A-module Structure Research Articles

Cyclicity of Drinfeld modules

Let X be a smooth, projective curve defined over a finite field F r , and ∞ a (closed) point of X. Let κ be the function field of X and A the elements of κ regular everywhere except possibly at ∞. Let ϕ : A → K { τ } be a Drinfeld module defined over K, where ι : A → K is an injective map that identifies K as a finite extension of κ. Suppose that the rank of ϕ is n ⩾ 2 . For a place ℘ of good reduction for ϕ, reducing the coefficients of ϕ modulo ℘ equips the residue field F ℘ with an A-module structure. We establish that the number of places ℘ of K of good reduction for ϕ, of degree x, and such that ϕ ( F ℘ ) is a cyclic A-module, has a natural density. Furthermore, this density is positive if and only if there are infinitely many primes ℘ such that ϕ ( F ℘ ) is cyclic as an A-module. We do not make further restrictions on either A or the ring of endomorphisms End K sep ( ϕ ) .

On noncommutative equivariant bundles

We discuss a possible noncommutative generalization of the notion of an equivariant vector bundle. Let A be a -algebra, M a left A-module, H a Hopf -algebra, an algebra coaction, and let denote with the right A-module structure induced by δ. The usual definitions of equivariant vector bundle naturally lead, in the context of -algebras, to an -module homomorphismthat fulfills some appropriate conditions. On the other hand, sometimes an (A, H)-Hopf module is considered instead, for the same purpose. When Θ is invertible, as is always the case when H is commutative, the two descriptions are equivalent. We point out that the two notions differ in general, by giving an example of a noncommutative Hopf algebra H for which there exists such a Θ that is not invertible and a left-right (A, H)-Hopf module whose corresponding homomorphism is not an isomorphism.

A function field analogue of Romanoff's theorem

We prove an analogue for Drinfeld modules of a theorem of Romanoff. Specifically, let ϕ be a Drinfeld A-module over a global function field L and denote by ϕ(L) the A-module structure on L coming from ϕ. Let Γ ⊂ ϕ (L) be a free A-submodule of finite rank. For each effective divisor [Formula: see text] of L, let fΓ(𝔇) be the cardinality of the image of the reduction map [Formula: see text] if all elements of Γ are relatively prime to the divisor 𝔇; otherwise, just set fΓ(𝔇) = ∞. We give explicit upper bounds for the series [Formula: see text] and [Formula: see text].

Primitive submodules for Drinfeld modules

AbstractThe ring A = $\mathbb{F}$r[T] and its fraction field k, where r is a power of a prime p, are considered as analogues of the integers and rational numbers respectively. Let K/k be a finite extension and let φ be a Drinfeld A-module over K of rank d and Γ ⊂ K be a finitely generated free A-submodule of K, the A-module structure coming from the action of φ. We consider the problem of determining the number of primes ℘ of K for which the reduction of Γ modulo ℘ is equal to $\mathbb{F}$℘ (the residue field of the prime ℘). We can show that there is a natural density of primes ℘ for which Γ mod ℘ is equal to $\mathbb{F}$℘. In certain cases, this density can be seen to be positive.

Cohomology of the de Rham complex twisted by the oscillatory representation

We introduce a Hilbert A-module structure on the higher oscillatory module, where A denotes the C⁎-algebra of continuous endomorphisms of the basic oscillatory module. We also define the notion of an exterior covariant derivative in an A-Hilbert bundle and use it for a construction of an A-elliptic complex of differential operators for certain symplectic manifolds equipped with a flat symplectic connection.

Erratum to “Support Varieties and Representation Type of Small Quantum Groups”

