The maps from loop suspensions to loop spaces are investigated using group representations in this article. The shuffle relations on the Cohen groups are given. By using these relations, a universal ring for functorial self maps of double loop spaces of double suspensions is given. Moreover the obstructions to the classical exponent problem in homotopy theory are displayed in the extension grou...

We develop a theory of quasi–coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological properties of such sheaves, and prove stack–theoretic versions of Grothendieck’s Fundamental Theorem for proper morphisms, Grothendieck’s Existence Theorem, Zariski’s Connectedness Theorem, as well as finiteness Theo...

in this work we consider conjugacy of elements and parabolic subgroups‎ ‎in details‎, ‎in a new class of artin groups‎, ‎introduced in an earlier work‎, ‎which may contain arbitrary parabolic subgroups‎. ‎in particular‎, ‎we find‎ ‎algorithmically minimal representatives of elements in a conjugacy class‎ ‎and also an algorithm to pass from one minimal representative to the others‎.

Contents 1. Introduction 1 2. Some basic facts 2 2.1. Results from algebraic number theory 3 2.2. Results from homological algebra 4 2.3. Ramification theory and the Artin representation 7 3. The perspective of curves 10 3.1. Jacobians and their generalizations 10 3.2. Class field theory 14 3.3. The Artin representation and algebraic curves 15 4. The perspective of arithmetic 16 4.1. Galois coh...

This result is a variation of a theorem of Duncan and O’Carroll [DO]: maximal ideals are replaced for any ideal using the unavoidable hypothesis dim(M/N) ≤ 1 (as an Example of Wang shows [W1]). Moreover it provides a partial positive answer to the question raised by Huneke in Conjecture 1.3 [H1]. We begin by recalling what is called uniform Artin-Rees properties. Let A be a noetherian ring, I b...

We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring ...

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a punctured sphere. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured disk together with a homomorphism to the integers. The technique, following Ivanov, is to prove that every superinjective map of the com...

Functional equations in number theory are relations between an L-function and some sort of dual L-function, and in general, the L-function and its dual need not coincide. For example, if χ is a primitive Dirichlet character then the functional equation relates L(s, χ) to L(1− s, χ), and L(s, χ) = L(s, χ) if and only if χ = 1. Or if f is a primitive cusp form of weight two for Γ1(N) and f∨ is th...

The algebraic closure of R is C, which is a finite extension. Are there other fields which are not algebraically closed but have an algebraic closure which is a finite extension? Yes. An example is the field of real algebraic numbers. Since complex conjugation is a field automorphism fixing Q, and the real and imaginary parts of a complex number can be computed using field operations and comple...

Let k be a complete discrete valuation field of equal characteristic p > 0 with possibly imperfect residue field. Let Gk be its Galois group. We will prove that the conductors computed by the arithmetic ramification filtrations on Gk defined in [2] coincide with the differential Artin conductors and Swan conductors of Galois representations of Gk defined in [15]. As a consequence, we give a Has...