We consider two overlapping classes of fields, IAC and VAC, which are defined using valuation theory but do not involve a distinguished valuation. Rather, each class is by condition that quantifies over all possible valuations on the field. In his thesis, Hong asked whether these equal (Hong, 2013, Question 5.6.8). this paper, we give an example negatively answers Hong's question. also explore ...

Let (A, m) be an excellent normal local ring with algebraically closed residue class field. Given integrally closed m-primary ideals I ⊃ J , we show that there is a composition series between I and J , by integrally closed ideals only. Also we show that any given integrally closed m-primary ideal I, the family of integrally closed ideals J ⊂ I, lA(I/J) = 1 forms an algebraic variety with dimens...

In this paper we study the Kummer extensions of a power series field K = k((X1, ...,Xr)), where k is an algebraically closed field of arbitrary characteristic. 1 Terminology and notation Let k be an algebraically closed field, X1, ...,Xr indeterminates formally independent over k, and let K and Lm be the fields K = k ((X1, ...,Xr)) , Lm = k (( X 1/m 1 , ...,X 1/m r )) , where m is a non negativ...

Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R4 can be described as certain 2-morphisms in the 2-category of ‘2-tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the ‘free semistrict braided monoidal 2-category with duals on one un...

In this paper, a group shift is an expansive action of Zd on a compact metrizable zero dimensional group by continuous automorphisms. All group shifts factor topologically onto equal-entropy Bernoulli shifts; abelian group shifts factor by continuous group homomorphisms onto canonical equalentropy Bernoulli group shifts; and completely positive entropy abelian group shifts are weakly algebraica...

We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If K is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra K: this is a refinement of Wang’s universality theorem for the (compact) quantum permutation group. We also prove a structural result for Hopf algebras having a non-...

X : f (x) = g(x)} is a compact subset of X\∂X. In case of codimension 0, the Nielsen number is the lower estimate of the number of points in this set. We suggest a generalization of the classical Nielsen theory for the coincidence and as well as the preimage problems in order to provide a framework for some recent Wecken type results for codimension 1. The Nielsen equivalence on the set of coin...

In the topology of coordinatewise convergence ∇∞ is a compact, metrizable and separable space. Denote by C(∇∞) the algebra of real continuous functions on ∇∞ with pointwise operations and the supremum norm. In C(∇∞) there is a distinguished dense subspace F := R [q1, q2, . . . ] generated (as a commutative unital algebra) by algebraically independent continuous functions qk(x) := ∑∞ i=1 x k+1 i...