Some of the general results in the paper require an additional hypothesis, such as quasitriangularity. Applications to specific types of Hopf algebras are correct, as some of these are quasitriangular, and for those that are not, the Hochschild support variety theory may be applied instead. Let A be a finite dimensional Hopf algebra over a field k. The vector space H∗(A, k) := ExtA(k, k) is an associative, graded commutative k-algebra under the cup product, or equivalently under Yoneda composition. If M and N are finitely generated left A-modules, then H∗(A, k) acts on ExtA(M,N) via the cup product, or equivalently by −⊗N followed by Yoneda composition. Let S be the composition inverse of the antipode S. The isomorphism towards the top of p. 1350 in [3], Ext q A(M,N) ∼= Ext q A(k,M ⊗N), assumes the following A-module structure on M∗ = Homk(M,k), which was not stated explicitly in the paper: (a · f)(m) = ∑ a2f(S(a1)m) for all a ∈ A, f ∈ Homk(M,k), and m ∈M . Under some finiteness assumptions as in [3], we recall the definition of support variety: Let M be a finitely generated left A-module. Let IA(M) be the annihilator of the action of H(A, k) on Ext q A(M,M), and let VA(M) denote the maximal ideal spectrum of the finitely generated commutative k-algebra H(A, k)/IA(M). In part (5) of [3, Proposition 2.4], it is stated that (*) VA(M ⊗N) ⊆ VA(M) ∩ VA(N). Our proof of this statement requires an additional hypothesis such as quasitriangularity of A, as we will explain. However, (*) holds under other hypotheses, for example, whenever VA(M) = VA(M) for all finitely generated A-modules M , as follows from the fact that dualization reverses the order of tensor products. For a counterexample to the general statement (*), see [1]. In the proof of (*) given in [3], the action of H(A, k) on Ext q A(M ⊗N,M ⊗N) does indeed factor through its action on Ext q A(M,M) as stated, since we may first apply − ⊗M , then − ⊗ N , then Yoneda composition. If A is quasitriangular, then M ⊗N ∼= N ⊗M , so the action factors through that on Ext q A(N,N) as well, Date: August 22, 2013. 1 2 JORG FELDVOSS AND SARAH WITHERSPOON and consequently (*) holds. However, it does not necessarily factor through its action on Ext q A(N,N) in general: Under the isomorphism on Ext that is used in the proof, Ext q A(M⊗N,M⊗N) ∼= Ext q A(N,M ∗⊗M⊗N), the actions of H(A, k) do not always coincide. The order of the tensor product affects the action. Thus we obtain VA(M ⊗N) ⊆ VA(M) in general, but not necessarily (*). Theorem 2.5, Corollary 2.6, Theorem 3.1, and Corollary 3.2 of [3] require an additional hypothesis such as quasitriangularity of A, since they use [3, Proposition 2.4(5)]. The proof of [3, Theorem 4.3] is incorrect, or at least incomplete; the Hopf algebras uq (g) are not in general quasitriangular. However, Hochschild support variety theory [2, 4] provides an alternative proof of this statement: By [4, Lemma 6.3] or [5, §2], the representation type of uq (g) is the same as that of u q (g), and we may apply [4, Theorem 5.4] to the latter algebra. The remainder of the proofs of the statements in [3, Section 4] are correct, as the Hopf algebras uq(g) are quasitriangular. We note that our results in [4] almost exclusively use the Hochschild support variety theory, and do not require any additional hypotheses. This theory is more general, and does not take advantage of the tensor products of modules one has at hand for a Hopf algebra. One may obtain stronger results by taking advantage of tensor products of modules, yet the constructions are necessarily one-sided in general. The one-sidedness affects results if the tensor product is not commutative up to isomorphism.

The mod 2 homology of infinite loopspaces

We study the spectral sequence that one obtains by applying mod 2 homology to the Goodwillie tower which sends a spectrum X to the suspension spectrum of its 0th space X_0. This converges strongly to H_*(X_0) when X is 0-connected. The E^1 term is the homology of the extended powers of X, and thus is a well known functor of H_*(X), including structure as a bigraded Hopf algebra, a right module over the mod 2 Steenrod algebra A, and a left module over the Dyer-Lashof operations. Hopf algebra considerations show that all pages of the spectral sequence are primitively generated, with primitives equal to a subquotient of the primitives in E^1. We use an operad structure on the tower and the Z/2 Tate construction to show how Dyer-Lashof operations and differentials interact. These then determine differentials that hold for any spectrum X. These universal differentials then lead us to construct, for every A-module M, an algebraic spectral sequence depending functorially on M. The algebraic spectral sequence for H_*(X) agrees with the topological spectral sequence for X for many spectra, including suspension spectra and almost all generalized Eilenberg-MacLane spectra, and appears to give an upper bound in general. The E^infty term of the algebraic spectral sequence has form and structure similar to E^1, but now the right A-module structure is unstable. Our explicit formula involves the derived functors of destabilization as studied in the 1980's by W. Singer, J. Lannes and S. Zarati, and P. Goerss.

The A-module structure induced by a Drinfeld A-module of rank 2 over a finite field

Let F q be a finite field and let L / F q be a finite extension. Let F be the Frobenius of L ( F : x ↦ x # L ) and let ( P ) be the F [ T ] -characteristic of F. Let m be the degree of the extension L / F q [ T ] / ( P ) . There exists then c ∈ F q [ T ] and μ ∈ F q such that the characteristic polynomial P F of F is equal to P F ( X ) = X 2 − c X + μ P m . Our main result is an analogue of Deuring's Theorem on elliptic curves: let M = F q [ T ] ( i 1 ) ⊕ F q [ T ] ( i 2 ) , where i 1 and i 2 are two polynomials of F q [ T ] such that i 2 | i 1 and i 2 | ( c − 2 ) , there exists an ordinary Drinfeld F q [ T ] -module Φ of rank 2 over L such that the structure of the finite F q [ T ] -module L Φ induced by Φ over L is isomorphic to M. To cite this article: M.-S. Mohamed-Ahmed, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Twisted Weyl groups of Lie groups and nonabelian cohomology

For a cyclic group A and a connected Lie group G with an A-module structure (with the additional assumptions that G is compact and the A-module structure on G is 1-semisimple if $$A\cong{\mathbb{Z}}$$ ), we define the twisted Weyl group W = W(G,A,T), which acts on T and H 1(A,T), where T is a maximal compact torus of $$G_0^A$$ , the identity component of the group of invariants G A . We then prove that the natural map $$W\backslash H^1(A,T)\rightarrow H^1(A,G)$$ is a bijection, reducing the calculation of H 1(A,G) to the calculation of the action of W on T. We also prove some properties of the twisted Weyl group W, one of which is that W is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.

