Jaffard-Ohm correspondence and Hochster duality

We study categorical aspects of the Jaffard–Ohm correspondencebetween abelian l-groups and Bézout domains and showthat this correspondence is close to a localization. For thispurpose, we establish a general extension theorem for valuationswith value group that is an abelian l-group. As an application,we prove Anderson's conjecture which refines the Jaffard–Ohmcorrespondence. We then extend the correspondence to sheaveson spectral spaces and show that the spectrum of a Bézoutdomain and the spectrum of its corresponding abelian l-groupprovide a concrete example for Hochster's duality of spectralspaces.     

Q-Curves and Abelian Varieties of GL2-Type

The relation between Q-curves and certain abelian varietiesof GL₂-type was established by Ribet (‘Abelian varietiesover Q and modular forms’, Proceedings of the KAIST MathematicsWorkshop (1992) 53–79) and generalized to building blocks,the higher-dimensional analogues of Q-curves, by Pyle in herPhD Thesis (University of California at Berkeley, 1995). Inthis paper we investigate some aspects of Q-curves with no complexmultiplication and the corresponding abelian varieties of GL₂-type,for which we mainly use the ideas and techniques introducedby Ribet (op. cit. and ‘Fields of definition of abelianvarieties with real multiplication’, Contemp.\ Math. 174(1994) 107–118). After the Introduction, in Sections 2and 3 we obtain a characterization of the fields where a Q-curveand all the isogenies between its Galois conjugates can be definedup to isogeny, and we apply it to certain fields of type (2,...,2).In Section 4 we determine the endomorphism algebras of all theabelian varieties of GL₂-type having as a quotient a given Q-curvein easily computable terms. Section 5 is devoted to a particularcase of Weil's restriction of scalars functor applied to a Q-curve,in which the resulting abelian variety factors over Q up toisogeny as a product of abelian varieties of GL₂-type. Finally,Section 6 contains examples: we parametrize the Q-curves comingfrom rational points of the modular curves X*N having genuszero, and we apply the results of Sections 2–5 to someof the curves obtained. We also give results concerning theexistence of quadratic Q-curves. 1991 Mathematics Subject Classification:primary 11G05; secondary 11G10, 11G18, 11F11, 14K02.     

Derived Functors of Inverse Limits Revisited

We prove, correct and extend several results of an earlier paperof ours (using and recalling several of our later papers) aboutthe derived functors of projective limit in abelian categories.In particular we prove that if C is an abelian category satisfyingthe Grothendieck axioms AB3 and AB4* and having a set of generatorsthen the first derived functor of projective limit vanisheson so-called Mittag-Leffler sequences in C. The recent examplesgiven by Deligne and Neeman show that the condition that thecategory has a set of generators is necessary. The conditionAB4* is also necessary, and indeed we give for each integerm 1 an example of a Grothendieck category C_m and a Mittag-Lefflersequence in C_m for which the derived functors of its projectivelimit vanish in all positive degrees except m. This leads toa systematic study of derived functors of infinite productsin Grothendieck categories. Several explicit examples of theapplications of these functors are also studied.     

Characterizing Automorphism Groups of Ordered Abelian Groups

A proof is given of the following theorem, which characterizesfull automorphism groups of ordered abelian groups: a groupH is the automorphism group of some ordered abelian group ifand only if H is right-orderable. 2000 Mathematics Subject Classification20K15, 20K20, 20F60, 20K30 (primary); 03E05 (secondary).     

Derived Subgroups of Products of an Abelian and a Cyclic Subgroup

Let G be a finite group and suppose that G = AB, where A andB are abelian subgroups. By a theorem of Ito, the derived subgroupG' is known to be abelian. If either of the subgroups A or Bis cyclic, then more can be said. The paper shows, for example,that G'/(G'A) is isomorphic to a subgroup of B in this case.     

Non-Nesting Actions On Real Trees

The theory of isometric group actions on R-trees is extendedto actions by homeomorphisms with the following non-nestingproperty: no group element maps an arc properly into itself.A finitely presented group acting freely by homeomorphisms onan R-tree is free abelian or splits over a (possibly trivial)cyclic group. 1991 Mathematics Subject Classification 20E08,20F32, 57M60.     

Hochschild cohomology of abelian categories and ringed spaces

This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild cohomology coincides with the one defined by Gerstenhaber, Schack and Swan in the case of module categories over diagrams and schemes and also with the Hochschild cohomology for exact categories introduced recently by Keller. In addition we show in complete generality that Hochschild cohomology satisfies a Mayer-Vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf.     

Deformations of locally abelian Galois representations and unramified extensions

We study the deformation theory of Galois representations whose restriction to every decomposition subgroup is abelian. As an application, we construct unramified non-solvable extensions over the field obtained by adjoining all p-power roots of unity to the field of rational numbers.     

Periodic Points for Expansive Actions of Zd on Compact Abelian Groups

In this note we show that the periodic points of an expansiveZ^d action on a compact abelian group are uniformly distributedwith respect to Haar measure if the action has completely positiveentropy. In the general expansive case, we show that any measureobtained as the distribution of periodic points along some sequenceof periods necessarily has maximal entropy but need not be Haarmeasure.     

