Noether's Problem Research Articles

Generalizations of noncommutative Noether's problem

Generalizations of noncommutative Noether's problem

Poisson Noether's Problem and Poisson rationality

In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all symplectic reflection groups — the analogue of complex reflection groups in the symplectic world. Our proofs are constructive, and generalize and refine previously known results. The results of this paper can be thought as analogues of the Noncommutative Noether Problem and the Gelfand-Kirillov Conjecture for rational Cherednik algebras in the quasi-classical limit. An abstract framework to understand these results is introduced. As a consequence for complex reflection groups, we obtain the Poisson rationality of the Calogero-Moser spaces associated to any of them, and we verify the Gelfand-Kirillov Conjecture for trigonometric Cherednik algebras and the Poisson rationality of their corresponding Calogero-Moser spaces.

Noether's Problem for Abelian Extensions of Cyclic p-groups of Nilpotency Class 2

Let $$K$$ be a field and $$G$$ be a finite group. Let $$G$$ act on the rational function field $$K(x(g):g\in G)$$ by $$K$$-automorphisms defined by $$g\cdot x(h)=x(gh)$$ for any $$g,h\in G$$. Denote by $$K(G)$$ the fixed field $$K(x(g):g\in G)^G$$. Noether's problem then asks whether $$K(G)$$ is rational over $$K$$. Let $$p$$ be prime and let $$G$$ be a $$p$$-group of exponent $$p^e$$. Assume also that (i) $$K = p>0$$, or (ii) $$K \ne p$$ and $$K$$ contains a primitive $$p^e$$-th root of unity. In this paper we prove that $$K(G)$$ is rational over $$K$$ if $$G$$ is any finite $$p$$-group of nilpotency class $$2$$ which is an abelian extension of a cyclic group.

Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds

We provide the first known example of a finite group action on an oriented surface $T$ that is free, orientation-preserving, and does not extend to an arbitrary (in particular, possibly non-free) orientation-preserving action on any compact oriented 3-manifold $N$ with boundary $\partial N = T$. This implies a negative solution to a conjecture of Dom\'inguez and Segovia, as well as Uribe's evenness conjecture for equivariant unitary bordism groups. We more generally provide sufficient conditions that imply infinitely many such group actions on surfaces exist. Intriguingly, any group with such a non-extending action is also a counterexample to the Noether problem over the complex numbers $\mathbb{C}$. In forthcoming work with Segovia we give a complete homological characterization of those finite groups admitting such a non-extending action, as well as more examples and non-examples. We do not address here the analogous question for non-orientation-preserving actions.

The noncommutative Noether's problem is almost equivalent to the classical Noether's problem

Motivated by the classical Noether's problem, J. Alev and F. Dumas proposed the following question, commonly referred to as the noncommutative Noether's problem: Let a finite group G act linearly on C n inducing the action on Frac ( A n ( C ) ) -the skew field of fractions of the n -th Weyl algebra A n ( C ) , is Frac ( A n ( C ) ) G isomorphic to Frac ( A n ( C ) ) ? In this note we show that if Frac ( A n ( C ) ) G ≅ Frac ( A n ( C ) ) then for any algebraically closed field k of large enough characteristic, field k ( x 1 , ⋯ , x n ) G is stably rational. This result allows us to produce counterexamples to the noncommutative Noether's problem based on well-known counterexamples to the Noether's problem for algebraically closed fields.

An extension of [formula omitted] related to the alternating group and Galois orders

In 2010, V. Futorny and S. Ovsienko gave a realization of $U(\mathfrak{gl}_n)$ as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of $S_1\times S_2\times\cdots\times S_n$, where $S_j$ is the symmetric group on $j$ variables. An interesting question is what a similar algebra would be in the invariant ring with respect to a product of alternating groups. In this paper we define such an algebra, denoted $\mathscr{A}(\mathfrak{gl}_n)$, and show that it is a Galois ring. For $n=2$, we show that it is a generalized Weyl algebra, and for $n=3$ provide generators and a list of verified relations. We also discuss some techniques to construct Galois orders from Galois rings. Additionally, we study categories of finite-dimensional modules and generic Gelfand-Tsetlin modules over $\mathscr{A}(\mathfrak{gl}_n)$. Finally, we discuss connections between the Gelfand-Kirillov Conjecture, $\mathscr{A}(\mathfrak{gl}_n)$, and the positive solution to Noether's problem for the alternating group.

