scholarly journals Finite domination and Novikov rings: Laurent polynomial rings in two variables

2015 ◽  
Vol 14 (04) ◽  
pp. 1550055
Author(s):  
Thomas Hüttemann ◽  
David Quinn
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Homotopy Equivalent ◽  
Polynomial Rings ◽  
Cochain Complex ◽  
Finitely Generated ◽  
Bounded Complex ◽  
Laurent Polynomial Ring ◽  
Free Modules

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x-1, y, y-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C ⊗L R〚x, y〛[(xy)-1] and C ⊗L R[x, x-1]〚y〛[y-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.

scholarly journals FINITE DOMINATION AND NOVIKOV RINGS. ITERATIVE APPROACH

2012 ◽  
Vol 55 (1) ◽  
pp. 145-160 ◽  
Author(s):  
THOMAS HÜTTEMANN ◽  
DAVID QUINN
Keyword(s):  
Laurent Series ◽  
Chain Complex ◽  
Laurent Polynomial ◽  
Projective Modules ◽  
Finitely Generated ◽  
Formal Laurent Series ◽  
Chain Complexes ◽  
Laurent Polynomial Ring ◽  
Bounded Below ◽  
Free Modules

AbstractSuppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,x−1]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C ⊗LR((x)) and C ⊗LR((x−1)) are acyclic, as has been proved by Ranicki (A. Ranicki, Finite domination and Novikov rings, Topology34(3) (1995), 619–632). Here R((x)) = R[[x]][x−1] and R((x−1)) = R[[x−1]][x] are rings of the formal Laurent series, also known as Novikov rings. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.

scholarly journals Prime ideals in skew Laurent polynomial rings

1993 ◽  
Vol 36 (2) ◽  
pp. 299-317 ◽  
Author(s):  
K. W. Mackenzie
Keyword(s):  
Prime Ideal ◽  
Polynomial Ring ◽  
Commutative Algebra ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Ideal Structure ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Laurent Polynomial Ring

Let R be a commutative ring and {σ1,…,σn} a set of commuting automorphisms of R. Let T = be the skew Laurent polynomial ring in n indeterminates over R and let be the Laurent polynomial ring in n central indeterminates over R. There is an isomorphism φ of right R-modules between T and S given by φ(θj) = xj. We will show that the map φ induces a bijection between the prime ideals of T and the Γ-prime ideals of S, where Γ is a certain set of endomorphisms of the ℤ-module S. We can study the structure of the lattice of Γ-prime ideals of the ring S by using commutative algebra, and this allows us to deduce results about the prime ideal structure of the ring T. As an example, if R is a Cohen-Macaulay ℂ-algebra and the action of the σj on R is locally finite-dimensional, we will show that the ring T is catenary.

scholarly journals Primitive skew Laurent polynomial rings

1978 ◽  
Vol 19 (1) ◽  
pp. 79-85 ◽  
Author(s):  
D. A. Jordan
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Primitive Group ◽  
Polynomial Rings ◽  
Group Rings ◽  
Ore Extension ◽  
Laurent Polynomial Ring ◽  
Necessary And Sufficient

In [8] the author studied the question of the primitivity of an Ore extension R[x, δ], where δ is a derivation of the ring R. If a is an automorphism of R then it can be shown that R[x, α] is primitive if the following conditions are satisfied: (i) no power αsS ≥ 1, of α is inner; (ii) the only ideals of R invariant under α are 0 and R. These conditions are also known to be necessary and sufficient for the skew Laurent polynomial ring R[x, x−1, α] to be simple [9]. The object of this paper is to find conditions which are sufficient for R[x, x−1, α] to be primitive. The results obtained are remarkably similar to those of [8]. Two logically independent conditions are each found to be sufficient for the primitivity of R[x, x−1, α]. Of these, one is also shown to be sufficient for R[x, α] to be primitive. Included in the examples illustrating these results are some applications to the theory of primitive group rings. The basic techniques involved are also applied to produce a counterexample to the converse of a theorem of Goldie and Michler [3] on when R[x, x−1, α] is a Jacobson ring.

