. * Results of Mather-Yau/Scherk/Gaffney-Hauser type. How does the module T^1_G f, or related algebras, determine the G-equivalence type of f?

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arXiv2021

Pairs of Lie-type and large orbits of group actions on filtered modules. (A characteristic-free approach to finite determinacy.)

Finite determinacy for mappings has been classically thoroughly studied in numerous scenarios in the real- and complex-analytic category and in the differentiable case. It means that the map-germ is determined, up to a given equivalence relation, by a finite part of its Taylor expansion. The equivalence relation is usually given by a group action and the first step is always to reduce the determinacy question to an "infinitesimal determinacy", i.e., to the tangent spaces at the orbits of the group action. In this work we formulate a universal, characteristic-free approach to finite determinacy, not necessarily over a field, and for a large class of group actions. We do not restrict to pro-algebraic or Lie groups, rather we introduce the notion of "pairs of (weak) Lie type", which are groups together with a substitute for the tangent space to the orbit such that the orbit is locally approximated by its tangent space, in a precise sense. This construction may be considered as a kind of replacement of the exponential resp. logarithmic maps. It is of independent interest as it provides a general method to pass from the tangent space to the orbit of a group action in any characteristic. In this generality we establish the "determinacy versus infinitesimal determinacy" criteria, a far reaching generalization of numerous classical and recent results, together with some new applications.

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arXiv2021

Orbits of the left-right equivalence of maps in arbitrary characteristic

The germs of maps (k^n,o)\to(k^p,o) are traditionally studied up to the right, left-right or contact equivalence. Various questions about the group-orbits are reduced to their tangent spaces. Classically the passage from the tangent spaces to the orbits was done by vector fields integration, hence it was bound to the real/complex-analytic or C^r-category. The purely-algebraic (characteristic-free) approach to the group-orbits of right and contact equivalence has been developed during the last decades. But those methods could not address the (essentially more complicated) left-right equivalence. Moreover, the characteristic-free results (in the right/contact cases) were weaker than those in characteristic zero, because of the (inevitable) pathologies of positive characteristic. In this paper we close these omissions. * We establish the general (characteristic-free) passage from the tangent spaces to the groups orbits for the groups of right, contact and let-right equivalence. Submodules of the tangent spaces ensure (shifted) submodules of the group-orbits. For the left-right equivalence this extends (and strengthens) various classical results of Mather, Gaffney, du Plessis, and others. * A filtration on the space of maps induces the filtration on the group and on the tangent space. We establish the criteria of type "$T_{G^{(j)}}f$ vs $G^{(j)} f$" in their strongest form, for arbitrary base field/ring, provided the characteristic is zero or high for a given map. This brings the "inevitably weaker" results of char>0 to the level of char=0. * As an auxiliary step, important on its own, we develop the mixed-module structure of the tangent space to the left-right group and establish various properties of the annihilator ideal (that defines the instability locus of the map).

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arXiv2020

Surjectivity of the completion map for rings of $C^\infty$-functions. (Whitney extension theorem for general filtrations)

The classical lemma of Borel reads: any power series with real coefficients is the Taylor series of a smooth function. Algebraically this means the surjectivity of the completion map at a point, $C^\infty(\Bbb{R}^n) \twoheadrightarrow \Bbb{R}[[\underline{x}]]$. Similarly, Whitney extension theorem implies the surjectivity of the completion at closed subsets of $\Bbb{R}^n$. For various applications one needs the surjectivity of completion for general $C^\infty$-rings and general filtrations. We establish the necessary and sufficient conditions for this surjectivity. Moreover, we prove: any element of the completion admits a $C^\infty$-representative that is real-analytic outside of the locus of completion, has any prescribed vanishing rate "at infinity", and the prescribed positivity behaviour at the finite part. Alternatively, one can impose on the smooth representative a set of (compatible) linear conditions.

math.AC↗
arXiv2020

Approximation results of Artin-Tougeron-type for general filtrations and for $C^r$-equations

Artin approximation and other related approximation results are used in various areas. The traditional formulation of such results is restricted to filtrations by powers of ideals, $\{I^j\}$, and to Noetherian rings. In this paper we extend several approximation results both to rather general filtrations and to $C^r$-rings, for $2\le r\le\infty$. As an auxiliary step we establish the surjectivity of the completion map for rings of $C^\infty$ functions, for a very broad class of filtrations.

math.AC↗
arXiv2019

Some genericity results over Noetherian rings

Let M be a filtered module. Some properties of elements of M are "generic" in the following sense: (being open/stable) if an element z of M has a property P then any approximation of z has P; (being dense) any element of M is approximated by an element that has P. (Here the approximation is taken in the filtered sense.) \\ Moreover, one can often ensure an approximation with further special properties, e.g. avoiding a prescribed set of submodules. We prove that being a regular sequence is a generic property. As immediate applications we get corollaries on the generic grades of modules, heights of ideals, properties of determinantal ideals, acyclicity of generalized Eagon-Northcott complexes and vanishing of Tor/Ext.

