Crossed Product Orders Over Valuation Rings II: Tamely Ramified Crossed Product Algebras

Let V be a commutative valuation domain of arbitrary Krull-dimension,with quotient field F, let K be a finite Galois extension ofF with group G, and let S be the integral closure of V in K.Suppose that one has a 2-cocycle on G that takes values in thegroup of units of S. Then one can form the crossed product ofG over S, S*G, which is a V-order in the central simple F-algebraK*G. If S*G is assumed to be a Dubrovin valuation ring of K*G,then the main result of this paper is that, given a suitabledefinition of tameness for central simple algebras, K*G is tamelyramified and defectless over F if and only if K is tamely ramifiedand defectless over F. The residue structure of S*G is alsoconsidered in the paper, as well as its behaviour upon passageto Henselization. 2000 Mathematics Subject Classification 16H05,16S35.     

Crossed Product Orders over Valuation Rings

Let V be a commutative valuation domain of arbitrary Krull-dimension(rank), with quotient field F, and let K be a finite Galoisextension of F with group G, and S the integral closure of Vin K. If, in the crossed product algebra K * G, the 2-cocycletakes values in the group of units of S, then one can form,in a natural way, a ‘crossed product order’ S *G K * G. In the light of recent results by H. Marubayashi andZ. Yi on the homological dimension of crossed products, thispaper discusses necessary and/or sufficient valuation-theoreticconditions, on the extension K/F, for the V-order S * G to besemihereditary, maximal or Azumaya over V. 2000 MathematicsSubject Classification 16H05, 16S35.     

The Intersection of Two Infinite Matroids

Conjecture: Let M and N be two matroids (possibly of infiniteranks) on the same set S. Then there exists a set I independentin both M and N, which can be partitioned as I=HK, where sp_M(H)sp_N(K)=S.This conjecture is an extension of Edmonds' matroid intersectiontheorem to the infinite case. We prove the conjecture when oneof the matroids (say M) is the sum of countably many matroidsof finite rank (the other matroid being general). For the proofwe have also to answer the following question: when does thereexist a subset of S which is spanning for M and independentin N?     

Finitary Power Semigroups of Infinite Groups are not Finitely Generated

For a semigroup S, the finitary power semigroup of S, denotedP_f(S), consists of all finite subsets of S under the usual multiplication.The main result of this paper asserts that P_f(G) is not finitelygenerated for any infinite group G. 2000 Mathematics SubjectClassification 20M05 (primary), 20M30, 20F99 (secondary).     

Finiteness Results in Descent Theory

It is shown that a -curve of genus g and with stable reduction (in some generalized sense)at every finite place outside a finite set S can be definedover a finite extension L of its field of moduli K dependingonly on g, S and K. Furthermore, there exist L-models that inheritall places of good and stable reduction of the original curve(except possibly for finitely many exceptional places dependingon g, K and S). This descent result yields this moduli formof the Shafarevich conjecture: given g, K and S as above, onlyfinitely many K-points on the moduli space M_g correspond to-curves of genus g and with good reduction outside S. Other applications to arithmetic geometry,like a modular generalization of the Mordell conjecture, aregiven.     

Noetherian Centralizing Hopf Algebra Extensions and Finite Morphisms of Quantum Groups

We study finite centralizing extensions AH of noetherian Hopfalgebras. Our main results provide necessary and sufficientconditions for the fibres of the surjection spec Hspec A tocoincide with the X-orbits in spec H, where X denotes the finitegroup of characters of H that restrict to the counit of A. Inparticular, all of the fibres are X-orbits if and only if thefibre over the augmentation ideal of A is an X-orbit. An applicationto the representation theory of quantum function algebras, atroots of unity, is presented. 1991 Mathematics Subject Classification16D30, 16S20, 16P40, 16W30, 81R50.     

Finsler Metrics of Constant Positive Curvature on the Lie Group S3

Guided by the Hopf fibration, a family (indexed by a positiveconstant K) of right invariant Riemannian metrics on the Liegroup S³ is singled out. Using the Yasuda–Shimada paperas an inspiration, a privileged right invariant Killing fieldof constant length is determined for each K > 1. Each suchRiemannian metric couples with the corresponding Killing fieldto produce a y-global and explicit Randers metric on S³. Employingthe machinery of spray curvature and Berwald's formula, it isproved directly that the said Randers metric has constant positiveflag curvature K, as predicted by Yasuda–Shimada. It isexplained why this family of Finslerian space forms is not projectivelyflat.     