scholarly journals Recent results on weakly factorial domains

2018 ◽  
Vol 20 ◽  
pp. 01001
Author(s):  
Chang Gyu Whan
Keyword(s):  
Polynomial Ring ◽  
Quotient Group ◽  
Integral Domain ◽  
Laurent Polynomial ◽  
Semigroup Ring ◽  
Prime Element ◽  
Cancellative Monoid ◽  
Factorial Domain ◽  
Laurent Polynomial Ring

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

scholarly journals Multivariate Alexander colorings

2018 ◽  
Vol 27 (14) ◽  
pp. 1850076 ◽  
Author(s):  
Lorenzo Traldi
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Knot Theory ◽  
Laurent Polynomial ◽  
Module Structure ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Linear Maps ◽  
Laurent Polynomial Ring ◽  
Special Case

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module [Formula: see text] over the Laurent polynomial ring [Formula: see text]. If [Formula: see text] is a diagram of a link [Formula: see text] with [Formula: see text] components, then the colorings of [Formula: see text] with values in [Formula: see text] form a [Formula: see text]-module [Formula: see text]. Extending a result of Inoue [Knot quandles and infinite cyclic covering spaces, Kodai Math. J. 33 (2010) 116–122], we show that [Formula: see text] is isomorphic to the module of [Formula: see text]-linear maps from the Alexander module of [Formula: see text] to [Formula: see text]. In particular, suppose [Formula: see text] is a field and [Formula: see text] is a homomorphism of rings with unity. Then [Formula: see text] defines a [Formula: see text]-module structure on [Formula: see text], which we denote [Formula: see text]. We show that the dimension of [Formula: see text] as a vector space over [Formula: see text] is determined by the images under [Formula: see text] of the elementary ideals of [Formula: see text]. This result applies in the special case of Fox tricolorings, which correspond to [Formula: see text] and [Formula: see text]. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine [Formula: see text]; this observation corrects erroneous statements of Inoue [Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications 10 (2001) 813–821; op. cit.].

Restrictions on Homflypt and Kauffman polynomials arising from local moves

2020 ◽  
Vol 29 (06) ◽  
pp. 2050036
Author(s):  
Sandy Ganzell ◽  
Mercedes V. Gonzalez ◽  
Chloe’ Marcum ◽  
Nina Ryalls ◽  
Mariel Santos
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Laurent Polynomial Ring

We study the effects of certain local moves on Homflyptand Kauffman polynomials. We show that all Homflypt(or Kauffman) polynomials are equal in a certain nontrivial quotient of the Laurent polynomial ring. As a consequence, we discover some new properties of these invariants.

scholarly journals ON MODULES OVER LAURENT POLYNOMIAL RINGS

2012 ◽  
Vol 21 (01) ◽  
pp. 1250007 ◽  
Author(s):  
DANIEL S. SILVER ◽  
SUSAN G. WILLIAMS
Keyword(s):  
Knot Theory ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Finitely Generated ◽  
The Absolute

A finitely generated ℤ[t, t-1]-module without ℤ-torsion and having nonzero order Δ(M) of degree d is determined by a pair of sub-lattices of ℤd. Their indices are the absolute values of the leading and trailing coefficients of Δ(M). This description has applications in knot theory.

scholarly journals Twisted Vertex Operators and Unitary Lie Algebras

2015 ◽  
Vol 67 (3) ◽  
pp. 573-596 ◽  
Author(s):  
Fulin Chen ◽  
Yun Gao ◽  
Naihuan Jing ◽  
Shaobin Tan
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Polynomial Ring ◽  
Central Extension ◽  
Laurent Polynomial ◽  
New Method ◽  
Vertex Operators ◽  
Irreducible Decomposition ◽  
Laurent Polynomial Ring

AbstractA representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral ℤ2–lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by–product, some fundamental representations of affine Kac–Moody Lie algebra of type A(2)n are recovered by the new method.

COLORINGS AND ALEXANDER POLYNOMIALS FOR RIBBON 2-KNOTS

2002 ◽  
Vol 11 (03) ◽  
pp. 403-412
Author(s):  
AKIKO SHIMA
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Alexander Polynomials ◽  
Laurent Polynomial Ring

Let K be a ribbon 2-knot. We show that for any ideal J of the Laurent polynomial ring Λ, if Alexander polynomail of K is trivial, i.e., ΔK(t) = 1, then all colorings of K on Λ/J are trivial.

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