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M. Rovinsky

Publications and source records attributed to M. Rovinsky.

17 recordsLinked to original sources
arXiv2024

L\"uroth's theorem for fields of rational functions in infinitely many permuted variables

L\"uroth's theorem describes the dominant maps from rational curves over a field. In this note we study those dominant rational maps from cartesian powers $X^{\Psi}$ of geometrically irreducible varieties $X$ over a field $k$ for infinite sets $\Psi$ that are equivariant with respect to all permutations of the factors $X$. At least some of such maps arise as compositions $h:X^{\Psi}\xrightarrow{f^{\Psi}}Y^{\Psi}\to H\backslash Y^{\Psi}$, where $X\xrightarrow{f}Y$ is a dominant $k$-map and $H$ is a group of birational automorphisms of $Y|k$, acting diagonally on $Y^{\Psi}$. In characteristic 0, we show that this construction, when properly modified, gives all dominant equivariant maps from $X^{\Psi}$, if $\dim X=1$. For arbitrary $X$, the results are only partial. Also, a somewhat similar problem of describing the equivariant integral schemes over $X^{\Psi}$ of finite type is touched very briefly.

math.AG↗
arXiv2023

A remark on 0-cycles as modules over algebras of finite correspondences

Given a smooth projective variety $X$ over a field, consider the $\mathbb Q$-vector space $Z_0(X)$ of 0-cycles (i.e. formal finite $\mathbb Q$-linear combinations of the closed points of $X$) as a module over the algebra of finite correspondences. Then the rationally trivial 0-cycles on $X$ form an absolutely simple and essential submodule of $Z_0(X)$.

math.AG↗
arXiv2022

Partial fraction decompositions, and semilinear representations of infinite symmetric groups

Let $F|k$ be a non-trivial regular field extension, $S$ be an infinite (discrete) set, $G$ be the group of all permutations of $S$ endowed with the compact-open topology, $L$ be the fraction field of the tensor product over $k$ of the copies of $F$ labeled by $S$. The field $L$ is endowed with the natural $G$-action. For each $G$-invariant subfield $K$ of $L$, let $Sm_K$ denote the category of smooth (i.e. with open stabilizers) $K$-semilinear representations of $G$. The categories $Sm_K$ (especially, their simple and injective objects) are the principal object of the present study, though only in some particular cases. It is known that the indecomposable injective objects of the category $Sm_L$ are the $L$-exterior powers $L\langle\binom{S}{s}\rangle$ ($s\ge 0$) of the $L$-vector space with the basis $S$, while $L$ is the only simple object. It turns out that the objects $K\langle\binom{S}{s}\rangle$ are injective quite generally. Let $K=L^H\subset L$ be the fixed field of an algebraic automorphism $k$-group $H$ of $F|k$ acting on $L$ diagonally. The question is: what could be a relation (a kind of the Schur--Weyl duality) between representations of $H$ and the indecomposable injectives or simple objects of $Sm_K$? In this paper we consider several examples, where $H$ is either a subgroup of $PGL_{2,k}$ or a torus. In these examples: a) a natural bijection between the finite-dimensional simple objects of $Sm_K$ and the irreducible rational representations of $H$ is constructed; b) for $H\neq PGL_{2,k}$, the indecomposable injectives and the simple objects of $Sm_K$ are described completely. For $H=PGL_{2,k}$, an infinite list of infinite-dimensional simple objects is produced, which is shown to be complete if $F\neq k$; a system of indecomposable injective generators is described.

math.RT↗
arXiv2018

0-cycles on Grassmannians as representations of projective groups

Let $F$ be an infinite division ring, $V$ be a left $F$-vector space, $r>0$ be an integer. We study the structure of the representation of the linear group $\mathrm{GL}_F(V)$ in the vector space of formal finite linear combinations of $r$-dimensional vector subspaces of $V$ with coefficients in a field $K$. This gives a series of natural examples of irreducible infinite-dimensional representations of projective groups. These representations are non-smooth if $F$ is locally compact and non-discrete.