An Euler-Type Version of the Local Steiner Formula for Convex Bodies

The purpose of this note is to establish a new version of thelocal Steiner formula and to give an application to convex bodiesof constant width. This variant of the Steiner formula generalizesresults of Hann [3] and Hug [6], who use much less elementarytechniques than the methods of this paper. In fact, Hann askedfor a simpler proof of these results [4, Problem 2, p. 900].We remark that our formula can be considered as a Euclideananalogue of a spherical result proved in [2, p. 46], and thatour method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensionalMinkowski spaces, see Hann [5, p. 363]. For further information about the notions used below, we referto Schneider's book [9]. Let Kⁿ be the set of all convex bodiesin Euclidean space Rⁿ, that is, the set of all compact, convex,non-empty subsets of Rⁿ. Let S^n–1 be the unit sphere.For KKⁿ, let NorK be the set of all support elements of K, thatis, the pairs (x, u)RⁿxS^n–1 such that x is a boundarypoint of K and u is an outer unit normal vector of K at thepoint x. The support measures (or generalized curvature measures)of K, denoted by ₀(K.), ..., _n–1(K.), are the unique Borelmeasures on RⁿxS^n–1 that are concentrated on NorK andsatisfy [formula] for all integrable functions f:RⁿR; here denotes the Lebesguemeasure on Rⁿ. Equation (1), which is a consequence and a slightgeneralization of Theorem 4.2.1 in Schneider [9], is calledthe local Steiner formula. Our main result is the following.1991 Mathematics Subject Classification 52A20, 52A38, 52A55.     

Chord Diagrams and Coxeter Links

The paper presents a construction of fibered links (K, ) outof chord diagrams L. Let be the incidence graph of L. Undercertain conditions on L the symmetrized Seifert matrix of (K,) equals the bilinear form of the simply-laced Coxeter system(W, S) associated to and the monodromy of (K, ) equals minusthe Coxeter element of (W, S). Lehmer's problem is solved forthe monodromy of these Coxeter links.     

Gorenstein Modules and Dualizing Modules

In this article, we study the characterizations of Gorenstein injective left S-modules and finitely generated Gorenstein projective left R-modules when there is a dualizing S-R-bimodule associated with a right noetherian ring R and a left noetherian ring S.     

The tower of K-theory of truncated polynomial algebras

Let A be a regular noetherian F_p-algebra. The relative K-groupsK_q(A[x]/(x^m),(x)) and the Nil-groups Nil_q(A[x]/(x^m)) were evaluatedby the author and Ib Madsen in terms of the big de Rham–Wittgroups W_rA^q of the ring A. In this paper, we evaluate the mapsof relative K-groups and Nil-groups induced by the canonicalprojection f: A[x]/(x^m) A[x]/(xⁿ). The result depends stronglyon the prime p. It generalizes earlier work by Stienstra onthe groups in degrees 2 and 3. Received February 28, 2007.     

Reduction of derived Hochschild functors over commutative algebras and schemes

We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analogous isomorphisms for any essentially finite-type flat map of noetherian schemes, with f^!OY in place of D.     

Cofiniteness of Local Cohomology Modules for Principal Ideals

In this note we show that if an ideal I of a noetherian ringis principal, up to radical, then the local cohomology moduleswith support in V(I) are I-cofinite. 1991 Mathematics SubjectClassification 14B15, 13D03, 18G15.     

Non-Complex Symplectic 4-Manifolds with Formula

Simply connected closed symplectic 4-manifolds with and K² = 0 are investigated. As aresult, it is confirmed that most of homotopy elliptic surfaces{E(1)_k|K is a fibred knot in S³} constructed by R. Fintusheland R. Stern in Invent. Math. 134 (1998) 363–400 are simplyconnected closed minimal symplectic 4-manifolds that do notadmit a complex structure. 2000 Mathematics Subject Classification57R17, 57R57 (primary), 14J26 (secondary).     

Rank Properties of Endomorphisms of Infinite Partially Ordered Sets

The relative rank (S : U) of a subsemigroup U of a semigroupS is the minimum size of a set V S such that U together withV generates the whole of S. As a consequence of a result ofSierpiski, it follows that for U T_X, the monoid of all self-mapsof an infinite set X, rank(T_X : U) is either 0, 1 or 2, or uncountable.In this paper, the relative ranks rank(T_X : O_X) are considered,where X is a countably infinite partially ordered set and O_Xis the endomorphism monoid of X. We show that rank(T_X : O_X) 2 if and only if either: there exists at least one elementin X which is greater than, or less than, an infinite numberof elements of X; or X has |X| connected components. Four examplesare given of posets where the minimum number of members of T_Xthat need to be adjoined to O_X to form a generating set is,respectively, 0, 1, 2 and uncountable. 2000 Mathematics SubjectClassification 08A35 (primary), 06A07, 20M20 (secondary).     

Hopf-Galois Structures on Field Extensions with Simple Galois Groups

Let L/K be a finite Galois extension of fields whose Galoisgroup is a nonabelian simple group. It is shown that L/K admitsexactly two Hopf–Galois structures. 2000 Mathematics SubjectClassification 12F10, 16W30.     

The finite images of finitely generated groups

Given any sequence of non-abelian finite simple primitive permutationgroups S_n, we construct a finitely generated group G whose profinitecompletion is the infinite permutational wreath product ...S_n S_n–1 ... S₀. It follows that the upper compositionfactors of G are exactly the groups S_n. By suitably choosingthe sequence S_n we can arrange that G has any one of a continuousrange of slow, non-polynomial subgroup growth types. We alsoconstruct a 61-generator perfect group that has every non-abelianfinite simple group as a quotient. 2000 Mathematics SubjectClassification: 20E07, 20E08, 20E18, 20E32.     

On Closed Orbits of Morse-Smale Flows on 3-Manifolds

Any link in a 3-manifold is the closed orbits of a non-singularMorse-Smale flow after taking the split sum with the unlinkand the connected sum with S² x S¹s. Current address: Department of Mathematics, University of Toronto,Toronto, Ontario M5S 1A1, Canada     

Finiteness Properties of Certain Metabelian Arithmetic Groups in the Function Field Case

Consider the group scheme where R is an arbitrary commutative ring with 1 0 and a unitx R* acts on R by multiplication. We will study the finiteness properties of subgroups of G(O_S)where O_S is an S-arithmetic subring of a global function field.The subgroups we are interested in are of the form where Q is a subgroup of O_S*. The finiteness propertiesof these metabelian groups can be expressed in terms of the-invariant due to R. Bieri and R. Strebel. Theorem A. Let S be a finite set of places of a global functionfield (regarded as normalized discrete valuations) and O_S thecorresponding S-arithmetic ring. Let Q be a subgroup of O_S*.Then Q is finitely generated and for all integers n 1 the followingare equivalent:

(1) O_S Q is of type FP_n;

(2) O_S is n-tameas a ZQ-module;

(3) each p S restricts to a non-trivial homomorphism and the set is n-tame.

If these conditions hold for at least one n 1 then the identity holds.} Theorem B. Let r denote the rank of Q. Then the followinghold:

(1) the group O_S Q is not of type FP_r+1};

(2) if Qhas maximum rank r = |S| –1 then the group O_S Q is oftype FP_r.

In particular, is of type FP_{|S| –1} but not of type FP_|S|. 1991 Mathematics SubjectClassification: 20E08, 20F16, 20G30, 52A20.     

Common Multiples of Complete Graphs

A graph H is said to divide a graph G if there exists a setS of subgraphs of G, all isomorphic to H, such that the edgeset of G is partitioned by the edge sets of the subgraphs inS. Thus, a graph G is a common multiple of two graphs if eachof the two graphs divides G. This paper considers common multiples of a complete graph oforder m and a complete graph of order n. The complete graphof order n is denoted K_n. In particular, for all positive integersn, the set of integers q for which there exists a common multipleof K₃ and K_n having precisely q edges is determined. It is shown that there exists a common multiple of K₃ and K_nhaving q edges if and only if q 0 (mod 3), q 0 (mod n2) and (1) q 3 n2 when n 5 (mod 6); (2) q (n + 1) n2 when n is even; (3) q {36, 42, 48} when n = 4. The proof of this result uses a variety of techniques includingthe use of Johnson graphs, Skolem and Langford sequences, andequitable partial Steiner triple systems. 2000 MathematicalSubject Classification: 05C70, 05B30, 05B07.