Noether's problem for some subgroups of S14: The modular case

Let $G$ be a subgroup of $S_{n}$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_{1},\cdots,x_{n})$ via $k$-automorphisms defined by $\sigma\cdot x_{i}:=x_{\sigma\cdot i}$ for any $\sigma\in G$ and $1\leq i\leq n$. In this article, we will show that if $G$ is a solvable transitive subgroup of $S_{14}$ and $\text{char}(k)=7$, then the fixed subfield $k(x_{1},\cdots,x_{14})^{G}$ is rational (i.e., purely transcendental) over $k$. In proving the above theorem, we rely on the Kuniyoshi-Gasch\"utz Theorem or some ideas in its proof.

On the Noether Problem for torsion subgroups of tori

We consider the Noether Problem for stable and retract rationality for the sequence of $d$-torsion subgroups $T[d]$ of a torus $T$, $d\geq 1$. We show that the answer to these questions only depends on $d\pmod{e(T)}$, where $e(T)$ is the period of the generic $T$-torsor. When $T$ is the norm one torus associated to a finite Galois extension, we find all $d$ such that the Noether Problem for retract rationality has a positive solution for $d$. We also give an application to the Grothendieck ring of stacks.

Noether's problem for some semidirect products

Let k be a field, G be a finite group, k ( x ( g ) : g ∈ G ) be the rational function field with the variables x ( g ) where g ∈ G . The group G acts on k ( x ( g ) : g ∈ G ) by k -automorphisms where h ⋅ x ( g ) = x ( h g ) for all h , g ∈ G . Let k ( G ) be the fixed field defined by k ( G ) : = k ( x ( g ) : g ∈ G ) G = { f ∈ k ( x ( g ) : g ∈ G ) : h ⋅ f = f for all h ∈ G } . Noether's problem asks whether the fixed field k ( G ) is rational (= purely transcendental) over k . Let m and n be positive integers and assume that there is an integer t such that t ∈ ( Z / m Z ) × is of order n . Define a group G m , n : = 〈 σ , τ : σ m = τ n = 1 , τ − 1 σ τ = σ t 〉 ≃ C m ⋊ C n . Assume furthermore that (i) m is an odd integer, and (ii) for any e | n , the ideal 〈 ζ e − t , m 〉 in Z [ ζ e ] is a principal ideal (where ζ e is a primitive e -th root of unity). Theorem. If k is a field with ζ m , ζ n ∈ k , then k ( G m , n ) is rational over k . Consequently, it may be shown that, for any positive integer n , the set S : = { p : p is a prime number such that C ( G p , n ) is rational over C } is of positive Dirichlet density; in particular, S is an infinite set.

The Noether Problem for spinor groups of small rank

Building on prior work of Bogomolov, Garibaldi, Guralnick, Igusa, Kordonskiĭ, Merkurjev and others, we show that the Noether Problem for Spin n has a positive solution for every n ≤ 14 over an arbitrary field of characteristic ≠2.

On a Rationality Problem for Fields of Cross-ratios

Let $k$ be a field, $n \geqslant 5$ be an integer, $x_1, \dots, x_n$ be independent variables and $L_n = k(x_1, \dots, x_n)$. The symmetric group $\Sigma_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\text{PGL}_2$ acts by applying (the same) fractional linear transformation to each variable. The fixed field $K_n = L_n^{\text{PGL}_2}$ is called ``the field of cross-ratios''. Let $S \subset \Sigma_n$ be a subgroup. The Noether Problem asks whether the field extension $L_n^S/k$ is rational, and the Noether Problem for cross-ratios asks whether $K_n^S/k$ is rational. In an effort to relate these two problems, H.~Tsunogai posed the following question: Is $L_n^S$ rational over $K_n^S$? He answered this question in several interesting situations, in particular in the case where $S = \Sigma_n$. In this paper we extend his results by recasting the problem in terms of Galois cohomology. Our main theorem asserts that the following conditions on a subgroup $S \subset \Sigma_n$ are equivalent: (a) $L_n^S$ is rational over $K_n^S$, (b) $L_n^S$ is unirational over $K_n^S$, (c) $S$ has an orbit of odd order in $\{ 1, \dots, n \}$.