math.RT↗
arXiv2017

An analogue of Hilbert's Theorem 90 for infinite symmetric groups

Let $K$ be a field and $G$ be a group of its automorphisms. If $G$ is precompact then $K$ is a generator of the category of smooth (i.e. with open stabilizers) $K$-semilinear representations of $G$. There are non-semisimple smooth semilinear representations of $G$ over $K$ if $G$ is not precompact. In this note the smooth semilinear representations of the group $G$ of all permutations of an infinite set $S$ are studied. Let $k$ be a field and $k(S)$ be the field freely generated over $k$ by the set $S$ (endowed with the natural $G$-action). One of principal results describes the Gabriel spectrum of the category of smooth $k(S)$-semilinear representations of $G$. It is also shown, in particular, that (i) for any smooth $G$-field $K$ any smooth finitely generated $K$-semilinear representation of $G$ is noetherian, (ii) for any $G$-invariant subfield $K$ in the field $k(S)$, the object $k(S)$ is an injective cogenerator of the category of smooth $K$-semilinear representations of $G$, (iii) if $K\subset k(S)$ is the subfield of rational homogeneous functions of degree 0 then there is a one-dimensional $K$-semilinear representation of $G$, whose integral tensor powers form a system of injective cogenerators of the category of smooth $K$-semilinear representations of $G$, (iv) if $K\subset k(S)$ is the subfield generated over $k$ by $x-y$ for all $x,y\in S$ then there is a unique isomorphism class of indecomposable smooth $K$-semilinear representations of $G$ of each given finite length.

math.RT↗
arXiv2015

On semilinear representations of the infinite symmetric group

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite field) are studied. Many results here are well-known to the experts, at least in the case of {\sl linear representations} of symmetric group. The presented results suggest, in particular, that an analogue of Hilbert's Theorem 90 should hold: in the case of faithful action of the group on the base field the irreducible smooth semilinear representations are one-dimensional (and trivial in appropriate sense).

math.RT↗
arXiv2009

Smooth representations and sheaves

The paper is concerned with `geometrization' of smooth (i.e. with open stabilizers) representations of the automorphism group of universal domains, and with the properties of `geometric' representations of such groups. As an application, we calculate the cohomology groups of several classes of smooth representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of an algebraically closed field.

math.AG↗
arXiv2009

On maximal proper subgroups of field automorphism groups

Let $G$ be the automorphism group of an extension $F|k$ of algebraically closed fields of characteristic zero and of transcendence degree $n$, $1\le n\le\infty$. In this paper we (i) construct some maximal closed non-open subgroups $G_v$, and some (all, in the case of countable transcendence degree) maximal open proper subgroups of $G$; (ii) describe, in the case of countable transcendence degree, the automorphism subgroups over the intermediate subfields (a question of Krull, \cite[\S4, question 3b)]{krull}); (iii) construct, in the case $n=\infty$, a fully faithful subfunctor $(-)_v$ of the forgetful functor from the category of smooth representations of $G$ to the category of smooth representations of $G_v$; (iv) construct, using the functors $(-)_v$, a subfunctor $Γ$ of the identity functor on the category of smooth representations of $G$, coincident (via the forgetful functor) with the functor $Γ$ on the category of smooth admissible semilinear representations of $G$ constructed in \cite{adm} in the case $n=\infty$ and $k=\bar{\mathbb Q}$. The study of open subgroups is motivated by the study of (the stabilizers of the) smooth representations undertaken in \cite{repr,adm}. The functor $Γ$ is an analogue of the global sections functor on the category of sheaves on a smooth proper algebraic variety. Another result is that `interesting' semilinear representations are `globally generated'.

math.RT↗
arXiv2006

Admissible semi-linear representations

The category of admissible (in the appropriately modified sense of representation theory of totally disconnected groups) semi-linear representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of the field of algebraic complex numbers is described.

math.RT↗
arXiv2006

Representations of field automorphism groups

This is a common introduction to math.RT/0101170, math.RT/0306333, math.RT/0506043, math.RT/0601028. Compared to these references there are new results including (i) a description of a separable closure of an extension of transcendence degree one of an algebraically closed field of positive characteristic; (ii) a "Künneth formula" for the products with curves; (iii) the semi-simplicity of the module of regular forms of top degree $Ω^n_{F/k}$.

