Loop torsors. Theory and applications

Hereditary atomicity in integral domains

For a given pair of s-dimensional real Laurent polynomials ( a → ( z ) , b → ( z ) ) , which has a certain type of symmetry and satisfies the dual condition b → ( z ) T a → ( z ) = 1 , an s × s Laurent polynomial matrix A ( z ) (together with its inverse A - 1 ( z ) ) is called a symmetric Laurent polynomial matrix extension of the dual pair ( a → ( z ) , b → ( z ) ) if A ( z ) has similar symmetry, the inverse A - 1 ( Z ) also is a Laurent polynomial matrix, the first column of A ( z ) is a → ( z ) and the first row of A - 1 ( z ) is ( b → ( z ) ) T . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks.

The first chapter contains some known facts and some novel results on Commutative Algebra which are crucial for the proofs of the results of Chapters 3 and 4. The former are presented here without their proofs (with the exception of Theorem 8) for the convenience of the reader. In the first section of this chapter, we define the localization of a ring and give some main properties. The second section is dedicated to integrally closed rings. We study particular cases of integrally closed rings, such as valuation rings, discrete valuation rings and Krull rings. We use their properties in order to obtain results on Laurent polynomial rings over integrally closed rings. We state briefly some results on the completions of rings in Section 1.3. In the fourth section, we introduce the notion of morphisms associated with monomials. They are morphisms which allow us to pass from a Laurent polynomial ring A in m+1 indeterminates to a Laurent polynomial ring B in m indeterminates, while mapping a specific monomial to 1. Moreover, we prove (Proposition 15) that every surjective morphism from A to B which maps each indeterminate to a monomial is associated with a monomial. We call adapted morphisms the compositions of morphisms associated with monomials. They play a key role in the proof of the main results of Chapters 3 and 4. Finally, in the last section of the first chapter, we give a criterion (Theorem 10) for a polynomial to be irreducible in a Laurent polynomial ring with coefficients in a field.

Power-closed ideals of polynomial and Laurent polynomial rings

In [1] the first and last authors studied a decomposition ofH *(R P ∞×…×R P ∞;F 2) into modules over the Steenrod algebra obtained from an action of the cyclic group $$F_{2^n }^* $$ . Here a minimal set of generators for the ring of invariants is characterized and counted by analyzing the associated ring of Laurent polynomials. A structure theorem for the ring of invariant Laurent polynomials is given and a ‘destabilisation cancels localisation’ theorem is obtained.

Many connections and dualities in representation theory and Lie theory can be explained using quasi-hereditary covers in the sense of Rouquier. Recent work by the first-named author shows that relative dominant (and codominant) dimensions are natural tools to classify and distinguish distinct quasi-hereditary covers of a finite-dimensional algebra. In this paper, we prove that the relative dominant dimension of a quasi-hereditary algebra, possessing a simple preserving duality, with respect to a direct summand of the characteristic tilting module is always an even number or infinite and that this homological invariant controls the quality of quasi-hereditary covers that possess a simple preserving duality. To resolve the Temperley–Lieb algebras, we apply this result to the class of Schur algebras $S(2, d)$ and their $q$ -analogues. Our second main result completely determines the relative dominant dimension of $S(2, d)$ with respect to $Q=V^{\otimes d}$ , the $d$ -th tensor power of the natural two-dimensional module. As a byproduct, we deduce that Ringel duals of $q$ -Schur algebras $S(2,d)$ give rise to quasi-hereditary covers of Temperley–Lieb algebras. Further, we obtain precisely when the Temperley–Lieb algebra is Morita equivalent to the Ringel dual of the $q$ -Schur algebra $S(2, d)$ and precisely how far these two algebras are from being Morita equivalent, when they are not. These results are compatible with the integral setup, and we use them to deduce that the Ringel dual of a $q$ -Schur algebra over the ring of Laurent polynomials over the integers together with some projective module is the best quasi-hereditary cover of the integral Temperley–Lieb algebra.

We give an algebraic interpretation of the well-known “zero-condition” or “sum rule” for multivariate refinable functions with respect to an arbitrary scaling matrix. The main result is a characterization of these properties in terms of containment in a quotient ideal, however not in the ring of polynomials but in the ring of Laurent polynomials.

Descriptions of radicals of skew polynomial and skew Laurent polynomial rings

Let R_n be the ring of Laurent polynomials in n variables over a field k of characteristic zero and let K_n be its fraction field. Given a linear algebraic k -group G , we show that a K_n -torsor under G which is unramified with respect to X = Spec (R_n) extends to a unique toral R_n -torsor under G . This result, in turn, allows us to classify all G -torsors over R_n .

Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{\pm 1},\ldots,x_n^{\pm 1}] . We prove that G is isotropic over R if and only if it is isotropic over the field of fractions k(x_1,\ldots,x_n) of R , and if this is the case, then the natural map H^1_{\acute{e}t}(R,G)\to H^1_{\acute{e}t}(k(x_1,\ldots,x_n),G) has trivial kernel and G is loop reductive. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that H^1_{Zar}(R,G)=\ast for such groups G . We also deduce that if G is a reductive group over R of isotropic rank \ge 2 , then the natural map of non-stable K_1 -functors K_1^G(R)\to K_1^G\bigl( k((x_1))\ldots((x_n)) \bigr) is injective, and an isomorphism if G is moreover semisimple.

In this paper, we develop a novel and effective algorithm for the construction of perfect reconstruction filter banks (PRFBs) with linear phase. In the algorithm, the key step is the symmetric Laurent polynomial matrix extension (SLPME). There are two typical problems in the construction: (1) For a given symmetric finite low-pass filter \(\mathbf {a}\) with the polyphase, to construct a PRFBs with linear phase such that its low-pass band of the analysis filter bank is \(\mathbf {a}\). (2) For a given dual pair of symmetric finite low-pass filters, to construct a PRFBs with linear phase such that its low-pass band of the analysis filter bank is \(\mathbf {a}\), while its low-pass band of the synthesis filter bank is \(\mathbf {b}\). In the paper, we first formulate the problems by the SLPME of the Laurent polynomial vector(s) associated to the given filter(s). Then we develop a symmetric elementary matrix decomposition algorithm based on Euclidean division in the ring of Laurent polynomials, which finally induces our SLPME algorithm.

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.

Algebraic jump loci for rank and Betti numbers over Laurent polynomial rings

Finite domination and Novikov rings. Laurent polynomial rings in several variables

Suppose A A is a local ring and R = A [ X , X − 1 ] R = A[X,{X^{ - 1}}] is a Laurent polynomial ring. We prove that for projective R R -modules P P and Q Q with rank Q > Q > rank P P , if Q f {Q_f} is a direct summand of P f {P_f} for a doubly monic polynomial f f then Q Q is also a direct summand of P P . We also prove the analogue of the Horrock’s theorem for Laurent polynomials rings.