Degree three unramified cohomology groups and Noether's problem for groups of order 243

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $k$-automorphisms defined as $h(x_g)=x_{hg}$ for any $g,h\in G$. We denote the fixed field $k(x_g : g\in G)^G$ by $k(G)$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is well-known that if $C(G)$ is stably rational over $C$, then all the unramified cohomology groups $H_[nr}^i(C(G),Q/Z)=0$ for $i \ge 2$. Hoshi, Kang and Kunyavskii [HKK] showed that, for a $p$-group of order $p^5$ ($p$: an odd prime number), $H_[nr}^2(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to the isoclinism family $\Phi_{10}$. When $p$ is an odd prime number, Peyre [Pe3] and Hoshi, Kang and Yamasaki [HKY1] exhibit some $p$-groups $G$ which are of the form of a central extension of certain elementary abelian $p$-group by another one with $H_[nr}^2(C(G),Q/Z)=0$ and $H_[nr}^3(C(G),Q/Z)\neq 0$. However, it is difficult to tell whether $H_[nr}^3(C(G),Q/Z)$ is non-trivial if $G$ is an arbitrary finite group. In this paper, we are able to determine $H_[nr}^3(C(G),Q/Z)$ where $G$ is any group of order $p^5$ with $p=3, 5, 7$. Theorem 1. Let $G$ be a group of order $3^5$. Then $H_[nr}^3(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to $\Phi_7$. Theorem 2. If $G$ is a group of order $3^5$, then the fixed field $C(G)$ is rational if and only if $G$ does not belong to $\Phi_{7}$ and $\Phi_{10}$. Theorem 3. Let $G$ be a group of order $5^5$ or $7^5$. Then $H_[nr}^3(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to $\Phi_6$, $\Phi_7$ or $\Phi_{10}$. Theorem 4. If $G$ is the alternating group $A_n$, the Mathieu group $M_{11}$, $M_{12}$, the Janko group $J_1$ or the group $PSL_2(F_q)$, $SL_2(F_q)$, $PGL_2(F_q)$ (where $q$ is a prime power), then $H_[nr}^d(C(G),Q/Z)=0$ for any $d\ge 2$. Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups.

Retract Rationality and Algebraic Tori

AbstractFor any prime number$p$and field$k$, we characterize the$p$-retract rationality of an algebraic$k$-torus in terms of its character lattice. We show that a$k$-torus is retract rational if and only if it is$p$-retract rational for every prime $p$, and that the Noether problem for retract rationality for a group of multiplicative type $G$has an affirmative answer for$G$if and only if the Noether problem for$p$-retract rationality for$G$has a positive answer for all $p$. For every finite set of primes$S$we give examples of tori that are$p$-retract rational if and only if$p\notin S$.

Noncommutative Noether’s problem for complex reflection groups

We solve the noncommutative Noether's problem for the reflection groups by showing that the skew field of the invariants of the Weyl algebra under the action of any reection group is a Weyl field, that is isomorphic to a skew field of some Weyl algebra over a transcendental extension of the ground field. We also extend this result to the invariants of the ring of differential operators on any dimensional torus.The results are applied to obtain analogs of the Gelfand-Kirillov Conjecture for Cherednik algebras and Galois algebras.

Noether's problem on semidirect product groups

Let $K$ be a field, $G$ a finite group. Let $G$ act on the function field $L = K(x_{\sigma} : \sigma \in G)$ by $\tau \cdot x_{\sigma} = x_{\tau\sigma}$ for any $\sigma, \tau \in G$. Denote the fixed field of the action by $K(G) = L^{G} = \left\{ \frac{f}{g} \in L : \sigma(\frac{f}{g}) = \frac{f}{g}, \forall \sigma \in G \right\}$. Noether's problem asks whether $K(G)$ is rational (purely transcendental) over $K$. It is known that if $G = C_m \rtimes C_n$ is a semidirect product of cyclic groups $C_m$ and $C_n$ with $\mathbb{Z}[\zeta_n]$ a unique factorization domain, and $K$ contains an $e$th primitive root of unity, where $e$ is the exponent of $G$, then $K(G)$ is rational over $K$. In this paper, we give another criteria to determine whether $K(C_m \rtimes C_n)$ is rational over $K$. In particular, if $p, q$ are prime numbers and there exists $x \in \mathbb{Z}[\zeta_q]$ such that the norm $N_{\mathbb{Q}(\zeta_q)/\mathbb{Q}}(x) = p$, then $\mathbb{C}(C_{p} \rtimes C_{q})$ is rational over $\mathbb{C}$.