Ergodicity and mixing of non-commuting epimorphisms

We study mixing properties of epimorphisms of a compact connectedfinite-dimensional abelian group X. In particular, we show thata set F, with |F| > dim X, of epimorphisms of X is mixingif and only if every subset of F of cardinality (dim X) + 1is mixing. We also construct examples of free non-abelian groupsof automorphisms of tori which are mixing, but not mixing oforder 3, and show that, under some irreducibility assumptions,ergodic groups of automorphisms contain mixing subgroups andfree non-abelian mixing subsemigroups.     

Every projective Schur algebra is Brauer equivalent to a radical abelian algebra

We prove that any projective Schur algebra over a field K isequivalent in Br(K) to a radical abelian algebra. This was conjecturedin 1995 by Sonn and the first author of this paper. As a consequence,we obtain a characterization of the projective Schur group bymeans of Galois cohomology. The conjecture was known for algebrasover fields of positive characteristic. In characteristic zerothe conjecture was known for algebras over fields with a Henselianvaluation over a local or global field of characteristic zero.     

Abelian Subgroups of Finitely Generated Kleinian Groups are Separable

By a Kleinian group we mean a discrete subgroup of PSL(2, C).We prove that abelian subgroups of finitely generated Kleiniangroups are separable. In other words, if M = H³/ is a hyperbolic3-orbifold, with finitely generated, then abelian subgroupsof are separable in . 1991 Mathematics Subject Classification20E26, 51M10, 57M05.     

Connected Lie Groups that Act Freely on a Product of Linear Spheres

In this note we prove that a compact connected Lie group G admitsa free action on some product of linear spheres if and onlyif it is isomorphic to (T^k x SU(2)^l)/Z for some k and l andfor some central elementary abelian 2-subgroup Z with Z SU(2)M^l= 1.     

On the non-existence of exceptional automorphisms on Shimura curves

We study the group of automorphisms of Shimura curves X₀(D,N) attached to an Eichler order of square-free level N in anindefinite rational quaternion algebra of discriminant D>1.We prove that, when the genus g of the curve is greater thanor equal to 2, Aut (X₀(D, N)) is a 2-elementary abelian groupwhich contains the group of Atkin–Lehner involutions W₀(D,N) as a subgroup of index 1 or 2. It is conjectured that Aut(X₀(D, N))=W₀(D, N) except for finitely many values of (D, N)and we provide criteria that allow us to show that this is indeedoften the case. Our methods are based on the theory of complexmultiplication of Shimura curves and the Cerednik–Drinfeldtheory on their rigid analytic uniformization at primes p| D.     

On A Property Of Minimal Zero-Sum Sequences And Restricted Sumsets

Let G be an additively written abelian group, and let S be asequence in G \ {0} with length |S| 4. Suppose that S is aproduct of two subsequences, say S = BC, such that the elementg + h occurs in the sequence S whenever g.h is a subsequenceof B or of C. Then S contains a proper zero-sum subsequence,apart from some well-characterized exceptional cases. This resultis closely connected with restricted set addition in abeliangroups. Moreover, it solves a problem on the structure of minimalzero-sum sequences, which recently occurred in the theory ofnon-unique factorizations. 2000 Mathematics Subject Classification11B50, 11B75, 11P99.     

Partial Difference Sets with Paley Parameters

Partial difference sets with parameters (,k,,µ) = (,(– 1)/2, ( – 5)/4,( – 1)/4) are called Paleypartial difference sets. By using finite local rings, we constructa family of Paley PDSs for abelian p-groups with any given exponent.Furthermore, we prove some non-existence results on Paley PDSs.Using these results, we prove that Paley PDSs exist in a rank2 abelian group if and only if the group is isomorphic to Z_p^r x Z_{p ^r} where p is an odd prime.     

Broue's Conjecture for the Hall-Janko Group and its Double Cover

Broué's abelian defect conjecture suggests a deep linkbetween the module categories of a block of a group algebraand its Brauer correspondent, viz. that they should be derivedequivalent. We are able to verify Broué's conjecturefor the Hall–Janko group, even its double cover 2.J₂,as well as for U₃(4) and Sp₄(4). In fact we verify Rickard'srefinement to Broué's conjecture and show that the derivedequivalence can be chosen to be a splendid equivalence for theseexamples. 2000 Mathematical Subject Classification: 20C20, 20C34.     

A Geometric Invariant for Metabelian Pro-p Groups

In [2] Bieri and Strebel introduced a geometric invariant forfinitely generated abstract metabelian groups that determineswhich groups are finitely presented. For a valuable survey oftheir results, see [6]; we recall the definition briefly inSection 4. We shall introduce a similar invariant for pro-pgroups. Let F be the algebraic closure of F_p and U be the formal powerseries algebra F[T], with group of units U^x. Let Q be a finitelygenerated abelian pro-p group. We write Z_p[Q] for the completedgroup algebra of Q over Z_p. Let T(Q) be the abelian group Hom(Q,U^x) of continuous homomorphisms from Q to U^x. We write 1 forthe trivial homomorphism. Each vT(Q) extends to a unique continuousalgebra homomorphism from Z_p[Q]to U.