math.RT↗
arXiv2005

Semi-linear representations of PGL

Let $L$ be the function field of a projective space ${\mathbb P}^n_k$ over an algebraically closed field $k$ of characteristic zero, and $H$ be the group of projective transformations. An $H$-sheaf ${\mathcal V}$ on ${\mathbb P}^n_k$ is a collection of isomorphisms ${\mathcal V} \longrightarrow g^{\ast}{\mathcal V}$ for each $g\in H$ satisfying the chain rule. We construct, for any $n>1$, a fully faithful functor from the category of finite-dimensional $L$-semi-linear representations of $H$ extendable to the semi-group ${\rm End}(L/k)$ to the category of coherent $H$-sheaves on ${\mathbb P}^n_k$. The paper is motivated by a study of admissible representations of the automorphism group $G$ of an algebraically closed extension of $k$ of countable transcendence degree undertaken in \cite{rep}. The semi-group ${\rm End}(L/k)$ is considered as a subquotient of $G$, hence the condition on extendability. In the appendix it is shown that, if $\tilde{H}$ is either $H$, or a bigger subgroup in the Cremona group (generated by $H$ and a standard involution), then any semi-linear $\tilde{H}$-representation of degree one is an integral $L$-tensor power of $\det_LΩ^1_{L/k}$. It is shown also that this bigger subgroup has no non-trivial representations of finite degree if $n>1$.

math.RT↗
arXiv2004

On certain representations of automorphism groups of an algebraically closed field

Let k be an algebraically closed field of characteristic zero, F its algebraically closed extension, and G be the group of k-automorphisms of F endowed with a natural topology. One of the purposes of this paper is to show that any non-faithful continuous representation of G factors through a discrete quotient of G. Properties of representation of G arising from geometry are studied. In some cases the groups of morphisms between geometric objects are identified with the groups of morphisms between corresponding G-modules, and the ${\rm Ext}^1$'s are related. In particular, the category of abelian varieties over k with morphisms tensored with the rationals can be described as a category of G-modules.

math.RT↗
arXiv2001

The Gauss-Manin connection on the Hodge structures

Pour tout schéma simplicial complexe $X_{\bullet}$ il existe une application canonique $\nabla:H^{\ast}(X_{\bullet})\longrightarrow Ω^1_{{\mathbb C}/{\mathbb Q}}\otimes H^{\ast}(X_{\bullet})$, appelée la connexion de Gauß-Manin. Nous montrons qu'il existe une unique connexion fonctorielle sur toute structure de Hodge-Tate mixte ayant certaines propriétés de la connexion de Gauß-Manin. Cette connexion n'est pas intégrable en général, et alors son intégrabilité est une condition non triviale pour qu'une structure de Hodge soit géométrique. Dans des cas particuliers, je donne des formules explicites pour la connexion de Gauß-Manin sur la cohomologie singulière des variétés algébriques sur ${\mathbb C}$ dans les termes de la structure de Hodge.

math.AG↗
arXiv2000

On certain isomorphisms between absolute Galois groups

Let $k$ be an algebraically closed field of characteristic zero, $F$ be an algebraically closed extension of $k$ of transcendence degree one, and $G$ be the group of automorphisms over $k$ of the field $F$. The purpose of this note is to calculate the group of continuous automorphisms of $G$.

math.AG↗
arXiv1998

Refining the Abel--Jacobi maps

Given a smooth projective variety $X$ over a field $k$ of characteristic zero, we consider the composition of the de Rham cohomology cycle class map over $k$ from the Chow group $CH^q(X\times_kK)$, where $K$ is the field of fractions of henselization $A^h$ of the local ring of a smooth closed point of a variety over the field $k$ with an appropriate projection: $CH^q(X\times_kK)\longrightarrow\bigoplus_{p=1}^qgr_F^{q-p}N^{q-p} H^{2q-p}_{dR/k}(X)\otimes_kΩ^p_{A^h/k,{\rm closed}},$ where $F^{\bullet}$ and $N^{\bullet}$ are the Hodge and the coniveau filtrations on the de Rham cohomology, respectively. The classical Abel--Jacobi map corresponds to the composition of this homomorphism with the projection to the summand $p=1$. This homomorphism is not injective, however, its composition with the embedding into the space $\bigoplus_{p=1}^qgr_F^{q-p}N^{q-p}H^{2q-p}_{dR/k}(X)\otimes_k \lim_{\longleftarrow_M}d(Ω^{p-1}_{A_M/k}),$ where $A_M=A^h/{\frak m}^M$ and ${\frak m}$ is the maximal ideal, is dominant for any $q$ for which the inverse Lefschetz operator $H^{2\dim X-q}(X)(\dim X)\stackrel{\sim}{\longrightarrow}H^q(X)(q)$ is induced by a correspondence.

math.AG↗
arXiv1996

A connectivity lemma for the Albanese map

I prove that for any complex projective variety $X$ and a sufficiently large integer $N$ all the fibers of Albanese map of the $N$-th configuration space of $X$ are dominated by smooth connected projective varieties with vanishing ${\rm H}^1$.

alg-geom↗