Toward Noether's Problem for the Fields of Cross-ratios

In this article, we consider an analogue of Noether's problem for the fields of cross-ratios, and discuss on a rationality problem which connects this with Noether's problem. We show that the affirmative answer of the analogue implies the affirmative answer for Noether's Problem for any permutation group with odd degree. We also obtain some negative results for various permutation groups with even degree.

Ulrich sheaves and higher-rank Brill–Noether theory

We show that the existence of an Ulrich sheaf on a projective variety X ⊆ P N is equivalent to the solution of a (possibly) higher-rank Brill–Noether problem for a curve on X that is rarely general in moduli. In addition, we exhibit a large family of curves for which this Brill–Noether problem admits a solution, and we show that existence of an Ulrich sheaf for a finite morphism of smooth projective varieties of any dimension implies sharp numerical constraints involving the degree of the map and the ramification divisor.

The q-difference Noether problem for complex reflection groups and quantum OGZ algebras

ABSTRACTFor any complex reflection group G = G(m,p,n), we prove that the G-invariants of the division ring of fractions of the n:th tensor power of the quantum plane is a quantum Weyl field and give explicit parameters for this quantum Weyl field. This shows that the q-difference Noether problem has a positive solution for such groups, generalizing previous work by Futorny and the author [10]. Moreover, the new result is simultaneously a q-deformation of the classical commutative case and of the Weyl algebra case recently obtained by Eshmatov et al. [8].Second, we introduce a new family of algebras called quantum OGZ algebras. They are natural quantizations of the OGZ algebras introduced by Mazorchuk [18] originating in the classical Gelfand–Tsetlin formulas. Special cases of quantum OGZ algebras include the quantized enveloping algebra of 𝔤𝔩n and quantized Heisenberg algebras. We show that any quantum OGZ algebra can be naturally realized as a Galois ring in the sense of Futorny-Ovsienko [11], with symmetry group being a direct product of complex reflection groups G(m,p,rk).Finally, using these results, we prove that the quantum OGZ algebras satisfy the quantum Gelfand–Kirillov conjecture by explicitly computing their division ring of fractions.

The Noether problem for Hopf algebras

In previous work, Eli Aljadeff and the first-named author attached an algebra \mathcal B_H of rational fractions to each Hopf algebra H . The generalized Noether problem is the following: for which finite-dimensional Hopf algebras H is \mathcal B_H the localization of a polynomial algebra? A positive answer to this question when H is the algebra of functions on a finite group G implies a positive answer to the classical Noether problem for G . We show that the generalized Noether problem has a positive answer for all finite-dimensional pointed Hopf algebras over a field of characteristic zero (we actually give a precise description of \mathcal B_H for such a Hopf algebra). A theory of polynomial identities for comodule algebras over a Hopf algebra H gives rise to a universal comodule algebra whose subalgebra of coinvariants \mathcal V_H maps injectively into \mathcal B_H . In the second half of this paper, we show that \mathcal B_H is a localization of \mathcal V_H when H is a finite-dimensional pointed Hopf algebra in characteristic zero.We also report on a result by Uma Iyer showing that the same localization result holds when H is the algebra of functions on a finite group.

Bogomolov multipliers for some p-groups of nilpotency class 2

The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of $G$. The triviality of the Bogomolov multiplier is an obstruction to Noether's problem. We show that if $G$ is a central product of $G_1$ and $G_2$, regarding $K_i\leq Z(G_i), i=1,2$, and $\theta:G_1\to G_2$ is a group homomorphism such that its restriction $\theta\vert_{K_1}:K_1\to K_2$ is an isomorphism, then the triviality of $B_0(G_1/K_1), B_0(G_1)$ and $B_0(G_2)$ implies the triviality of $B_0(G)$. We give a positive answer to Noether's problem for all $2$-generator $p$-groups of nilpotency class $2$, and for one series of $4$-generator $p$-groups of nilpotency class $2$ (with the usual requirement for the roots of unity).