The structure of smooth algebras in Kapranov's framework for noncommutative geometry

In [M. Kapranov, Noncommutative geometry based on commutator expansions, J. Reine Angew. Math. 505 (1998) 73–118] a theory of noncommutative algebraic varieties was proposed. Here we prove a structure theorem for the noncommutative coordinate rings of affine open subsets of such of those varieties which are smooth (Theorem 3.4). The theorem describes the local ring of a point as a truncation of a quantization of the enveloping Poisson algebra of a smooth commutative local algebra. An explicit description of this quantization is given in Theorem 2.5. A description of the A-module structure of the Poisson envelope of a smooth commutative algebra A was given in loc. cit., Theorem 4.1.3. However the proof given in loc. cit. has a gap. We fix this gap for A local (Theorem 1.4) and prove a weaker global result (Theorem 1.6).

Module Amenability for Semigroup Algebras

We extend the concept of amenability of a Banach algebra A to the case that there is an extra \A-module structure on A, and show that when S is an inverse semigroup with subsemigroup E of idempotents, then A = \ell1(S) as a Banach module over \A = \ell1(E) is module amenable if and only if S is amenable. When S is a discrete group, \ell1(E) = ℂ and this is just the Johnson’s theorem.

Fast Algorithms for Zero-Dimensional Polynomial Systems using Duality

Many questions concerning a zero-dimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A=k[X 1 ,…,X n ]/ℐ, where ℐ is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the A-module structure of the dual space $\widehat{A}$ . An important feature of our algorithms is that we do not require $\widehat{A}$ to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 nD 5/2 ) and O(n2 nD 5/2 ) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.

Local–global problem for Drinfeld modules

Let K be a function field with an A-algebra structure. The ring A arises in the definition of the Drinfeld module φ over K. By E( K) we denote K together with the A-module structure induced on it by φ. For any principal prime ideal ( a)⊂ A, we study the question whether an element x∈ E( K) which is an a-fold in E( K ν ) for every place ν of K, is an a-fold in E( K). In particular, we study the group S(a,K)≔ ker E(K)/aE(K)→ ∏ ν E(K ν)/aE(K ν) for Drinfeld modules of rank 2. We show that this finite group is trivial in many cases, but can become arbitrarily large.

On some congruences for units of localp-cyclotomic fields

Let p be a fixed odd prime number, and let kn (n > 0) be the local pn+l-st cyclotomic field Qp(#p,+~) and k~ = t2 kn. Let lIn be the group of principal units of kn and let ~B, be the intersection of Nm/nlIm of all m with m >_ n, Nm/n being the norm map from km x to kn x. Further, let 1I = [~lI, = ~_~Bn be the projective limit w.r.t, the relative norms. The Galois groups A = Gal(ko/lI~p) and F = Gal(k~/ko) act on these groups in a natural way. We fix the topological generator 7 of F such that ~ = ~l+p for all pa-th roots r of unity (a >_ 1). We identify, as usual, the completed group ring 7Zv[[F]] with the power series ring A = Zp[[S]] by ~ = 1 + s. For a (Qp-valued) character X of A, denote by U(X), Iln(X) and ~Bn(X) the x-eigenspaces of 1[, lI. and ~Bn respectively. They are considered as modules over A in a natural way. In [8], the A-module structures of U(X) and ~Bn(X) were determined (see the Fact in Section 2). The purpose of the present article is to determine the A-module structure of the submodule

Circular bar construction

Let A be a CDG (commutative differential graded) algebra over a field of characteristic 0, and B(A), the bar construction of A. If C is a DG (differential graded) A @ Amodule, define B(A; C) = C BAOR B(A). If C’ and C” are DG A-modules, then B(A; C’ @ C”) coincides with the usual two sided bar construction B(C’, A, C”). (See [12].) In this sense, B(A; C) closes the two sides of the bar construction and is therefore “circular”. The geometric significance of B(A; C) lies in the case where A = A(M) and C = A(N) are the Rham complexes and the A @ A-module structure of C is induced by a differentiable map f : N-t M x M. Theorem 0.1[6] (Theorem 4.3.1[5]) asserts that, if M and N are differentiable manifolds and if M is simply connected with finite Betti numbers, then H(B(A; C)) is the real cohomology of the pullback space Ef of the following pullback diagram of the free path fibration of M: