Torsion Modules Research Articles

Noncommutative localisation in algebraicK–theory I

This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic K-theory. The main result goes as follows. Let A be an associative ring and let A-->B be the localisation with respect to a set sigma of maps between finitely generated projective A-modules. Suppose that Tor_n^A(B,B) vanishes for all n>0. View each map in sigma as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes D^perf(A). Denote by <sigma> the thick subcategory generated by these complexes. Then the canonical functor D^perf(A)-->D^perf(B) induces (up to direct factors) an equivalence D^perf(A)/<sigma>--> D^perf(B). As a consequence, one obtains a homotopy fibre sequence K(A,sigma)-->K(A)-->K(B) (up to surjectivity of K_0(A)-->K_0(B)) of Waldhausen K-theory spectra. In subsequent articles we will present the K- and L-theoretic consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of Tor_n^A(B,B), we also assume that every map in sigma is a monomorphism, then there is a description of the homotopy fiber of the map K(A)-->K(B) as the Quillen K-theory of a suitable exact category of torsion modules.

Affine equivalence and Gorensteinness

Recently, Dwyer and Greenless established a Morita-like equivalence between categories consisting of complete modules and torsion modules. It turns out that these categories contain certain full subcategories which may be viewed as "perturbed" Auslander and Bass classes; Auslander and Bass classes are used in the study of so-called Gorenstein dimensions. This observation allows us to prove that any ideal in a commutative, local, Noetherian ring can detect whether or not the underlying ring is Gorenstein.

Noncommutative knot theory

The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S^3-K, considered as a module over the (commutative) Laurent polynomial ring, and the Blanchfield linking pairing defined on this module. From the perspective of the knot group, G, these invariants reflect the structure of G^(1)/G^(2) as a module over G/G^(1) (here G^(n) is the n-th term of the derived series of G). Hence any phenomenon associated to G^(2) is invisible to abelian invariants. This paper begins the systematic study of invariants associated to solvable covering spaces of knot exteriors, in particular the study of what we call the n-th higher-order Alexander module, G^(n+1)/G^(n+2), considered as a Z[G/G^(n+1)$-module. We show that these modules share almost all of the properties of the classical Alexander module. They are torsion modules with higher-order Alexander polynomials whose degrees give lower bounds for the knot genus. The modules have presentation matrices derived either from a group presentation or from a Seifert surface. They admit higher-order linking forms exhibiting self-duality. There are applications to estimating knot genus and to detecting fibered, prime and alternating knots. There are also surprising applications to detecting symplectic structures on 4-manifolds. These modules are similar to but different from those considered by the author, Kent Orr and Peter Teichner and are special cases of the modules considered subsequently by Shelly Harvey for arbitrary 3-manifolds.

Stacked submodules of torsion modules over discrete valuation domains

A submodule W of a torsion module M over a discrete valuation domain is called stacked in M if there exists a basis ℬ of M such that multiples of elements of ℬ form a basis of W. We characterise those submodules which are stacked in a pure submodule of M.

Principal Rings and their Invariant Factors

In this article we put forward a new look at the theory of principal and Invariant-Factor rings, with a view toward facilitating the formalization, automation, and archiving of results and their proofs. We take an elementary and constructive approach: standard techniques such as prime ideals and factorization of elements are avoided, and determinant constructions are minimized. Using such ‘‘computationally friendly’’ methods, the main existence and uniqueness results on invariant factors for a f.g. torsion module are derived, and several new algebraic constructions and results are found. The lattice of principal integral ideals for any commutative Bezoutian ring is explicitly constructed based on a first-order proof overlooked in the literature, together with a proof that this lattice is distributive. A ‘‘Lagrange quotient’’ theorem for finitely generated modules over any principal ring is stated for the first time. A very constructive new proof is given that a principal ring has the Hermite property, so is also an Invariant-Factor ring. A calculus that is needed in the ideal lattice, naturally yields a number of formulas valid for a function lattice.

Links between associated additive Galois modules and computation of H1 for local formal group modules

Using the methods described in the papers (Documenta Math. 5 (2000) 657; Local Leopoldt's problem for ideals in p-extensions of complete discrete valuation fields, to appear), we prove that a cocycle for a formal group in a Galois p-extension of a complete discrete valuation field is a coboundary if and only if the corresponding group algebra elements increase valuations by a number that is sufficiently large. We also calculate the valuation of the splitting element of a coboundary. A special case of the main theorem allows us to determine when a p-extension of a complete discrete valuation fields contains a root of a Kummer equation for a formal group. The theorem of Coates–Greenberg for formal group modules in deeply ramified extensions is generalized to noncommutative formal groups. Some results concerning finite torsion modules for formal groups are obtained.

MODULES OVER IWASAWA ALGEBRAS

Let be a compact -valued -adic Lie group, and let be its Iwasawa algebra. The present paper establishes results about the structure theory of finitely generated torsion -modules, up to pseudo-isomorphism, which are largely parallel to the classical theory when is abelian (except for basic differences which occur for those torsion modules which do not possess a non-zero global annihilator). We illustrate our general theory by concrete examples of such modules arising from the Iwasawa theory of elliptic curves without complex multiplication over the field generated by all of their -power torsion points.AMS 2000 Mathematics subject classification: Primary 11G05; 11R23; 16D70; 16E65; 16W70

THE DUAL OF THE TORSION MODULE OF DIFFERENTIALS

ABSTRACT In this paper we will establish a duality between the torsion module of differentials, , and the highest non-zero exterior power of the module of Kähler differentials, , of reduced affine hypersurfaces with only isolated singularities in , where is an algebraically closed field of characteristic zero. In an earlier paper [5] we have shown that: However by[5] the two sides are not isomorphic as -modules. Let denote the polynomial ring in variables over , and let denote the Jacobian ideal of the hypersurface. Both -modules also have an -module structure. We will show that when considered as -module, is dual to the torsion module of differentials, .

Foxby equivalence, complete modules, and torsion modules

Taking the idea from classical Foxby equivalence, we develop an equivalence theory for derived categories over differential graded algebras. Both classical Foxby equivalence and the Morita equivalence for complete modules and torsion modules developed by Dwyer and Greenlees arise as special cases.

Affine hypersurfaces with Gorenstein singular loci

The main result of this paper presents necessary and sufficient criteria for the torsion module of differentials of an affine hypersurface with isolated singularities to be cyclic.

Abelian groups that are torsion over their endomorphism rings

Using Lambek torsion as the torsion theory, we investigate the question of when an Abelian group G is torsion as a module over its endomorphism ring E. Groups that are torsion modules in this sense are called ℒ-torsion. Among the classes of torsion and truly mixed Abelian groups, we are able to determine completely those groups that are ℒ-torsion. However, the case when G is torsion free is more complicated. Whereas no torsion-free group of finite rank is ℒ-torsion, we show that there are large classes of torsion-free groups of infinite rank that are ℒ-torsion. Nevertheless, meaningful definitive criteria for a torsion-free group to be ℒ-torsion have not been found.

On the Homflypt skein module of $S^1 \times S^2$

Let k be a subring of the field of rational functions in x, v,s which contains $x^{\pm 1}, v^{\pm 1}, s^{\pm 1}$ . If M is an oriented 3-manifold, let $S(M)$ denote the Homflypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the Homflypt skein relations: (1) $x^{-1}L_{+}-xL_{-}=(s-s^{-1})L_{0}$ ; (2) L with a positiv e twist $=(xv^{-1})L$ ; (3) $L\sqcup O=(\frac{v-v^{-1}}{s-s^{-1}})L$ where O is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the boundary. The first basis is related to the basis of $S(S^1\times D^2)$ given by J. Hoste and M. Kidwell and also V. Turaev; the second basis is related to a Young idempotent basis for $S(S^1\times D^2)$ based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements $s^{2n}-1$ , for n a nonzero integer, and the elements $s^{2m}-v^{2}$ , for any integer m, are invertible ink, then $S(S^{1} \times S^2)=k$ -torsion module $\oplus k$ . Here the free part is generated by the empty link $\phi$ . In addition, if the elements $s^{2m}-v^{4}$ , for m an integer, are invertible in k, then $S(S^{1} \times S^2)$ has no torsion. We also obtain some results for more general k.

DIFFERENTIALS OF THE FAMILY Y 2 − X 3 − tX 2

In this paper we study the K-vector space dimension of the torsion module of differentials of the family of affine, singular, plane curves: The main result proves that among all members of the family, the quasi-homogeneous member the cusp has maximum torsion. *Dr. Ruth Ingrid Michler, Associate Professor of Mathematics at the University of North Texas, was killed in a tragic accident on November 1, 2000 while visiting at Northeastern University for the year. †Correspondence and reprint requests can be directed to Prof. Anthony Iarrobino, E-mail: iarrobin@neu.edu or Prof. Marc Levine, E-mail: marc@neu.edu

Eigenvalues of majorized Hermitian matrices and Littlewood–Richardson coefficients

Answering a question raised by S. Friedland, we show that the possible eigenvalues of Hermitian matrices (or compact operators) A, B, and C with C⩽A+B are given by the same inequalities as in Klyachko's theorem for the case where C=A+B, except that the equality corresponding to tr(C)= tr(A)+ tr(B) is replaced by the inequality corresponding to tr(C)⩽ tr(A)+ tr(B) . The possible types of finitely generated torsion modules A, B, and C over a discrete valuation ring such that there is an exact sequence B→C→A are characterized by the same inequalities.

On the number of generators of the torsion module of differentials

On the number of generators of the torsion module of differentials

Witt group of Hermitian forms over a noncommutativediscrete valuation ring

We investigate Hermitian forms on finitely generated torsion modules over a noncommutative discrete valuation ring. We also give some results for lattices, which still are satisfied even if the base ring is not commutative. Moreover, for a noncommutative discrete-valued division algebraD with valuation ring R and residual division algebra , we prove that where WT(R) denotes the Witt group of regular Hermitian forms on finitely generated torsion R‐modules.

Novikov–Shubin Signatures, II

This paper continues the author's previous paper, published inAnn. Global Anal. Geom.18 (2000), 477–515. Here weconstruct a linking form on the torsion part of middle-dimensionalextended L2 homology and cohomology of odd-dimensionalmanifolds. We give a geometric necessary condition when this linkingform is hyperbolic. We compute this linking form in case, when themanifold bounds. We introduce and study new numerical invariants of thelinking form: the Novikov–Shubin signature and the torsion signature;we compute these invariants explicitly for manifolds withπ1 = Z in terms of the Blanchfield form. We develop anotion of excess for extensions of torsion modules and show how thisconcept can be used to guarantee vanishing of the torsion signature.

Torsion Modules, Lattices and P-Points

Answering a long-standing question in the theory of torsion modules, we show that weakly productively bounded domains are nec- essarily productively bounded. (See the introduction for deflnitions.) Moreover, we prove a twin result for the ideal lattice L of a domain equating weak and strong global intersection conditions for families (Xi)i2I of subsets of L with the property that T i2I Ai 6 0 whenever Ai 2 Xi. Finally, we show that, for domains with Krull dimension (and countably generated extensions thereof), these lattice-theoretic condi- tions are equivalent to productive boundedness.

Cyclic homology of affine hypersurfaces with isolated Singularities

We consider reduced, affine hypersurfaces with only isolated singularities. We give an explicit computation of the Hodge-components of their cyclic homology in terms of de Rham cohomology and torsion modules of differentials for large n. It turns out that the vector spaces HC n ( A) are finite dimensional for n ≥ N − 1.

On the quotients of cubic Hecke algebras

Between the rank 3 quotients of cubic Hecke algebras there is essentially one of maximal dimension. We prove it has a unique Markov trace having values in a torsion module. Therefore the description of a Markov trace on the cubic Hecke algebra corresponding tox 3+1 and having the parameters (1, 1) is derived. Thus we obtain a numerical link invariant of finite degree, and define a whole sequence of 3rd order Vassiliev invariants.