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arXiv2019

Finite determinacy of matrices over local rings. Tangent modules to the miniversal deformation for R-linear group actions

We consider matrices with entries in a local ring, Mat(m,n,R). Fix a group action, G on Mat(m,n,R), and a subset of allowed deformations, Σ\subseteq Mat(m,n,R). The standard question of Singularity Theory is the finite-(Σ,G)-determinacy of matrices. Finite determinacy implies algebraizability and is equivalent to a stronger notion: stable algebraizability. In our previous work this determinacy question was reduced to the study of the tangent spaces to Σand to the orbit, T_{(Σ,A)}, T_{(GA,A)} , and their quotient, the tangent module to the miniversal deformation. In particular, the order of determinacy is controlled by the annihilator of this tangent module. In this work we study this tangent module for the group action GL(m,R)\times GL(n,R) on Mat(m,n,R) and various natural subgroups of it. We obtain ready-to-use criteria of determinacy for deformations of (embedded) modules, (skew-)symmetric forms, filtered modules, filtered morphisms of filtered modules, chains of modules etc.

math.AG↗
arXiv2019

Group actions on matrices over local rings. Annihilators of T^1-modules for the groups \mathcal{G}_{lr} , \mathcal{G}_{congr}

We consider matrices with entries in a local ring, Mat(R). Fix a group action, G on Mat(R), and a subset of allowed deformations, Σ. The traditional objects of study in Singularity Theory and Algebraic Geometry are the tangent spaces T_{(Σ,A)}, T_{(GA,A)}, and their quotient, the tangent module to the miniversal deformation, T^1_{(Σ,G,A)}. This module plays the key role in various deformation problems, e.g., deformations of maps, of modules, of (skew-)symmetric forms. In particular, the first question is to determine the support/annihilator of this tangent module. In [Belitski-Kerner.1] we have studied this tangent module for various R-linear group actions. In the current work we study the support of the module T^1_{(Σ,G,A)} for group actions that involve automorphisms of the ring. (Geometrically, these are group actions that involve the local coordinate changes.) We obtain various bounds on localizations of T^1_{(Σ,G,A)} and compute the radical of the annihilator of T^1_{(Σ,G,A)}, i.e., the set-theoretic support. This brings the definition of an (apparently new) type of singular locus, the "essential singular locus" of a map/sub-scheme. It reflects the "unexpected" singularities of a subscheme, ignoring those imposed by the singularities of the ambient space. Unlike the classical singular locus (defined by a Fitting ideal of the module of differentials) the essential is defined by the annihilator ideal of the module of derivations.

math.AC↗
arXiv2017

Discriminant of the ordinary transversal singularity type. The global equivalence class

Consider a space X with the singular locus of positive dimension, Z=Sing(X). Suppose both Z and X are locally complete intersections at each point. The transversal type of X along Z is generically constant but at some points of Z it degenerates. In the previous work we have introduced (locally) the discriminant of the transversal type, a subscheme of Z, that reflects these degenerations whenever the generic transversal type is "ordinary". We have established the basic local properties of the discriminant. In the current paper we consider the global case. We compute the equivalence class of the discriminant in the Picard group, Pic(Z). If $X$ is a hypersurface, the discriminant is naturally stratified by the singularities of fibres in the projectivized normal cone $\mathbb{P}$N$_{X/Z}$. In this case (under some additional assumptions) we compute the classes of low codimension strata in the Chow group, A$^2$(Z). As immediate applications, we (re)derive the multi-degrees of the classical discriminant of projective complete intersections and bound the jumps of multiplicity of X along Z (when the singular locus is one-dimensional).

math.AG↗
arXiv2017

Discriminant of the ordinary transversal singularity type. The local aspects

Consider a space X with the singular locus, Z=Sing(X), of positive dimension. Suppose both Z and X are locally complete intersections. The transversal type of X along Z is generically constant but at some points of Z it degenerates. We introduce (under certain conditions) the discriminant of the transversal type, a subscheme of Z, that reflects these degenerations whenever the generic transversal type is `ordinary'. The scheme structure of this discriminant is imposed by various compatibility properties and is often non-reduced. We establish the basic properties of this discriminant: it is a Cartier divisor in Z, functorial under base change, flat under some deformations of (X,Z), and compatible with pullback under some morphisms, etc. Furthermore, we study the local geometry of this discriminant, e.g. we compute its multiplicity at a point, and we obtain the resolution of its structure sheaf (as module on Z) and study the locally defining equation.

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